{ "entries": [ { "abstract": "Quantitative permanence and recovery bounds are established for mass-action systems with vanishing reverse rates. For a finite weakly reversible network with rates proportional to powers of epsilon, eventual concentration bounds depend on the spread of the rate exponents, with geometric constants chosen before the individual rates. For an explicit three-species network, a prepared region admits global positive solutions with three distinct concentration scales. Every prepared solution reaches and remains in a smaller target by time (31/10)/(p^2 h^7), while an explicit prepared solution cannot enter before 400/(1313 p^2 h^7). Inventory inequalities give additional necessary restoration and depletion times from arbitrary positive initial states. A continuous reverse-exponent family retains arbitrary admitted prefactors and yields matching worst-case prepared recovery orders, together with exact common-clock transport, preparation identities and cumulative reverse-reaction bounds. Abstract constant-capacity and open-range state-dependent inventory comparisons are included without assuming monotone inventories.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "37N25", "92C42" ] }, "formalization": { "theorem_names": [ "ReverseRateLimits.general_quantitative_root", "ReverseRateLimits.SharpRecovery.prepared_recovery_sharp", "ReverseRateLimits.SharpRecovery.witness_sharp", "ReverseRateLimits.GlobalInventory.global_inventory_clocks", "ReverseRateLimits.SharpRecovery.family_root_sharp", "ReverseRateLimits.ExponentFamily.identities", "ReverseRateLimits.ExponentFamily.original_cutoff_bounds", "ReverseRateLimits.ExponentFamily.cutoff_decimal", "ReverseRateLimits.CommonClock.rescale_iff", "ReverseRateLimits.RecoveryCorollaries.scale_tradeoff", "ReverseRateLimits.ReactionExtent.restoration_extent", "ReverseRateLimits.SharpRecovery.inventory_bottleneck", "ReverseRateLimits.SharpRecovery.inventory_unreachable", "ReverseRateLimits.GlobalInventory.inventory_bottleneck_scaling", "ReverseRateLimits.RecoveryCorollaries.no_linear_inventory", "QuantitativeRegistry.family_parameters", "QuantitativeRegistry.admitted_parameters", "QuantitativeRegistry.state_capacity_clock" ] }, "id": "PALOMAR-2026-10-11-000020", "path": "entries/PALOMAR-2026-10-11-000020-v1.json", "preservation": { "repositories": [ { "commit": "e542422946035f4509e92c963b494e586105d09d", "fork_repository": "PalomarArchive/mmislan--palomar-quantitative-permanence--b8c5a1503b2d", "source_repository": "mmislan/palomar-quantitative-permanence" } ] }, "published_at": "2026-10-11T16:49:33Z", "source": { "commit": "e542422946035f4509e92c963b494e586105d09d", "project_path": null, "repository": "mmislan/palomar-quantitative-permanence" }, "status": "registered", "title": "Quantitative permanence and recovery with vanishing reverse rates", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Exact worst squared spectral gaps for finite open chiral chains with inhomogeneous strong-bond lower bounds and weak-bond ratio intervals. The scalar optimum is the minimum over the ordered one-switch endpoint configurations, with a two-certificate reduction when cuts are allowed. Strictly positive weak lower bounds give a complete point-or-edge classification of scalar minimizers and an exact aligned-channel equality criterion for arbitrary complex matrix disorder. The sharp channel bound includes the finite chiral Hamiltonian spectral bridge.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.SP", "math-ph" ], "msc2020": [ "15A18", "15A42", "47A75", "81Q10" ] }, "formalization": { "theorem_names": [ "ChiralGap.robustMinimum_spec", "ChiralGap.one_switch_value", "ChiralGap.two_certificate_value", "ChiralGap.scalarGap_eigenvector", "ChiralGap.scalarGap_le_eigenvalue", "ChiralGap.scalar_optimizer_classification", "ChiralGap.scalar_minimizers_vertex_or_edge", "ChiralGap.robust_hopping_bound", "ChiralGap.hopping_chiral_zero_multiplicity", "ChiralGap.robust_chiral_sharpness_constrained", "ChiralGap.hopping_nonzero_spectrum", "ChiralGap.robust_chiral_complex_gap", "ChiralGap.matrix_equality_criterion", "ChiralGap.chiral_eigenbasis" ] }, "id": "PALOMAR-2026-10-11-000019", "path": "entries/PALOMAR-2026-10-11-000019-v1.json", "preservation": { "repositories": [ { "commit": "3414bf564e0afd92d713ced8ba51c28db96834b2", "fork_repository": "PalomarArchive/JD-Jones-ASES--chiral-gap--a9168b8ac388", "source_repository": "JD-Jones-ASES/chiral-gap" } ] }, "published_at": "2026-10-11T16:33:06Z", "source": { "commit": "3414bf564e0afd92d713ced8ba51c28db96834b2", "project_path": null, "repository": "JD-Jones-ASES/chiral-gap" }, "status": "registered", "title": "Exact robust gaps in finite open chiral chains", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every positive generating degree d, there is a variable threshold depending only on d such that, over every infinite field and for every number of general degree-d generators, the quotient polynomial ring has Fr\u00f6berg's predicted Hilbert function through degree 2d. The threshold is independent of both the field and the number of generators. The prediction uses positive truncation at the first nonpositive coefficient of (1-t^d)^r/(1-t)^n. The development includes a direct construction for d at least three and a separate quadratic argument in arbitrary characteristic.", "authors": [ { "name": "Vlado Vojdanovski" } ], "classification": { "arxiv": [ "math.AC" ], "msc2020": [ "13D40", "13F20", "13D02", "68V15" ] }, "formalization": { "theorem_names": [ "FrobergPaper.main_result" ] }, "id": "PALOMAR-2026-10-11-000018", "path": "entries/PALOMAR-2026-10-11-000018-v1.json", "preservation": { "repositories": [ { "commit": "b32bc82f7eea38479abab80b6b74649026c15d24", "fork_repository": "PalomarArchive/Vlado-V--froberg-2d--66bf7f387855", "source_repository": "Vlado-V/froberg-2d" } ] }, "published_at": "2026-10-11T15:58:52Z", "source": { "commit": "b32bc82f7eea38479abab80b6b74649026c15d24", "project_path": null, "repository": "Vlado-V/froberg-2d" }, "status": "registered", "title": "Fr\u00f6berg's conjecture through degree 2d in sufficiently many variables", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization, with Mathlib, of the main results of the paper \"Prime-power structure of the stable regime for modular Schur numbers\" (docs/paper/modular-schur.pdf in this repository). The modular Schur number S_m(k,\u2113), introduced by Chappelon, Revuelta Marchena and Sanz Dom\u00ednguez (2013), is the largest N such that 1, \u2026, N can be coloured with k colours so that no colour class contains a solution of x_1 + \u22ef + x_\u2113 \u2261 y (mod m). For m \u2265 2 and \u2113 \u2265 2, put n = m/gcd(m,\u2113\u22121). The main theorem is the closed form S_m(k,\u2113) = n \u2212 1 for every k \u2265 n \u2212 1. D'orville, Sim, Wong and Ho (Integers 25 (2025) #A62) gave it when gcd(m,\u2113\u22121) = 1 (their Corollary 5). The general case is a short corollary of the singleton criterion and their Corollary 3; to the author's knowledge it is first stated in the paper. The proof turns on one value: \u2113 copies of n sum to n modulo m, so the class {n} alone breaks the rule, while each of 1, \u2026, n \u2212 1 is safe in a class of its own. The formalization also proves the one-colour closed form S_m(1,\u2113) = min(\u2113 \u2212 1, \u230am/\u2113\u230b) for 2 \u2264 \u2113 \u2264 m, which answers Problem 1.3 of D'orville et al.; the bound S_m(k,\u2113) \u2264 m \u2212 1 of Chappelon et al.; and the reduction from integers to residues modulo m. Twelve theorems with Mathlib-only statements are compared by lake comparator (statement identity, permitted axioms, and replay through the Lean, nanoda and con-ron kernels); their only axioms are propext, Classical.choice and Quot.sound. The 33 project modules also contain further results, listed in LEAN_STATUS.md, that are proved and axiom-audited but are not compared, among them the bound m/p, with p the least prime factor of m, on the size of a set of residues whose pairwise differences all lie in a proper subgroup of \u2124/m. An earlier computational part of the project, which uses native_decide, is not in this repository and is not part of these results. The informal proofs, the Lean formalization and the paper were produced with AI systems (Claude, Aristotle and GPT-5) under the direction of the author.", "authors": [ { "name": "Adam McKenna" } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05D10", "11B13", "11B75", "68V20" ] }, "formalization": { "theorem_names": [ "ComparatorClaims.schurMod_integerClosedForm", "ComparatorClaims.schurMod_integerCap_isGreatest", "ComparatorClaims.noValidIntegerPartition_of_ge_modulus", "ComparatorClaims.schurMod_integer_eq_residue", "ComparatorClaims.schurModResidue_closedForm", "ComparatorClaims.schurModResidue_upperBound", "ComparatorClaims.schurModResidue_lowerBound", "ComparatorClaims.singleton_sumFree_iff_nonzeroMultiple", "ComparatorClaims.criticalResidue_isUnsafe", "ComparatorClaims.zeroMem_notSumFree", "ComparatorClaims.schurModResidue_oneColorClosedForm_of_le_modulus", "ComparatorClaims.schurModResidue_oneColorClosedForm" ] }, "id": "PALOMAR-2026-09-01-000001", "path": "entries/PALOMAR-2026-09-01-000001-v2.json", "preservation": { "repositories": [ { "commit": "7c91cf3c5ca9b50354a74344dce71a28c9fefa01", "fork_repository": "PalomarArchive/mysticflounder--modular-schur--d5f9358df5a5", "source_repository": "mysticflounder/modular-schur" } ] }, "published_at": "2026-10-11T15:18:09Z", "source": { "commit": "7c91cf3c5ca9b50354a74344dce71a28c9fefa01", "project_path": "lean", "repository": "mysticflounder/modular-schur" }, "status": "registered", "title": "Modular Schur numbers: S_m(k,\u2113) = n \u2212 1 for k \u2265 n \u2212 1, where n = m/gcd(m,\u2113\u22121)", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A Lean formalization of open phases with every prescribed finite replica symmetry breaking depth in mixed even-spin Ising models at zero external field and positive temperature. The first selected result constructs strictly increasing even degrees beginning at four and a nonempty open set of positive effective covariance coefficients, with an optimizing probability measure having exactly k+1 support points. The independent statement uses concrete Gaussian Cole\u2013Hopf recursions and comparison against all finite schemes at every competing depth. The substantive development constructs minimizers over all probability measures and includes the revised quadratic-inclusive universal transition, exact phase geometry and asymptotics, atomic Hessian, entropy, and analytic comparison results.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "math.PR", "math-ph" ], "msc2020": [ "60K35", "82B44" ] }, "formalization": { "theorem_names": [ "ArbitraryRSB.arbitrary_finite_depth_open_set", "ArbitraryRSB.universal_gardner_transition" ] }, "id": "PALOMAR-2026-10-11-000017", "path": "entries/PALOMAR-2026-10-11-000017-v1.json", "preservation": { "repositories": [ { "commit": "5162e2d15b440ce46fd322f20eb429468d601948", "fork_repository": "PalomarArchive/pmaronow--arbitrary-rsb-palomar--0236bf82e169", "source_repository": "pmaronow/arbitrary-rsb-palomar" } ] }, "published_at": "2026-10-11T15:02:55Z", "source": { "commit": "5162e2d15b440ce46fd322f20eb429468d601948", "project_path": null, "repository": "pmaronow/arbitrary-rsb-palomar" }, "status": "registered", "title": "Replica symmetry breaking at arbitrary depth", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of the solved parts, extremal bounds, and finite verification up to 10000 for Erd\u0151s problem 261 on representing n / 2^n as a finite sum \u2211_{k=1}^t a_k / 2^(a_k) of distinct positive integers a_k. Proves the Borwein\u2013Loring telescoping family n = 2^(m+1) - m - 2 (resolving Part (i) that infinitely many positive integers n admit such a representation), verifies every integer 1 \u2264 n \u2264 10000 via a coalesced dyadic block certificate forest (Theorem 3.5 of Tengely, Ulas, and Zygad\u0142o 2020), constructs infinitely many rationals with at least 2^m distinct infinite representations for every m \u2208 \u2115, and proves the Tengely\u2013Ulas\u2013Zygad\u0142o sharp extremal bound n \u2264 2^(t+1) - t - 2 and its equality uniqueness, first-term localization a_k \u2208 [n+1, n+3], 2-adic last-gap divisibility 2^(a_{t+1} - a_t) \u2223 a_{t+1}, the punctured-interval infinite family n_k = (2^(4k+2) + 1)/5 - 4k, and arbitrary-length splicing.", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11D61", "11B75" ] }, "formalization": { "theorem_names": [ "Erdos261.Palomar.borwein_loring", "Erdos261.Palomar.borwein_loring_property", "Erdos261.Palomar.erdos_261_parts_i", "Erdos261.Palomar.le_10000", "Erdos261.Palomar.two_representations", "Erdos261.Palomar.infinite_rationals_two_pow_reps", "Erdos261.Palomar.max_n_bound", "Erdos261.Palomar.exists_term_le_add_three", "Erdos261.Palomar.last_term_gap_dvd", "Erdos261.Palomar.borwein_loring_unique", "Erdos261.Palomar.punctured_family_erdos261Prop", "Erdos261.Palomar.arbitrary_length_rep_of_all_erdos261Prop" ] }, "id": "PALOMAR-2026-10-11-000016", "path": "entries/PALOMAR-2026-10-11-000016-v1.json", "preservation": { "repositories": [ { "commit": "b53791ff6294b4b90e3653970367f127a52fd16e", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-11T14:53:34Z", "source": { "commit": "b53791ff6294b4b90e3653970367f127a52fd16e", "project_path": "erdos-261", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 261: dyadic weighted representations n / 2^n = \u2211 a_k / 2^(a_k)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Machine-checked upper bound on Erd\u0151s problem 607. For n \u2265 2 let F(n) be the number of distinct multiplicity sets A(P) that can arise from a collection P of n points of the plane, where A(P) is the set of block sizes (point counts) of the lines determined by P. The proof shows that the lines of P form a pairwise-balanced design on the n points (every pair of points lies on exactly one line, and each line contains at least two points), so A(P) is exactly the set of block sizes of such a PBD. Hence F(n) is bounded by the number of PBD profile sets on n points, and the Colbourn\u2013Phelps\u2013R\u00f6dl (1984) Theorem 2.4 bound exp(c\u00b7\u221an) on that quantity gives F(n) \u2264 exp(c\u00b7\u221an) for every n \u2265 2. The bound is stated for n \u2265 2 \u2014 the range where the bound is meaningful (a single point determines no line) \u2014 so the hypothesis is non-vacuous. The CPR-1984 PBD bound is a published theorem and is taken as the hypothesis hCPR; it is cited, not re-proved here. Together with the Szemer\u00e9di\u2013Trotter (1983) lattice-grid lower bound F(n) \u2265 exp(c'\u00b7\u221an) (also cited, not re-proved), the order of growth is F(n) = exp(\u0398(\u221an)). This repository contains zero axioms and zero sorry in the proof development (the only sorry is the deliberate statement surface in Challenge.lean).", "authors": [ { "name": "Raphael Delors" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D10", "52C10" ] }, "formalization": { "theorem_names": [ "E607.main_result" ] }, "id": "PALOMAR-2026-10-11-000015", "path": "entries/PALOMAR-2026-10-11-000015-v1.json", "preservation": { "repositories": [ { "commit": "45ef2462f0a9f139734ac590238456229bca068b", "fork_repository": "PalomarArchive/raphaeldelors-code--palomar-erdos607--6888ed550193", "source_repository": "raphaeldelors-code/palomar-erdos607" } ] }, "published_at": "2026-10-11T14:37:18Z", "source": { "commit": "45ef2462f0a9f139734ac590238456229bca068b", "project_path": null, "repository": "raphaeldelors-code/palomar-erdos607" }, "status": "registered", "title": "Erdos607", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of a deterministic rational nested-interval construction defining a particular transcendental real x in (1/4,1/3). For this x, no strictly increasing infinite sequence of positive integer denominators is optimal among all distinct-denominator strict Egyptian underapproximations at every order from any finite starting order onward. The formalization includes the fully specified terminating construction, all prefix and polynomial enumerations, a total executable rational approximation function, and the error bound |x-q_n| <= 2^(-n-1) for every n >= 0. It also proves attainment of the best finite underapproximation values. The independent Palomar statement quantifies directly over all sequences and all competing positive increasing lists. This is a specific effective example; the measure-zero and prior existence results are not claimed as new.", "authors": [ { "name": "Yuhang Shi" } ], "classification": { "arxiv": [ "math.NT", "math.LO" ], "msc2020": [ "11D68", "03D78" ] }, "formalization": { "theorem_names": [ "EgyptianPalomar.explicit_counterexample" ] }, "id": "PALOMAR-2026-10-11-000014", "path": "entries/PALOMAR-2026-10-11-000014-v1.json", "preservation": { "repositories": [ { "commit": "b0c255255c1b8ba45fd0ab6c693ecce747fd602f", "fork_repository": "PalomarArchive/yuhangshi888--erdos-206-explicit-counterexample--385ee6dc63f2", "source_repository": "yuhangshi888/erdos-206-explicit-counterexample" } ] }, "published_at": "2026-10-11T10:19:22Z", "source": { "commit": "b0c255255c1b8ba45fd0ab6c693ecce747fd602f", "project_path": "formalization", "repository": "yuhangshi888/erdos-206-explicit-counterexample" }, "status": "registered", "title": "An explicit computable transcendental counterexample for Erd\u0151s Problem #206", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of Michael (1968), Theorems 1.2 and 1.3: arbitrary indexed products of biquotient continuous surjections are biquotient, without separation assumptions; for a Hausdorff target, biquotience is equivalent to quotientness of the product with every identity map and to the same property for paracompact test spaces. The cover definition, filter and ultrafilter bridges, and an actual Hausdorff paracompact counterexample topology on the underlying target set are proved. Reverse tests use the target universe; the forward implication is separately universe-polymorphic.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54C10", "54B10", "54A20", "54D20" ] }, "formalization": { "theorem_names": [ "Michael.IsBiquotientMap.piMap", "Michael.IsBiquotientMap.prod_id_isQuotientMap", "Michael.isBiquotientMap_iff_all_tests", "Michael.isBiquotientMap_iff_paracompact_tests", "Michael.isBiquotientMap_iff_liftsClusterPoints", "Michael.isBiquotientMap_iff_liftsUltrafilters", "Michael.exists_t2_paracompact_counterexample" ] }, "id": "PALOMAR-2026-10-11-000013", "path": "entries/PALOMAR-2026-10-11-000013-v1.json", "preservation": { "repositories": [ { "commit": "dac5d2d127d25d61e972fc5d9fc442146c67440d", "fork_repository": "PalomarArchive/Arthur742Ramos--Math-Formalization--4c88ddbef95a", "source_repository": "Arthur742Ramos/Math-Formalization" } ] }, "published_at": "2026-10-11T10:09:05Z", "source": { "commit": "dac5d2d127d25d61e972fc5d9fc442146c67440d", "project_path": "topology/michael-biquotient", "repository": "Arthur742Ramos/Math-Formalization" }, "status": "registered", "title": "Michael's biquotient-map product and characterization theorems", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib formalization of the theorem that the independence sequence of every finite forest is unimodal: the numbers of independent sets with 0, 1, 2, ... vertices weakly increase up to some index and weakly decrease after it. This is the statement listed as Erd\u0151s Problem 993; later sources attribute the question to a 1987 paper of Alavi, Malde, Schwenk and Erd\u0151s. The first claimed proof for all finite forests is by T. Zhang and W. Li (self-archived preprints, September 2026). This repository formalizes a second proof, by W. Li, K. Vallier and T. Zhang: it adapts the public arXiv:2610.07943v1 and follows version 2.15 of that manuscript, deposited with restricted access and under revision. The Lean proof is computer-assisted and in two parts (at most 24 vertices; at least 25). About 13,700 finite computations are evaluated by the Lean kernel, without native_decide, on the three standard axioms. AI systems produced and reviewed all of the Lean; no human has read the Lean proofs. No manuscript cited is peer reviewed. No priority and no first formalization is claimed.", "authors": [ { "name": "Kevin Vallier" }, { "name": "Tong Zhang" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C69", "05C05", "05C31", "05A20" ] }, "formalization": { "theorem_names": [ "Erdos993.erdos_993_forest", "Erdos993.erdos_993" ] }, "id": "PALOMAR-2026-10-11-000012", "path": "entries/PALOMAR-2026-10-11-000012-v1.json", "preservation": { "repositories": [ { "commit": "c020ce114a4b21ee164946547093e40452af898a", "fork_repository": "PalomarArchive/selfreferencing--erdos993-palomar--53067af9d958", "source_repository": "selfreferencing/erdos993-palomar" } ] }, "published_at": "2026-10-11T09:59:30Z", "source": { "commit": "c020ce114a4b21ee164946547093e40452af898a", "project_path": null, "repository": "selfreferencing/erdos993-palomar" }, "status": "registered", "title": "Unimodality of the independence sequence of a finite forest (Erd\u0151s Problem 993)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We construct uniformly bounded orthonormal bases for homogeneous holomorphic polynomials on complex unit spheres. Fix d >= 2 and N >= 0, and put m = d-1 and L = N+d-1. For each multi-index alpha = (alpha_1,...,alpha_d) of nonnegative integers with sum N, define s_b(alpha) = b + alpha_1 + ... + alpha_b, 1 <= b <= m, and the normalized monomial M_alpha(z) = sqrt((N+d-1)! / ((d-1)! alpha_1! ... alpha_d!)) z_1^alpha_1 ... z_d^alpha_d. Let F be the normalized Fourier matrix, with entries F_(t,s) = L^(-1/2) exp(2 pi i t s/L), for 1 <= t,s <= L. For each increasing m-element subset T = {t_1,...,t_m} of {1,...,L}, define P_T(z) = sum_(|alpha|=N) det[F_(t_a,s_b(alpha))] exp(2 pi i sqrt(2) sum_r alpha_r^2) M_alpha(z), where the determinant is over 1 <= a,b <= m. These polynomials form an orthonormal basis with respect to surface measure of total mass one on the complex unit sphere. Their absolute values on the closed unit ball are bounded by a constant depending only on d, independently of N and T. The construction and proof are formalized in Lean.", "authors": [ { "name": "Mark Lewko" } ], "classification": { "arxiv": [ "math.CV", "math.FA" ], "msc2020": [] }, "formalization": { "theorem_names": [ "BourgainStatement.manuscript_main" ] }, "id": "PALOMAR-2026-10-09-000004", "path": "entries/PALOMAR-2026-10-09-000004-v2.json", "preservation": { "repositories": [ { "commit": "a91b7a0f08c5771f1c66435eac4a456a918ff384", "fork_repository": "PalomarArchive/mrklwk--uniformly-bounded-holomorphic-bases--6ee7b12e3633", "source_repository": "mrklwk/uniformly-bounded-holomorphic-bases" } ] }, "published_at": "2026-10-11T09:52:29Z", "source": { "commit": "a91b7a0f08c5771f1c66435eac4a456a918ff384", "project_path": null, "repository": "mrklwk/uniformly-bounded-holomorphic-bases" }, "status": "registered", "title": "Explicit uniformly bounded bases of homogeneous holomorphic polynomials in every dimension", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "Ordinary mass-action realizations are characterized for a finite family of two-species modules coupled through a shared resource and buffer. Every positive parameter vector admits a strongly connected realization of the same polynomial field. A complex-balanced realization at a prescribed positive state exists exactly when the state is stationary and two explicit local flux inequalities hold, even when arbitrary finite auxiliary complexes on the same species are allowed. An explicit endpoint sign test determines the untransported heterogeneous parameter locus, and the state criterion extends to directed transfers of both private species. A homogeneous specialization has the sharp sharing threshold v <= n/8, including equality. Global positive solutions and permanence uniform over compact sets of initial conditions are established. For d >= 1, the positive equilibrium is unique and is locally exponentially stable in the homogeneous family. Complex-balanced realizations yield global convergence, including with transport, and local exponential stability is established for the transported complex-balanced family. An exact example at d = 1/10 has three positive equilibria, two locally exponentially stable and one unstable.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "34D20", "37N25", "92C42" ] }, "formalization": { "theorem_names": [ "SharedResourceSubmission.realization_core", "SharedResourceSubmission.sharing_threshold", "SharedResourceSubmission.sharing_equality_witness", "SharedResourceSubmission.unique_positive_equilibrium", "SharedResourceSubmission.uniform_permanence", "SharedResourceSubmission.homogeneous_nonlinear_stability", "SharedResourceSubmission.complex_balanced_global_convergence", "SharedResourceSubmission.exact_bistability_and_instability", "SharedResourceSubmission.transported_uniform_permanence", "SharedResourceSubmission.transported_complex_balanced_global_convergence", "SharedResourceSubmission.transported_complex_balanced_nonlinear_stability" ] }, "id": "PALOMAR-2026-10-11-000011", "path": "entries/PALOMAR-2026-10-11-000011-v1.json", "preservation": { "repositories": [ { "commit": "d985d04a6cee575f8ac6dc32db98a390393d9b39", "fork_repository": "PalomarArchive/mmislan--palomar-shared-resource-realizations--ef3d07bf71b0", "source_repository": "mmislan/palomar-shared-resource-realizations" } ] }, "published_at": "2026-10-11T09:43:16Z", "source": { "commit": "d985d04a6cee575f8ac6dc32db98a390393d9b39", "project_path": null, "repository": "mmislan/palomar-shared-resource-realizations" }, "status": "registered", "title": "Realizations and dynamics of shared-resource mass-action systems", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Let N(\u03c3,T) count the nontrivial zeros of the Riemann zeta function with real part at least \u03c3 and imaginary part of absolute value at most T, counted with multiplicity. For every fixed \u03c3 \u2208 [1/2,1] and every \u03b5 > 0, we prove N(\u03c3,T) \u2264 C\u03c3,\u03b5 T^(A(\u03c3)(1\u2212\u03c3)+\u03b5) for all T \u2265 2, where\n\n| \u03c3 interval | A(\u03c3) |\n| --- | --- |\n| [1/2,9/13] | 3/(2\u2212\u03c3) |\n| (9/13,5/7] | 18/(7\u03c3+3) |\n| (5/7,3/4] | 9/4 |\n| (3/4,1] | 3/(2\u03c3) |\n\nThe maximum of this coefficient bound is 39/17, improving the coefficient 30/13 in the Guth\u2013Maynard bound. We also prove the separate estimate N(\u03c3,T) \u2264 C\u03b5 T^((39/17)(1\u2212\u03c3)+\u03b5) for every \u03b5 > 0, simultaneously for \u03c3 \u2208 [1/2,1] and T \u2265 2, with a constant independent of \u03c3 and T. The improved large-values estimate and its zero-density consequences are formalized in Lean.", "authors": [ { "name": "Mark Lewko" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [] }, "formalization": { "theorem_names": [ "GuthMaynardStatement.full_curve", "GuthMaynardStatement.global_uniform", "GuthMaynardStatement.uniform_away_boundary", "GuthMaynardStatement.middle_large_values" ] }, "id": "PALOMAR-2026-10-11-000010", "path": "entries/PALOMAR-2026-10-11-000010-v1.json", "preservation": { "repositories": [ { "commit": "ee2a8a32be49a4f73c7386e9dffdea24d7e3d8c0", "fork_repository": "PalomarArchive/mrklwk--zeta-density-39-over-17-lean--524c4cd16d2b", "source_repository": "mrklwk/zeta-density-39-over-17-lean" } ] }, "published_at": "2026-10-11T09:00:10Z", "source": { "commit": "ee2a8a32be49a4f73c7386e9dffdea24d7e3d8c0", "project_path": "lean", "repository": "mrklwk/zeta-density-39-over-17-lean" }, "status": "registered", "title": "Zero-density estimates from an improved large-values bound", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of the paper \"Anomalous scaling limit of a Brownian particle in a log-correlated potential\" by Armstrong, Bou-Rabee and Kuusi: the subdiffusive scaling limit of a Brownian particle whose drift is the gradient of a log-correlated random potential (Theorem A, in its introduction form and in its precise quenched form), quantitative homogenization of the Dirichlet problem uniformly in the infrared cutoff (Theorem B), and large-scale H\u00f6lder regularity with a Liouville theorem (Theorem C), together with the supporting coarse-graining, elliptic-regularity and Markov-process theory. The theorems hold for every random environment satisfying the paper's standing assumptions (a GMCModel). The development builds on Mathlib and on the public CoarseGraining and MarkovProcess libraries. The three headline theorems are independently checked by the comparator pairs under SubdiffusiveProcessAudit/.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Ahmed Bou-Rabee" }, { "name": "Tuomo Kuusi" } ], "classification": { "arxiv": [ "math.PR", "math.AP" ], "msc2020": [ "60K37", "35R60", "82B28" ] }, "formalization": { "theorem_names": [ "SubdiffusiveProcessAudit.ProcessConvergence.theoremA" ] }, "id": "PALOMAR-2026-10-11-000009", "path": "entries/PALOMAR-2026-10-11-000009-v1.json", "preservation": { "repositories": [ { "commit": "4c6b8fa60357a07fd5713e5104a6cc70f8e432dc", "fork_repository": "PalomarArchive/nitromannitol--subdiffusive-process--7f56d1d55d23", "source_repository": "nitromannitol/subdiffusive-process" } ] }, "published_at": "2026-10-11T08:00:33Z", "source": { "commit": "4c6b8fa60357a07fd5713e5104a6cc70f8e432dc", "project_path": null, "repository": "nitromannitol/subdiffusive-process" }, "status": "registered", "title": "Anomalous scaling limit of a Brownian particle in a log-correlated potential", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the theorem of A. Kupiainen, R. Rhodes and V. Vargas (Annals of Mathematics, 2020) that the three-point structure constants of Liouville conformal field theory on the Riemann sphere are given by the DOZZ formula of Dorn\u2013Otto and Zamolodchikov\u2013Zamolodchikov. The structure constants are defined probabilistically: for 0 < gamma < 2, cosmological constant mu > 0 and charges satisfying the extended Seiberg bounds, C_gamma(alpha_1, alpha_2, alpha_3) = 2 mu^(-s) gamma^(-1) Gamma(s) E[rho^(-s)], where rho integrates explicit power weights against the Gaussian multiplicative chaos of the Gaussian free field pinned on the equator. The formalization proves C_gamma = C_gamma^DOZZ, expressed through the Zamolodchikov function Upsilon_{gamma/2}, at every point of the domain where Gamma(s) is finite, and at every point including the poles for the unit-volume constant; and it proves that the probabilistic reflection coefficient, built from Brownian motion conditioned to stay negative and lateral chaos, equals R^DOZZ (Theorem 3.5). The theorems hold for every model of the field and chaos, and models are constructed. All prerequisites are proved: the construction, uniqueness and moment theory of Gaussian multiplicative chaos and the extended Seiberg bounds (David, Kupiainen, Rhodes and Vargas, 2016); the BPZ equations for degenerate four-point functions and the classification of their real single-valued solutions by the Gauss hypergeometric function (Kupiainen, Rhodes and Vargas, 2019); analytic continuation of correlation functions, tail estimates for multiplicative chaos, fusion asymptotics, the shift equations and the quotient argument. Errata found in the three papers, and the places where the Lean statements differ from them, are recorded in docs/ERRATA-AND-DEVIATIONS.md.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Am\u00e9lie Loher" } ], "classification": { "arxiv": [ "math.PR", "math-ph", "math.MP", "hep-th" ], "msc2020": [ "60G60", "60G57", "81T40", "60J65", "33C05", "33B15" ] }, "formalization": { "theorem_names": [ "DOZZChallenge.dozz_formula", "DOZZChallenge.dozz_unitVolume", "DOZZChallenge.dozz_existence", "DOZZChallenge.IsUpsilon.unique", "DOZZChallenge.isCircleAverage_law", "DOZZChallenge.equatorGFF_GMC_unique_ae", "DOZZChallenge.equatorGFF_GMC_nontrivial", "DOZZChallenge.equatorGFF_GMC_law_transport", "DOZZChallenge.equatorGFF_extendedSeiberg_rho_moment" ] }, "id": "PALOMAR-2026-10-11-000008", "path": "entries/PALOMAR-2026-10-11-000008-v1.json", "preservation": { "repositories": [ { "commit": "5d770d1a6d69fac6dbb63060e8a89861bf3ee330", "fork_repository": "PalomarArchive/scottnarmstrong--DOZZ--ab6295909671", "source_repository": "scottnarmstrong/DOZZ" } ] }, "published_at": "2026-10-11T07:06:08Z", "source": { "commit": "5d770d1a6d69fac6dbb63060e8a89861bf3ee330", "project_path": null, "repository": "scottnarmstrong/DOZZ" }, "status": "registered", "title": "Integrability of Liouville theory: the Kupiainen\u2013Rhodes\u2013Vargas proof of the DOZZ formula", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "An integral extension of OpenAI's matrix multiplication exponent bound and its previous all-fields extension. The formalization proves that the exact integer coefficient tensor-rank exponent is at most 9/4, yielding division-free arithmetic programs over every associative unital ring, including noncommutative rings. For every positive epsilon, one constant bounds the operation count at all positive matrix sizes by C*n^(9/4+epsilon). The arithmetic model is preserved apart from Field-to-Ring assumptions. The extension uses a primitive tensor spectrum, coefficient extraction, fixed finite-free descent, and coprime patching of Fourier and interpolation constructions. It does not improve the numerical exponent or assert an endpoint O(n^(9/4)) algorithm, efficient uniform generation, or bit complexity.", "authors": [ { "name": "Sela Navot" } ], "classification": { "arxiv": [ "cs.CC", "cs.DS" ], "msc2020": [ "68Q25", "15A69", "68W30" ] }, "formalization": { "theorem_names": [ "ComparatorChecks.omega_bound", "ComparatorChecks.admissible_bddBelow", "ComparatorChecks.admissible_nonempty", "ComparatorChecks.omega_lower", "ComparatorChecks.epsilon_cost", "ComparatorChecks.exact_coefficients" ] }, "id": "PALOMAR-2026-10-11-000007", "path": "entries/PALOMAR-2026-10-11-000007-v1.json", "preservation": { "repositories": [ { "commit": "9d10116c89f9bdcdaccd58aacea353d4b7ae6e90", "fork_repository": "PalomarArchive/selanavot--omega_le_nine_quarters_all_rings--afa3d9923d65", "source_repository": "selanavot/omega_le_nine_quarters_all_rings" } ] }, "published_at": "2026-10-11T06:55:37Z", "source": { "commit": "9d10116c89f9bdcdaccd58aacea353d4b7ae6e90", "project_path": null, "repository": "selanavot/omega_le_nine_quarters_all_rings" }, "status": "registered", "title": "Matrix multiplication: omega at most 9/4 over every ring", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of a fixed-temperature N^(-7/10) bound for the finite-size mean bias of the zero-field Sherrington-Kirkpatrick model with independent standard Gaussian couplings and the unnormalized partition function. For every beta > 1, a positive constant depending on beta bounds the nonnegative pressure gap for every integer N >= 2. The selected results also establish the thermodynamic limit for beta > 0, the corresponding free-energy-density and expected log-partition corrections, a positive exponent saving over 2/3, and a full-sequence little-o improvement. The independently readable interface defines the limiting pressure by a thermodynamic limit; Solution identifies it with the all-probability-profile Parisi variational value.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "math.PR", "cond-mat.dis-nn" ], "msc2020": [ "82B44", "60K35", "60G15" ] }, "formalization": { "theorem_names": [ "SKMeanBias.thermodynamic_limit", "SKMeanBias.main_mean_bias", "SKMeanBias.free_energy_density_correction", "SKMeanBias.log_partition_mean_correction", "SKMeanBias.strictly_better_than_two_thirds", "SKMeanBias.mean_bias_little_o_two_thirds" ] }, "id": "PALOMAR-2026-10-11-000006", "path": "entries/PALOMAR-2026-10-11-000006-v1.json", "preservation": { "repositories": [ { "commit": "7601a9d5ade08066632bc68b9a615e163f2dd02c", "fork_repository": "PalomarArchive/pmaronow--sk-mean-bias-palomar--c8eac0abe48b", "source_repository": "pmaronow/sk-mean-bias-palomar" } ] }, "published_at": "2026-10-11T06:45:14Z", "source": { "commit": "7601a9d5ade08066632bc68b9a615e163f2dd02c", "project_path": null, "repository": "pmaronow/sk-mean-bias-palomar" }, "status": "registered", "title": "Finite-size mean bias of the Sherrington-Kirkpatrick free energy", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 development on the capacity of digraphs in the sense of Alon (1998): w(R^n) is the largest set of words of length n such that every ordered pair of distinct words has a coordinate carrying an arc from the first to the second, and C(R) is the supremum of w(R^n)^(1/n), which equals the limit; for a tournament T, t(T) is the order of its largest transitive subtournament and t(T) \u2264 C(T). Conjecture 1.1 of Alon's paper, attributed to K\u00f6rner and Simonyi, asserts C(T) = t(T) for every tournament; Alon disproved it with a random construction and with the Paley tournament on 67 vertices, and asked for the smallest tournament on which the equality fails. Let T7 be the subtournament of the Paley tournament on 23 vertices (x \u2192 y iff y \u2212 x is a nonzero square mod 23) induced on 0, 1, 2, 3, 9, 14, 18. Proved here: T7 is a tournament with t(T7) = 4; eighteen listed points of T7 \u00d7 T7 form a transitive clique of its second Sperner power; hence, by the lift lemma (N injective points of the k-th power, each earlier one dominating each later one in some coordinate, give for every m a clique of the km-th power of size at least N^m / ((N \u2212 1)m + 1), by words of fixed index sum), some Sperner power of T7 has a clique larger than 4^n, \u221a18 \u2264 C(T7), and t(T7) < C(T7); C(T7) \u2264 5, by an explicit rank-5 factorization over GF(2) of a matrix with nonzero diagonal and zeros on the arcs, through the rank bound (a clique of the n-th power indexes a nonsingular diagonal submatrix of the n-th Kronecker power, which factors through F^(r^n)); the general statements N^(1/k) \u2264 C(R), t(R) \u2264 C(R), C(R) \u2264 r for any such factorization, and C(R) \u2264 |V|; and, for the Paley tournament on 23 vertices, t = 5 and \u221a29 \u2264 C from a 29-point transitive clique of its second Sperner power; and every tournament on at most six vertices satisfies C(T) = t(T) (for each of the 1, 1, 1, 2, 4, 12, 56 isomorphism classes on 0 to 6 vertices, a representative with a rank-t factorization over GF(2) and a transitive t-chain, and a kernel check, by a bitset over all orientations, that every labelled tournament on at most six vertices is a relabelling of a representative), so a tournament with t < C has at least seven vertices and seven is the least order of a counterexample. The capacity is formalized in its supremum form. On paper, in the research note of the repository, with standard-library certificates: the square value 18 is exact; on seven vertices 455 of the 456 isomorphism classes have a rank-t matrix over GF(2) and the exception is T7, which is therefore the unique smallest counterexample up to isomorphism; every proper subtournament of T7 satisfies the equality; and the Paley tournament on 7 vertices has C = 3. The smallest example previously exhibited has 67 vertices (examples on 27 and 26 vertices follow from Alon's argument over GF(27) with a published value of t). Nothing in Lean is claimed about tournaments on seven or more vertices other than T7 and the Paley tournament on 23 vertices.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.CO", "cs.IT" ], "msc2020": [ "05C20", "05C35", "94A24" ] }, "formalization": { "theorem_names": [ "SpernerCapacity.T7_isTournament", "SpernerCapacity.transitiveNumber_T7", "SpernerCapacity.T7_square_witness", "SpernerCapacity.T7_exists_clique_gt_four_pow", "SpernerCapacity.four_lt_capacity_T7", "SpernerCapacity.sqrt18_le_capacity_T7", "SpernerCapacity.capacity_T7_le_five", "SpernerCapacity.transitiveNumber_lt_capacity_T7", "SpernerCapacity.lift_lemma", "SpernerCapacity.rpow_le_capacity", "SpernerCapacity.transitiveNumber_le_capacity", "SpernerCapacity.capacity_le_of_factorization", "SpernerCapacity.capacity_le_card", "SpernerCapacity.paley_isTournament", "SpernerCapacity.transitiveNumber_paley", "SpernerCapacity.five_lt_capacity_paley", "SpernerCapacity.sqrt29_le_capacity_paley", "SpernerCapacity.capacity_eq_transitiveNumber_of_card_le_six", "SpernerCapacity.seven_le_card_of_transitiveNumber_lt_capacity" ] }, "id": "PALOMAR-2026-10-11-000005", "path": "entries/PALOMAR-2026-10-11-000005-v1.json", "preservation": { "repositories": [ { "commit": "0e846b6144d2c5954a983d2d97e90e02169d9809", "fork_repository": "PalomarArchive/JD-Jones-ASES--sperner-t7-lean--f76f736b49c4", "source_repository": "JD-Jones-ASES/sperner-t7-lean" } ] }, "published_at": "2026-10-11T01:54:34Z", "source": { "commit": "0e846b6144d2c5954a983d2d97e90e02169d9809", "project_path": null, "repository": "JD-Jones-ASES/sperner-t7-lean" }, "status": "registered", "title": "A 7-vertex tournament whose Sperner capacity exceeds the order of its largest transitive subtournament: T\u2087, induced by the Paley tournament on 23 vertices on 0, 1, 2, 3, 9, 14, 18, has t = 4 and \u221a18 \u2264 C \u2264 5; every tournament on at most six vertices has C = t, so seven is the least order", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalisation of exponential separation for analytic iterated function systems on the real line. Strong exponential separation holds on an open and dense subset in the C2 topology and follows from an explicit criterion involving dual projections. The development characterises conjugacy to self-similar systems and proves dimension consequences conditional on cited results of Rapaport and Bowen, including a drop in the L^q dimensions of the natural measure of a concrete three-map example.", "authors": [ { "name": "Bal\u00e1zs B\u00e1r\u00e1ny" }, { "name": "Istv\u00e1n Kolossv\u00e1ry" }, { "name": "Sascha Troscheit" } ], "classification": { "arxiv": [ "math.DS", "math.CA", "math.MG" ], "msc2020": [ "28A80", "37C45" ] }, "formalization": { "theorem_names": [ "Challenge.audit_sesc_open_dense", "Challenge.audit_sesc_of_dualProj", "Challenge.audit_example_criterion", "Challenge.audit_example_sesc", "Challenge.audit_conj_similarity", "Challenge.audit_conj_iff", "Challenge.audit_subconj_iff", "Challenge.audit_dim_of_esc", "Challenge.audit_dim_open_dense", "Challenge.audit_example_dim", "Challenge.audit_example_localDim", "Challenge.audit_example_lq" ] }, "id": "PALOMAR-2026-10-11-000004", "path": "entries/PALOMAR-2026-10-11-000004-v1.json", "preservation": { "repositories": [ { "commit": "cf134207f9218a3eb3e0e4bcbd472af513eef9f6", "fork_repository": "PalomarArchive/sascha2718--lean-dual-analytic-ifs--016e527776b7", "source_repository": "sascha2718/lean-dual-analytic-ifs" } ] }, "published_at": "2026-10-11T00:53:46Z", "source": { "commit": "cf134207f9218a3eb3e0e4bcbd472af513eef9f6", "project_path": null, "repository": "sascha2718/lean-dual-analytic-ifs" }, "status": "registered", "title": "Exponential separation of analytic self-conformal sets on the real line: Lean formalisation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of the paper \"Coarse ellipticity and De Giorgi\u2013Nash\u2013Moser theory in the optimal range\" by S. Armstrong, B. Avelin, T. Kuusi and A. Turpeinen (arXiv:2610.11879). The paper extends the theory of De Giorgi\u2013Nash\u2013Moser to elliptic equations with symmetric coefficients a(x) which are possibly degenerate and unbounded but satisfy a coarse ellipticity condition: spatial moments of coarse-grained matrices, suitably discounted and summed across scales, are finite. Under this condition it proves local upper bounds for weak subsolutions, a weak Harnack inequality for nonnegative supersolutions, and a Harnack inequality for nonnegative solutions. The condition holds if a \u2208 W^{-\u03b1,p} \u2229 L^1, a^{-1} \u2208 W^{-\u03b2,q} \u2229 L^1 and (\u03b1+\u03b2)/2 + ((d-1)/2)(1/p + 1/q) < 1, for some \u03b1, \u03b2 \u2265 0 and 1 < p, q < \u221e. This range is sharp, including its boundary, in every dimension d \u2265 3: for \u03b1 = \u03b2 = 0, where it corresponds to the results of Bella and Sch\u00e4ffner, and for the corresponding Besov-type quasi-norms of negative order when \u03b1, \u03b2 > 0. The formalization covers these results, together with the optimality of the powers of the coarse ellipticity ratio in the local bounds and of the range of exponents in the weak Harnack inequality. The statements are for dimension d \u2265 3 and are formulated on the unit cube; the limiting cases p = \u221e or q = \u221e are not formalized. CORRESPONDENCE.md gives the status of every numbered statement of the paper. Built on Mathlib and the CoarseGraining library, using only Lean's standard axioms.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Benny Avelin" }, { "name": "Tuomo Kuusi" }, { "name": "Aatu Turpeinen" } ], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35J70", "35B65", "35J15", "35B45" ] }, "formalization": { "theorem_names": [ "CoarseDeGiorgiAudit.Harnack.harnack" ] }, "id": "PALOMAR-2026-10-11-000003", "path": "entries/PALOMAR-2026-10-11-000003-v1.json", "preservation": { "repositories": [ { "commit": "200c40c9c55a5e19188739d27466ddffcb2660c4", "fork_repository": "PalomarArchive/scottnarmstrong--CoarseDeGiorgi--c1c596d07d50", "source_repository": "scottnarmstrong/CoarseDeGiorgi" } ] }, "published_at": "2026-10-11T00:25:32Z", "source": { "commit": "200c40c9c55a5e19188739d27466ddffcb2660c4", "project_path": null, "repository": "scottnarmstrong/CoarseDeGiorgi" }, "status": "registered", "title": "De Giorgi\u2013Nash\u2013Moser theory under coarse ellipticity", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean development of upper and local lower minimax bounds for nonlinear functionals of conditional means under bounded, rough random design. The generic theorem covers the root-n regime and the subcritical regime with its density-dependent stretched-exponential correction, logarithmic upper and lower factors, and a one-quarter minimax probability lower bound. The substantive development also contains projection, polynomial-estimator, Poisson reduction, lattice-prior, application, and causal-identification results. The selected Comparator surface is the generic theorem's two parts. The remaining logarithmic gap is unresolved.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "stat.ME" ], "msc2020": [ "62G05", "62G08", "62G20", "62C20" ] }, "formalization": { "theorem_names": [ "RoughRegimeSubmission.mainUpper", "RoughRegimeSubmission.mainLower" ] }, "id": "PALOMAR-2026-10-11-000002", "path": "entries/PALOMAR-2026-10-11-000002-v1.json", "preservation": { "repositories": [ { "commit": "ec97825d7539266fd3ebd9d0527c34b0ba5d0f65", "fork_repository": "PalomarArchive/pmaronow--roughregime-palomar--9ee2236482f9", "source_repository": "pmaronow/roughregime-palomar" } ] }, "published_at": "2026-10-11T00:11:04Z", "source": { "commit": "ec97825d7539266fd3ebd9d0527c34b0ba5d0f65", "project_path": null, "repository": "pmaronow/roughregime-palomar" }, "status": "registered", "title": "Nearly Minimax Rates for Functional Estimation Under Rough Random Design", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalisation of Aleksandrov maximum principles for kinetic equations d_t u + v.grad_x u - A : D_v^2 u = f in non-divergence form, with measurable, uniformly elliptic diffusion acting only in the velocity. For coefficients A(t,v) the estimate holds for every source exponent p > 2d+1; for autonomous coefficients a(x,v) in one space dimension for every p above a threshold 1 + beta_*(Lambda/lambda) < 4, where no exponent below 4 works uniformly in the ellipticity ratio; and for general measurable coefficients A(t,x,v) for every p >= 1 + 128 d^2 (Lambda/lambda)^2 / 3. In each case the estimate implies interior H\u00f6lder continuity of solutions; for coefficients A(t,x,v) this is the kinetic Krylov\u2013Safonov theorem. The library also proves the parabolic Aleksandrov estimate of Krylov, the parabolic Harnack inequality of Krylov and Safonov, the terminal solution operators of the kinetic equation, and classical Dirichlet solvability on ellipsoidal cylinders. H\u00f6rmander's hypoellipticity theorem is taken from the formalisation hoermander-rothschild-stein.", "authors": [ { "name": "Am\u00e9lie Loher" } ], "classification": { "arxiv": [ "math.AP", "math.PR" ], "msc2020": [ "35K70", "35B50", "35B65", "35Q84", "35H10" ] }, "formalization": { "theorem_names": [ "KineticAleksandrovChallenge.kinetic_aleksandrov_timeVelocity", "KineticAleksandrovChallenge.kinetic_aleksandrov_timeVelocity_localised", "KineticAleksandrovChallenge.kinetic_holder_timeVelocity", "KineticAleksandrovChallenge.kinetic_aleksandrov_autonomous", "KineticAleksandrovChallenge.kinetic_aleksandrov_autonomous_localised", "KineticAleksandrovChallenge.kinetic_holder_autonomous", "KineticAleksandrovChallenge.kinetic_aleksandrov_autonomous_counterexample", "KineticAleksandrovChallenge.existsUnique_bellmanAdjointExponent", "KineticAleksandrovChallenge.bellmanAdjointExponent_characterization", "KineticAleksandrovChallenge.parabolic_aleksandrov", "KineticAleksandrovChallenge.parabolic_harnack_unit_cylinder" ] }, "id": "PALOMAR-2026-10-11-000001", "path": "entries/PALOMAR-2026-10-11-000001-v1.json", "preservation": { "repositories": [ { "commit": "d7e0125892d7bbd013c86c9c5d6a04e39e0168a6", "fork_repository": "PalomarArchive/amelieloher--hypoelliptic-aleksandrov--29cafd061e52", "source_repository": "amelieloher/hypoelliptic-aleksandrov" } ] }, "published_at": "2026-10-11T00:02:03Z", "source": { "commit": "d7e0125892d7bbd013c86c9c5d6a04e39e0168a6", "project_path": null, "repository": "amelieloher/hypoelliptic-aleksandrov" }, "status": "registered", "title": "Kinetic Aleksandrov estimates", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean formalization accompanying the paper An Exponent of 1.043 for the Unit Distance Problem by Eric Naslund (cited under sources), which proves the exponent 1.043171 unconditionally with a computer-assisted certificate. Conditional theorem on unit distances in the Euclidean plane. Let B = Q(sqrt 241) and let E be the number field of degree 512 generated over Q by sqrt 241 and square roots of the eight elements -1, -71011068 + 4574225 sqrt 241, (-6101 - 393 sqrt 241)/2, (6101 - 393 sqrt 241)/2, 31 - 2 sqrt 241, 31 + 2 sqrt 241, 326 - 21 sqrt 241 and 326 + 21 sqrt 241 of B (a basis of the {2,3,5}-units of B modulo squares). Let E_W be the field of degree 8192 obtained from E by adjoining square roots of four explicit elements beta_1, ..., beta_4 of E, each generating with two of the square roots a dihedral extension of B of degree 8 (the elements are written out in the Challenge). Hypothesis (H_W) is one displayed numerical inequality for the Dedekind zeta function of E_W: log(Re zeta_{E_W}(1 + 1/4411))/8192 + (1/4411)((L - gamma - log(4 pi))/4 - Re((zeta_{E_W}'/zeta_{E_W})(2))/8192) < 50969/1000000, where L = (9/4) log 2 + (1/2) log 3615 and gamma is Euler's constant. The compared theorem proves that (H_W) implies the existence of finite sets U_j in the plane with |U_j| -> infinity such that the number of unordered pairs of points of U_j at distance one, divided by |U_j|^(20863/20000), tends to infinity. The exponent is exactly 20863/20000 = 1.04315. The conclusion is a sequence statement only; no bound at every sufficiently large cardinality is asserted. The infinite pro-2 tower over B with Frobenius caps at both primes above 29 and at one prime above 41 (generators, relation bound, presentation, Golod-Shafarevich infinitude, local data), its fields, which are Galois over B and over a fixed base field but not over Q, their local types and signature, the root-discriminant bound, the field E_W with its degree, its embedding in the base field and its local data, the passage from (H_W) to the analytic ceiling, the numerical margin and the planar construction are proved in Lean over Mathlib; (H_W) itself is not proved. Outside Lean, a rigorous computation (certified values of the 576 L-functions of degrees 4 and 8 in the factorization of zeta_{E_W} and an Euler product with explicit tail) bounds the left side of (H_W) by 0.0509172; that computation is not part of the formal proof. The earlier formalization registered as PALOMAR-2026-10-01-000018 version 1 proved the exponent 10427/10000 from a different inequality, for the field E.", "authors": [ { "name": "Eric Naslund" } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "52C10", "11R37", "20C05" ] }, "formalization": { "theorem_names": [ "UnitDistanceSqrt241Submission.target_of_wide_zeta_bound" ] }, "id": "PALOMAR-2026-10-01-000018", "path": "entries/PALOMAR-2026-10-01-000018-v2.json", "preservation": { "repositories": [ { "commit": "edd88c0c175ab4d18cf6115e2f6ddf3865d36b12", "fork_repository": "PalomarArchive/enaslund--unit-distance-bound-0.0427--b3a8b62d2e9c", "source_repository": "enaslund/unit-distance-bound-0.0427" } ] }, "published_at": "2026-10-10T20:31:53Z", "source": { "commit": "edd88c0c175ab4d18cf6115e2f6ddf3865d36b12", "project_path": "lean", "repository": "enaslund/unit-distance-bound-0.0427" }, "status": "registered", "title": "An Exponent of 1.043 for the Unit Distance Problem: Lean formalization at exponent 1.04315, conditional on one zeta inequality", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "Minimax root-mean-square bounds for estimating a constant conditional variance from independent random-design regression observations. The design density has fixed positive lower and upper bounds but no smoothness; covariate-dependent errors have uniformly bounded conditional fourth moments. For 1 < s < d/4, the proved bracket identifies the polynomial exponent 2(s+1)/(d+4) and the common explicit stretched-exponential factor, with a remaining logarithmic gap. It also proves the exponent corollary, the exact low-smoothness rate, and the parametric regime.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "math.ST", "stat.TH" ], "msc2020": [ "62G05", "62G08", "62G20", "62C20" ] }, "formalization": { "theorem_names": [ "NearlyMinimax.Submission.paper_main", "NearlyMinimax.Submission.exponent_limit", "NearlyMinimax.Submission.polynomial_bounds", "NearlyMinimax.Submission.low_smoothness", "NearlyMinimax.Submission.parametric" ] }, "id": "PALOMAR-2026-10-10-000002", "path": "entries/PALOMAR-2026-10-10-000002-v1.json", "preservation": { "repositories": [ { "commit": "f3ef2ef312d44bbc693c1d5bea9cee654a29b598", "fork_repository": "PalomarArchive/pmaronow--ccv-palomar--47775cd59e5a", "source_repository": "pmaronow/ccv-palomar" } ] }, "published_at": "2026-10-10T04:35:14Z", "source": { "commit": "f3ef2ef312d44bbc693c1d5bea9cee654a29b598", "project_path": null, "repository": "pmaronow/ccv-palomar" }, "status": "registered", "title": "Nearly minimax variance estimation under rough random design", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean formalization of Patrick Lopatto's zero-field Sherrington\u2013Kirkpatrick results. The selected claims are full interval support, a nonnegative smooth density with a terminal atom for every inverse temperature beta > 1, and quantitative inequalities for that actual atom. The independent Challenge defines the Parisi functional through concrete bounded-gradient weak PDE solutions, with terminal log cosh and coefficient beta squared divided by two. Solution identifies that definition with the constructed Parisi potential. The repository retains the supporting PDE, diffusion, covariance, minimizer, and numbered-result development described in COVERAGE.md and results.json.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "math.PR", "math-ph" ], "msc2020": [ "82B44", "60G15" ] }, "formalization": { "theorem_names": [ "PalomarFRSB.fullRSB", "PalomarFRSB.quantitativeAtom" ] }, "id": "PALOMAR-2026-10-10-000001", "path": "entries/PALOMAR-2026-10-10-000001-v1.json", "preservation": { "repositories": [ { "commit": "b3ff38e8bc8c057b8ee2addf0fa2428db32f106b", "fork_repository": "PalomarArchive/pmaronow--frsb-palomar--66f552945dc4", "source_repository": "pmaronow/frsb-palomar" } ] }, "published_at": "2026-10-10T00:49:02Z", "source": { "commit": "b3ff38e8bc8c057b8ee2addf0fa2428db32f106b", "project_path": null, "repository": "pmaronow/frsb-palomar" }, "status": "registered", "title": "Full replica symmetry breaking in the Sherrington\u2013Kirkpatrick model", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean proof of the dimension-two case of Nielsen\u2013Okamura Conjecture 9.1. On any connected open planar Euclidean domain, complementarity of the actual canonical local affine-parameter midpoints of a smooth pair of dual torsion-free connections forces the symmetric cubic tensor to be constant. The proof constructs and identifies the short geodesic germs and derives the fourth-order obstruction, including the exceptional rank-deficient tensors. A second theorem needs only everywhere Frechet differentiability and vanishing of that obstruction; five fixed direction tests at every point suffice. Dimensions above two are outside the claim.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.DG", "math.CA" ], "msc2020": [ "53B12", "39B22" ] }, "formalization": { "theorem_names": [ "PlanarMidpoint.planar_dual_midpoint_rigidity", "PlanarMidpoint.planar_obstruction_rigidity", "PlanarMidpoint.planar_five_direction_rigidity" ] }, "id": "PALOMAR-2026-10-09-000016", "path": "entries/PALOMAR-2026-10-09-000016-v1.json", "preservation": { "repositories": [ { "commit": "148b39618710caf576189cbdb75551cec1735201", "fork_repository": "PalomarArchive/JD-Jones-ASES--PlanarMidpoint--8b5260b291db", "source_repository": "JD-Jones-ASES/PlanarMidpoint" } ] }, "published_at": "2026-10-09T22:55:50Z", "source": { "commit": "148b39618710caf576189cbdb75551cec1735201", "project_path": null, "repository": "JD-Jones-ASES/PlanarMidpoint" }, "status": "registered", "title": "Planar Euclidean midpoint rigidity", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Uniform permanence is established for finite endotactic reaction networks with arbitrary real complexes and measurable rate coefficients bounded above and away from zero almost everywhere. Each positive stoichiometric compatibility class contains a compact convex forward-invariant absorbing set, with entry times uniform over compact sets of initial conditions and all rate histories within a fixed rate box. Global positive Carath\u00e9odory solutions exist and are unique, and an exponential continuous-dependence estimate holds for initial data under a common rate history on a compact positive region. The result resolves the Endotactic Permanence Conjecture stated as Conjecture 1.3 by Johnston, Pantea and Donnell (https://arxiv.org/abs/1412.4662), using an ordinary-endotactic drift estimate together with the finite affine barrier construction from OpenAI\u2019s work on weakly reversible networks.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "34A12", "34D23", "92C42" ] }, "formalization": { "theorem_names": [ "EndotacticSubmission.uniform_permanence", "EndotacticSubmission.continuous_dependence" ] }, "id": "PALOMAR-2026-10-09-000015", "path": "entries/PALOMAR-2026-10-09-000015-v1.json", "preservation": { "repositories": [ { "commit": "c72c945264565397ecc3eee779e6bd5f3f3a061d", "fork_repository": "PalomarArchive/mmislan--palomar-endotactic-permanence--fcc3441bc2f5", "source_repository": "mmislan/palomar-endotactic-permanence" } ] }, "published_at": "2026-10-09T19:55:24Z", "source": { "commit": "c72c945264565397ecc3eee779e6bd5f3f3a061d", "project_path": null, "repository": "mmislan/palomar-endotactic-permanence" }, "status": "registered", "title": "Uniform permanence of endotactic reaction networks with measurable rates", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the regularity theory of H\u00f6rmander operators: second-order operators built from smooth vector fields X_1, ..., X_q (and possibly a drift X_0) whose iterated commutators span the tangent space at every point. It proves H\u00f6rmander's 1967 theorem that such operators are hypoelliptic: every locally integrable weak solution of X_0 u + sum X_i^2 u + c u = g with smooth data is smooth. It then develops the Rothschild\u2013Stein theory (Acta Math. 1976), following Bramanti and Brandolini's \"H\u00f6rmander Operators\" (2023): lifting and approximation of the fields by the generators of a homogeneous group; the homogeneous fundamental solution and singular-integral estimates on homogeneous groups; and the sharp interior estimates, by which a distributional solution of sum X_i^2 T = f with f in W^{k,p}_X or C^{k,alpha}_X is a function in W^{k+2,p}_X or C^{k+2,alpha}_X, for every k >= 0, and at k = 0 with a drift. The formalization also proves the formal Baker\u2013Campbell\u2013Hausdorff theorem, Chow\u2013Rashevskii connectivity, and the Nagel\u2013Stein\u2013Wainger theorems on control balls (ball-box, volume and doubling, comparison of distances), uniformly over compact sets.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Am\u00e9lie Loher" } ], "classification": { "arxiv": [ "math.AP", "math.CA", "math.DG" ], "msc2020": [ "35H10", "35H20", "43A80", "53C17", "22E25" ] }, "formalization": { "theorem_names": [ "HormanderChallenge.hormander_hypoellipticity", "HormanderChallenge.memSobolev_neg_of_integrable", "HormanderChallenge.exists_contDiff_of_forall_memSobolev", "HormanderChallenge.subelliptic_estimate", "HormanderChallenge.forall_memSobolev_of_localized", "HormanderChallenge.bch_lie_series", "HormanderChallenge.chow_rashevskii", "HormanderChallenge.ball_box", "HormanderChallenge.ball_volume_doubling", "HormanderChallenge.distance_comparison", "HormanderChallenge.lifting_approximation", "HormanderChallenge.lifting_approximation_drift" ] }, "id": "PALOMAR-2026-10-09-000014", "path": "entries/PALOMAR-2026-10-09-000014-v1.json", "preservation": { "repositories": [ { "commit": "d1462014b4ca69cf04b250582ea45021aa07b546", "fork_repository": "PalomarArchive/amelieloher--hoermander-rothschild-stein--a3f8df3a01f8", "source_repository": "amelieloher/hoermander-rothschild-stein" } ] }, "published_at": "2026-10-09T17:57:04Z", "source": { "commit": "d1462014b4ca69cf04b250582ea45021aa07b546", "project_path": null, "repository": "amelieloher/hoermander-rothschild-stein" }, "status": "registered", "title": "H\u00f6rmander's hypoellipticity theorem and the Rothschild\u2013Stein theory", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the positive-field Sherrington\u2013Kirkpatrick results in Patrick Lopatto's arXiv:2604.11921v2. The selected registry claims are convergence of the expected finite-volume free energy to the concrete replica-symmetric Gaussian formula for beta > 0, h > 0, and AT parameter at most one, including equality; and an exact description of the positive-field AT boundary as a positive strictly increasing smooth graph on beta > 1, tending to zero at beta = 1 from the right, together with (1,0). The repository also contains the numbered PDE, stochastic-process, moment, and variational results documented in COVERAGE.md and results.json.", "authors": [ { "name": "P. M. Aronow" }, { "name": "Patrick Lopatto" } ], "classification": { "arxiv": [ "math.PR", "math-ph" ], "msc2020": [ "82B44", "60G15" ] }, "formalization": { "theorem_names": [ "PalomarAT.replicaSymmetry", "PalomarAT.smoothATBoundary" ] }, "id": "PALOMAR-2026-10-09-000013", "path": "entries/PALOMAR-2026-10-09-000013-v1.json", "preservation": { "repositories": [ { "commit": "915bca99f46bad65629ef35c8be4bf8fc962af9a", "fork_repository": "PalomarArchive/pmaronow--at-palomar--1a4bf753f431", "source_repository": "pmaronow/at-palomar" } ] }, "published_at": "2026-10-09T17:37:41Z", "source": { "commit": "915bca99f46bad65629ef35c8be4bf8fc962af9a", "project_path": null, "repository": "pmaronow/at-palomar" }, "status": "registered", "title": "Replica symmetry up to the de Almeida\u2013Thouless line in the Sherrington\u2013Kirkpatrick model", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of MIP* = RE (Ji, Natarajan, Vidick, Wright and Yuen): the class of languages decidable by a classical polynomial-time verifier interacting with two provers who share entanglement, with completeness 1 and soundness 1/2, is exactly the class of recursively enumerable languages. The development also contains the halting reduction behind it (a computable map from Turing machines to nonlocal games whose synchronous and quantum values are 1 if the machine halts and at most 1/2 otherwise) and its corollaries that the synchronous and the quantum value of a game are uncomputable. It builds on three other formalizations: the soundness of the classical low individual degree test (MIPStarRE, by Sirui Lu, Ruixuan Deng and Zhengfeng Ji, registered as PALOMAR-2026-08-18-000001), imported as a pinned Lake dependency, and the parallel repetition theorem for entangled strategies (ten-proofs, OpenAI) and the direct parallel repetition theorem for commuting-operator strategies (commuting-repetition), which are vendored with their provenance recorded. Palomar/Challenge.lean states the four compared theorems on top of Mathlib alone, in a tree-program model of polynomial time that the fidelity section describes; Palomar/Solution.lean proves them from the library.", "authors": [ { "name": "Thomas Vidick" } ], "classification": { "arxiv": [ "quant-ph", "cs.CC" ], "msc2020": [ "68Q15", "81P45", "03D35", "46L10" ] }, "formalization": { "theorem_names": [ "MIPRE.Palomar.halting_reduces_to_value", "MIPRE.Palomar.syncValue_uncomputable", "MIPRE.Palomar.quantumValue_uncomputable", "MIPRE.Palomar.mipstar_eq_re" ] }, "id": "PALOMAR-2026-10-09-000012", "path": "entries/PALOMAR-2026-10-09-000012-v1.json", "preservation": { "repositories": [ { "commit": "7798a80300e1c63b783c214f9e0cd9c00a8721b6", "fork_repository": "PalomarArchive/vidick--MIPRE-formalization--0d44c812c686", "source_repository": "vidick/MIPRE-formalization" } ] }, "published_at": "2026-10-09T16:39:17Z", "source": { "commit": "7798a80300e1c63b783c214f9e0cd9c00a8721b6", "project_path": null, "repository": "vidick/MIPRE-formalization" }, "status": "registered", "title": "MIP* = RE", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We formalize an exact positional inverse classification in the elementary abelian group C3^3. For a sequence of 24 nonzero elements, every pair of nonempty zero-sum subsequences of length at most three has intersection sum zero if and only if the original sequence, up to a permutation of all 24 positions, consists of three copies of eight distinct elements having no short nonempty zero sum. The two implications, including the positional intersection condition, are proved in Lean. The existing literature review did not locate the exact inverse classification, but historical novelty, priority and significance are not certified.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B75" ] }, "formalization": { "theorem_names": [ "InverseZeroSum.Candidate3.frozenClassification" ] }, "id": "PALOMAR-2026-10-09-000011", "path": "entries/PALOMAR-2026-10-09-000011-v1.json", "preservation": { "repositories": [ { "commit": "c5f9e5e822020688c72b2a3bfacbf147a801e219", "fork_repository": "PalomarArchive/jfairfaxball-348--Inverse-Zero-Sum-Research--d48564ec88c4", "source_repository": "jfairfaxball-348/Inverse-Zero-Sum-Research" } ] }, "published_at": "2026-10-09T15:53:49Z", "source": { "commit": "c5f9e5e822020688c72b2a3bfacbf147a801e219", "project_path": null, "repository": "jfairfaxball-348/Inverse-Zero-Sum-Research" }, "status": "registered", "title": "Inverse classification for innerly joint short zero-sum subsequences in C3^3", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 proof of the PCP theorem and of its consequences for approximation, for an explicit single-tape Turing machine model. The compared theorems are: the PCP theorem in gap form, that for a fixed fraction num/den > 0 every NP language reduces in polynomial time to 3-SAT so that members give satisfiable formulas and non-members give formulas of which every assignment leaves at least that fraction of the clauses unsatisfied (Complexity.pcp_gap); the PCP theorem in verifier form, NP = PCP(O(log n), O(1)) (Complexity.pcp_theorem); that the problems of the syntactic class MaxSNP0 are in NPO (Complexity.npo_maxSNP0); that MaxSNP-hard problems have no polynomial-time approximation scheme unless P = NP (Complexity.maxSNPHard_not_hasPTAS), even when a separate algorithm is allowed for each precision (Complexity.maxSNPHard_no_ptas); that MAX-3-SAT and 3-OCCURRENCE MAX-3-SAT are MaxSNP-complete (Complexity.max3SAT_maxSNPComplete, Complexity.max3SAT3_maxSNPComplete); and that both are APX-complete (Complexity.max3SAT_apxComplete, Complexity.max3SAT3_apxComplete). The statement, Challenge.lean, imports only Lean core and defines everything it uses: the machine model, P, NP and polynomial-time functions; probabilistically checkable proofs and gap reductions; optimization problems, approximation algorithms and schemes; NPO; L-reductions, the syntactic class MaxSNP0 of Papadimitriou and Yannakakis (structures of fixed input and solution signatures on a finite universe, and a well-formed quantifier-free formula) and the class MaxSNP of NPO problems L-reducible to it; APX and PTAS-reductions in the sense of Crescenzi and Trevisan; and the two problems.\nThe gap form of the PCP theorem is proved by Dinur's gap amplification: the 3-CNF gap map, its soundness and its implementation as a polynomial-time multi-stack Turing machine are taken from the Lean formalization of the Unique Games theorem in the OpenAI mathematics repository (194 files, adapted to the Lean module system and to Lean v4.35.0-rc2), and composed with the Cook-Levin reduction of the Palomar entry PALOMAR-2026-10-02-000013 (whose library is included). A simulation theorem carries polynomial time from Mathlib's multi-stack machines to the single-tape model. The verifier form samples clauses of the gap formula with O(log n) random bits. MaxSNP-hard problems have no approximation scheme because MAX-3-SAT L-reduces to them and an approximation scheme for MAX-3-SAT would decide the gap. MAX-3-SAT is put in MaxSNP0 by an explicit quantifier-free formula, and every MaxSNP0 problem L-reduces to it by the Papadimitriou-Yannakakis clause construction (one clause per falsifying pattern of atoms, split into 3-clauses with fresh variables, with an averaging bound for the constant; assignments are mapped back to structures of the solution signature). The MaxSNP0 problems are in NPO by an explicit polynomial-time parser and feasibility check, and a size bound for structures of a fixed signature. 3-OCCURRENCE MAX-3-SAT is reached in two L-reductions: copies of each variable linked along an explicit constant-degree expander (from the same OpenAI development), and then a torus gadget of implications. APX-hardness under PTAS-reductions needs a witness-recovering PCP theorem, which is proved here: an explicit deterministic decoder, shown polynomial-time, turns every assignment of the gap formula that satisfies more than the fraction 1 - num/den of its clauses into an assignment of the original formula and then into an NP certificate. With it every APX problem reduces to MAX-3-SAT by a threshold construction. Membership in APX uses a 2-approximation. The proof uses only the axioms propext, Classical.choice and Quot.sound.", "authors": [ { "name": "Robert D. Barish" } ], "classification": { "arxiv": [ "cs.CC", "cs.LO" ], "msc2020": [ "68Q17", "68Q25", "68W25", "68Q15", "68V20" ] }, "formalization": { "theorem_names": [ "Complexity.pcp_gap", "Complexity.pcp_theorem", "Complexity.npo_maxSNP0", "Complexity.maxSNPHard_no_ptas", "Complexity.maxSNPHard_not_hasPTAS", "Complexity.max3SAT_maxSNPComplete", "Complexity.max3SAT3_maxSNPComplete", "Complexity.max3SAT_apxComplete", "Complexity.max3SAT3_apxComplete" ] }, "id": "PALOMAR-2026-10-09-000010", "path": "entries/PALOMAR-2026-10-09-000010-v1.json", "preservation": { "repositories": [ { "commit": "10079b2a14bd0df47d2a24364133f6008b9c207f", "fork_repository": "PalomarArchive/RBarish-UTokyo--Max3SAT-MaxSNP-APX-Lean4--da68bef5965e", "source_repository": "RBarish-UTokyo/Max3SAT-MaxSNP-APX-Lean4" } ] }, "published_at": "2026-10-09T11:47:04Z", "source": { "commit": "10079b2a14bd0df47d2a24364133f6008b9c207f", "project_path": null, "repository": "RBarish-UTokyo/Max3SAT-MaxSNP-APX-Lean4" }, "status": "registered", "title": "MaxSNP-completeness and APX-completeness of MAX-3-SAT and 3-OCCURRENCE MAX-3-SAT (the PCP theorem and hardness of approximation) in Lean 4", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "This repository proves Baker's Brill--Noether existence conjecture for finite graphs of genus at most six. The conjecture states that if G is a finite connected loopless multigraph, with genus defined to be g = E-V+1, then for all nonnegative integers d, r, there exists a configuration of d chips on G with Baker--Norine rank at least r provided that the Brill--Noether number rho = g - (r+1)(g-d+r) is nonnegative. To the author's knowledge, this is a new result, extending a result of Atanasov and Ranganathan from genus five to six. Two generalizations are also included. First, in genus at most six, the Brill--Noether rank, in the sense of Lim--Payne--Potashnik, is at least the expected dimension of the Brill--Noether locus, when d is at most g+r. This is a combinatorial analog of the classical lower bound on the dimension of Brill--Noether loci, usually established on algebraic curves using intersection theory. Second, the Brill--Noether existence conjecture for graphs with one marked vertex, as stated by Pflueger--Solomon, holds in genus up to five. The proof synthesizes recent work of Christ--Ma and Draisma--Vargas. The work of Christ--Ma predicts the subdivision required to obtain special divisors on graphs, using enumerative geometry. The work of Draisma--Vargas provides a combinatorial construction to realize these special divisors. The novel contribution in this repository is a descent argument, producing special divisors on the original graph from special divisors on an odd subdivision. This work has not yet been written up for publication; the present repository is the first announcement of it.", "authors": [ { "name": "Nathan Pflueger" } ], "classification": { "arxiv": [ "math.CO", "math.AG" ], "msc2020": [ "05C57", "14T15", "14H51" ] }, "formalization": { "theorem_names": [ "GenusSix.brill_noether_existence_through_six", "GenusSix.brill_noether_rank_through_six", "GenusSix.brill_noether_rank_ge_rho_through_six", "GenusSix.once_marked_brill_noether_existence_through_five" ] }, "id": "PALOMAR-2026-10-09-000009", "path": "entries/PALOMAR-2026-10-09-000009-v1.json", "preservation": { "repositories": [ { "commit": "bfa274b00e6e559f34f3307d33c4aae52825778e", "fork_repository": "PalomarArchive/npflueger--brill-noether-graphs--fa3c542b766f", "source_repository": "npflueger/brill-noether-graphs" } ] }, "published_at": "2026-10-09T09:45:24Z", "source": { "commit": "bfa274b00e6e559f34f3307d33c4aae52825778e", "project_path": null, "repository": "npflueger/brill-noether-graphs" }, "status": "registered", "title": "Brill--Noether existence and rank for graphs through genus six", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Whitehead's asphericity problem would have a negative answer if a non-aspherical two-complex began an infinite chain whose inclusions induce zero maps on second homotopy groups. This Lean formalization proves the article's main finite-length characterization: a connected two-complex K begins such chains of every finite length if and only if K has a connected acyclic regular cover. It also formalizes the finite acyclic enhancement.", "authors": [ { "name": "Laurent Bartholdi" }, { "name": "Roman Mikhailov" } ], "classification": { "arxiv": [ "math.GT", "math.GR" ], "msc2020": [ "55P15" ] }, "formalization": { "theorem_names": [ "Whitehead.TheoremA" ] }, "id": "PALOMAR-2026-10-09-000008", "path": "entries/PALOMAR-2026-10-09-000008-v1.json", "preservation": { "repositories": [ { "commit": "e0683c947a3120af725ac7a29fc95a059798b64e", "fork_repository": "PalomarArchive/laurentbartholdi--whitehead--359fad44528b", "source_repository": "laurentbartholdi/whitehead" } ] }, "published_at": "2026-10-09T06:49:29Z", "source": { "commit": "e0683c947a3120af725ac7a29fc95a059798b64e", "project_path": null, "repository": "laurentbartholdi/whitehead" }, "status": "registered", "title": "Finite Chains of Two-Complexes and Acyclic Covers", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the strict Neumann P\u00f3lya inequality for bounded simply connected Lipschitz domains in the plane. The plane is identified with \u2102; the Neumann eigenvalues \u03bc_0 \u2264 \u03bc_1 \u2264 ... are defined by the Courant\u2013Fischer min\u2013max formula for the form \u222b|\u2207u|\u00b2 on H\u00b9(\u03a9) (weak gradients in L\u00b2), indexed from 0 and repeated according to multiplicity, and N_N(E) = #{j \u2265 0 : \u03bc_j < E}. The principal results are: for every bounded simply connected Lipschitz domain \u03a9 and every integer j \u2265 1, |\u03a9| \u03bc_j(\u03a9) < 4\u03c0j; and, equivalently, N_N(E) > |\u03a9| E / (4\u03c0) for every E > 0. The proof development (about 100,000 lines, namespace PolyaNeumann) also identifies the variational eigenvalues with the spectrum of the constructed self-adjoint Neumann operator, proves a quantitative smooth-domain estimate via boundary transport, Cayley/monodromy rank bounds and a Herglotz-wave reduction, and passes to Lipschitz domains by smooth approximation.", "authors": [ { "name": "Quanyu Tang" }, { "name": "Zuoqin Wang" } ], "classification": { "arxiv": [ "math.SP", "math.AP" ], "msc2020": [ "35P15", "35P20", "35J05" ] }, "formalization": { "theorem_names": [ "PalomarPolyaNeumann.strict_neumann_polya", "PalomarPolyaNeumann.strict_neumann_polya_count" ] }, "id": "PALOMAR-2026-10-09-000007", "path": "entries/PALOMAR-2026-10-09-000007-v1.json", "preservation": { "repositories": [ { "commit": "f07f1c7051862ba146429db6eda5dc4e8b020f74", "fork_repository": "PalomarArchive/QuanyuTang--polya-neumann-simply-connected-formalization-Palomar-version-v2--51a609a05e54", "source_repository": "QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2" } ] }, "published_at": "2026-10-09T06:38:53Z", "source": { "commit": "f07f1c7051862ba146429db6eda5dc4e8b020f74", "project_path": null, "repository": "QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2" }, "status": "registered", "title": "Strict Neumann Polya inequality on simply connected planar domains", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every fixed integer k >= 3, the minimum number of colors in an edge-coloring of an n-vertex simple graph with at least floor(n^2/4)+1 edges, in which every copy of C_(2k+1) is rainbow, is (1/8 + o(1))n^2. The minimum is attained. The development combines Asad Shahab's seven-cycle proof, using an exact rational five-vertex certificate, with a formalization of the Buci\u0107-Chen-Ma argument for longer odd cycles.", "authors": [ { "name": "Asad Shahab" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C35", "05C15", "05D10" ] }, "formalization": { "theorem_names": [ "Erdos809Palomar.main_result" ] }, "id": "PALOMAR-2026-10-09-000006", "path": "entries/PALOMAR-2026-10-09-000006-v1.json", "preservation": { "repositories": [ { "commit": "15f8839e2a21f64fc5558714506ebba465926eda", "fork_repository": "PalomarArchive/Asad-Shahab--erdos-809-lean--5163a53d73cf", "source_repository": "Asad-Shahab/erdos-809-lean" } ] }, "published_at": "2026-10-09T06:28:55Z", "source": { "commit": "15f8839e2a21f64fc5558714506ebba465926eda", "project_path": null, "repository": "Asad-Shahab/erdos-809-lean" }, "status": "registered", "title": "Erd\u0151s Problem 809: an independent proof of the seven-cycle threshold", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every real sigma>3/4 and epsilon>0 there exists C>0 such that for all real T>=2, all actual Riemann zeta zeros with Re(rho)>=sigma and |Im(rho)|<=T, counted with analytic multiplicity, number at most C*T^(3*(1-sigma)/(2*sigma)+epsilon). The proof combines separately proved Type I Dirichlet-polynomial large-values and critical-line twelfth-moment bounds. The companion moment is the literal integral on [0,3T] bounded by K*T^(2+epsilon), eventually in T. No analytic premise remains in the final endpoints. Compared with Tao\u2013Trudgian\u2013Yang Table 2 and Kerr Theorem 6, the leading exponent improves for 3/4 0 and all sufficiently large N we prove D(N) >= N^(0.7580758318008816 - epsilon). The exact exponent is 473797394875551/625000000000000. The reusable criterion starts from finite digit sets in pairwise coprime square bases, with ordered intervals and width-moment inequalities. A strict surplus in the component exponents yields the asymptotic bound through finite stopping words, rank rounding, the Chinese remainder theorem, and interpolation between geometric scales. Both the general criterion and its numerical application are formalized in Lean. All nine concrete components are proved: six prime chains, a 25-state binary recursion, and two complete composite witnesses, at the primes 5 and 43 (three restricted digits at each) and at 19 and 23 (three at 19, two at 23). Every required ordering, numerical moment, lifting, and asymptotic argument is checked, with no remaining finite-certificate hypotheses. This version replaces the exponent 0.75806746 of version 1 with new finite witnesses and more precise certificates. The repository includes self-contained PDF and LaTeX write-ups of the full result and a simpler companion construction attaining exponent 0.758001 that explains several core ideas, together with exact computational certificate programs. The simpler construction is not separately formalized or a selected submission statement.", "authors": [ { "name": "Eric Naslund" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B75", "05D05" ] }, "formalization": { "theorem_names": [ "SarkozySubmission.interval_moment_bound", "SarkozySubmission.improved_bound" ] }, "id": "PALOMAR-2026-09-19-000006", "path": "entries/PALOMAR-2026-09-19-000006-v2.json", "preservation": { "repositories": [ { "commit": "5a83f4174578c242ff3ea9373c2d4b9e406a9ffb", "fork_repository": "PalomarArchive/enaslund--sarkozy-lower-bound-0.758--6006cb546a0b", "source_repository": "enaslund/sarkozy-lower-bound-0.758" } ] }, "published_at": "2026-10-09T01:33:54Z", "source": { "commit": "5a83f4174578c242ff3ea9373c2d4b9e406a9ffb", "project_path": null, "repository": "enaslund/sarkozy-lower-bound-0.758" }, "status": "registered", "title": "An interval-moment criterion for square-difference-free integer sets", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "Lean development of the sharp Poincar\u00e9 inequality for the complex Ginibre log-gas, its two sum-of-squares deficit identities, equality cases, Gaussian Hermite analysis, and the accompanying analytic and stochastic results.", "authors": [ { "name": "Djalil Chafa\u00ef" } ], "classification": { "arxiv": [ "math.PR", "math.FA" ], "msc2020": [ "60E15", "60B20", "60J60", "32W05", "82B21", "82C31" ] }, "formalization": { "theorem_names": [ "PalomarGinibre.theoremOneOne" ] }, "id": "PALOMAR-2026-10-09-000001", "path": "entries/PALOMAR-2026-10-09-000001-v1.json", "preservation": { "repositories": [ { "commit": "fb58b4fd765f19a65c46cb82fb647fb0d94e28ca", "fork_repository": "PalomarArchive/djalilchafai--ginibre-poincare--2abb6cec2279", "source_repository": "djalilchafai/ginibre-poincare" } ] }, "published_at": "2026-10-09T00:48:41Z", "source": { "commit": "fb58b4fd765f19a65c46cb82fb647fb0d94e28ca", "project_path": null, "repository": "djalilchafai/ginibre-poincare" }, "status": "registered", "title": "Ginibre Poincar\u00e9 formalization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A formalization of sharp balanced-law robustness and stability refinements of the classical four-row permanent inequalities of Carlen-Lieb-Loss and Bristiel-Caputo. It proves the exact total-variation radius 1/24, optimal variance-deficit coefficient (1-24r)/9, complete nonnegative equality cases, sharp endpoint entropy constants, and adaptive tensorization. The explicit interior exponent is sufficient rather than optimal; at radius zero it is weaker than the classical optimal uniform exponent. Lean checks exhaustive support coverage and exact rational Gram identities.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.CO", "math.PR" ], "msc2020": [ "15A15", "60E15", "26D15" ] }, "formalization": { "theorem_names": [ "FourRowPrincipal.endpoint", "FourRowPrincipal.sharp_stability", "FourRowPrincipal.stability_optimal", "FourRowPrincipal.explicit_extremal_family", "FourRowPrincipal.radius_sharp", "FourRowPrincipal.endpoint_equality", "FourRowPrincipal.explicit_exponent", "FourRowPrincipal.larger_exponents", "FourRowPrincipal.subcritical_threshold", "FourRowPrincipal.entropy", "FourRowPrincipal.observation_contraction", "FourRowPrincipal.entropy_optimal", "FourRowPrincipal.observation_optimal", "FourRowPrincipal.adaptive_tensorization", "FourRowPrincipal.adaptive_probability", "FourRowPrincipal.adaptive_marginals" ] }, "id": "PALOMAR-2026-10-08-000007", "path": "entries/PALOMAR-2026-10-08-000007-v1.json", "preservation": { "repositories": [ { "commit": "861abe5342847d3f32985d55403547e1832ffac4", "fork_repository": "PalomarArchive/JD-Jones-ASES--four-row--c4855362551d", "source_repository": "JD-Jones-ASES/four-row" } ] }, "published_at": "2026-10-08T20:26:30Z", "source": { "commit": "861abe5342847d3f32985d55403547e1832ffac4", "project_path": null, "repository": "JD-Jones-ASES/four-row" }, "status": "registered", "title": "Sharp robustness and stability of the four-row permanent inequality", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib formalization of the stability of non-relativistic Coulomb matter in three dimensions. For normalized antisymmetric L2 electron states with q >= 1 spin states and static nuclei with charges between zero and Z, it proves a lower energy bound -C(q,Z)(N+M), uniformly in particle counts and nuclear positions. The main estimate is an additive inequality between nonnegative extended energies, valid even for infinite kinetic energy and nuclear collisions. A real quadratic-form theorem proves Coulomb finiteness and the lower bound for states with finite kinetic energy and distinct nuclear positions. The library also proves the kinetic Lieb-Thirring inequality with Rumin's explicit coefficient (9/35)(6 pi^2)^(2/3) q^(-2/3), Baxter's electrostatic inequality with its positive nearest-other-nucleus correction, and equality of the Fourier kinetic form with the classical Dirichlet integral on spin-valued Schwartz functions. The kinetic normalization is -Delta. The proof combines Pauli projections and Rumin's layer cake, a Voronoi boundary-measure proof of Baxter's inequality, nearest-nucleus screening, and Hardy's inequality. Three extensions prove the atomic bound N < 2Z+1 for weak ground states and strict binding, the unique continuous convex canonical zero-temperature energy density of neutral matter with finite-mass bosonic nuclei in Dirichlet balls, and the molecular Thomas-Fermi large-charge limit for arbitrary fixed spin and every fixed TF mass nu>0. Thomas-Fermi theory also has a unique relaxed minimizer, mass saturation at the total nuclear charge, and exact-mass attainment precisely up to neutrality.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Am\u00e9lie Loher" } ], "classification": { "arxiv": [ "math-ph", "math.AP" ], "msc2020": [ "81V70", "81Q10", "35Q40" ] }, "formalization": { "theorem_names": [ "LiebThirringAudit.stability_of_matter", "LiebThirringAudit.stability_of_matter_real", "LiebThirringAudit.kinetic_lieb_thirring", "LiebThirringAudit.baxter", "LiebThirringAudit.electron_count_lt_of_atomic_binding", "LiebThirringAudit.electron_count_lt_of_atomic_weak_ground_state", "LiebThirringAudit.exists_unique_thermodynamic_energy_density", "LiebThirringAudit.tendsto_electronicGroundStateEnergy_tf", "LiebThirringAudit.tendsto_groundStateEnergy_tf" ] }, "id": "PALOMAR-2026-10-08-000006", "path": "entries/PALOMAR-2026-10-08-000006-v1.json", "preservation": { "repositories": [ { "commit": "8bf4656f07f5aaa227b0b850d77b16976915bbd4", "fork_repository": "PalomarArchive/scottnarmstrong--LiebThirring--405b8eacc42e", "source_repository": "scottnarmstrong/LiebThirring" } ] }, "published_at": "2026-10-08T16:52:47Z", "source": { "commit": "8bf4656f07f5aaa227b0b850d77b16976915bbd4", "project_path": null, "repository": "scottnarmstrong/LiebThirring" }, "status": "registered", "title": "Stability of Matter in Lean: Lieb\u2013Thirring, Ionization, the Thermodynamic Limit and Thomas\u2013Fermi Theory", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "This development extends OpenAI's Lean proof of the matrix multiplication exponent bound 9/4 from the complex numbers to every field, including finite fields and fields of positive characteristic. It generalizes Fourier separation and interpolation over each field's algebraic closure, descends tensor powers through one fixed finite coefficient algebra, and connects the resulting exact-rank bounds to the original division-free arithmetic-program model. The compared statements include the arithmetic exponent upper bound, nonvacuity and lower-bound checks, an explicit positive-epsilon operation-count theorem, and exact coordinatewise coefficient decompositions. The original numerical bound and complex-field construction are OpenAI's work; this fork contributes the arbitrary-field extension and its formal verification.", "authors": [ { "name": "Sela Navot" } ], "classification": { "arxiv": [ "cs.CC", "cs.DS" ], "msc2020": [ "68Q17", "68W30", "15A69" ] }, "formalization": { "theorem_names": [ "ComparatorChecks.omega_bound", "ComparatorChecks.admissible_bddBelow", "ComparatorChecks.admissible_nonempty", "ComparatorChecks.omega_lower", "ComparatorChecks.epsilon_cost", "ComparatorChecks.exact_coefficients" ] }, "id": "PALOMAR-2026-10-08-000005", "path": "entries/PALOMAR-2026-10-08-000005-v1.json", "preservation": { "repositories": [ { "commit": "8375e73ed454e3128500d10d87480c5dd597ff03", "fork_repository": "PalomarArchive/selanavot--matrix-multiplication-all-fields--d4d6744d47b0", "source_repository": "selanavot/matrix-multiplication-all-fields" } ] }, "published_at": "2026-10-08T15:59:16Z", "source": { "commit": "8375e73ed454e3128500d10d87480c5dd597ff03", "project_path": null, "repository": "selanavot/matrix-multiplication-all-fields" }, "status": "registered", "title": "Matrix multiplication exponent at most 9/4 over arbitrary fields", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 formalization of Bezdek's annulus theorem and the quantitative area stability of planar plank covers based on the paper \"Area stability of plank covers\" by Egor Bakaev and Amir Yehudayoff (arXiv:2609.30409). The formalization proves that there exists an explicit positive radius r > 0 such that if finitely many planar planks cover the closed annulus r <= ||x|| <= 1, their total width is at least 2. The core theorem established is the quantitative overlap theorem, which bounds the excess overlap cost on the unit disk from below by a positive constant times the hole area divided by r.", "authors": [ { "name": "Arseniy Akopyan" } ], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52C10", "52A10", "52A38" ] }, "formalization": { "theorem_names": [ "bezdek_annulus" ] }, "id": "PALOMAR-2026-10-08-000004", "path": "entries/PALOMAR-2026-10-08-000004-v1.json", "preservation": { "repositories": [ { "commit": "b2fab1d9af2fe6ca3ef93a167fea9ba6a83b671a", "fork_repository": "PalomarArchive/egor-bakaev--planks-lean--19f7a71a0d1b", "source_repository": "egor-bakaev/planks-lean" } ] }, "published_at": "2026-10-08T14:57:03Z", "source": { "commit": "b2fab1d9af2fe6ca3ef93a167fea9ba6a83b671a", "project_path": null, "repository": "egor-bakaev/planks-lean" }, "status": "registered", "title": "Area stability of planar plank covers and Bezdek's annulus theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The moving sofa problem asks for the largest area of a planar shape that can be moved around the right-angled corner of a hallway of unit width. Jineon Baek proved (arXiv:2411.19826) that Gerver's sofa G, of area 2.2195..., is optimal. His proof bounds the area of every large moving sofa by a concave functional Q of a triple (K, B, D) of convex bodies built from a cap K of the sofa, and shows that Q is largest at Gerver's triple, where it equals |G|.\n\nThis entry proves one estimate for Q, the coercive certificate, and derives three results from it. On an enlarged domain of triples, on which the cap need not satisfy Baek's injectivity condition, Q(K, B, D) <= |G|, and the cap K lies within Euclidean Hausdorff distance 2 sec(phi) sqrt(|G| - Q(K, B, D)) of the horizontal translate of Gerver's cap with the same leftmost point, where phi = 0.039... is Gerver's angle. The proof writes |G| - Q as a sum of nonnegative terms, and the energy terms among them also bound that distance. The bound is attained: Gerver's triple lies in the enlarged domain, and Q equals |G| there.\n\nFrom the certificate: optimality, Baek's theorem, proved without his Theorem 1.1.1 and without the balance argument of his Chapters 4 to 6; uniqueness, every moving sofa with the area of G is mapped onto G by a rotation and a translation (a translation suffices); and stability, after a translation a moving sofa of area |G| - epsilon lies within Euclidean Hausdorff distance C sqrt(epsilon) of G, their symmetric difference has area at most C' sqrt(epsilon), its rotation angle misses a right angle by at most C'' epsilon, and the exponent 1/2 cannot be improved. The Challenge also states the correspondence between Baek's definitions and those of Google DeepMind's formal-conjectures, and formal-conjectures' statements, among them the uniqueness, which it lists as open; here they are proved through the certificate.\n\nVersions 1 to 4 of this entry registered the formalization of Baek's paper with a first proof of uniqueness that uses his Theorem 1.1.1; they state the same optimality, uniqueness and formal-conjectures theorems, proved through Baek's argument, and the repository keeps their Challenge and Solution in baek/. This version proves those theorems through the certificate and adds the certificate and the stability. A script in the repository, scripts/AuditCoerciveRoute.lean, run by the continuous integration, checks that its proofs reach neither Baek's Theorem 1.1.1, nor the results of his balance argument, nor the main theorem and the equality analysis of the first proof of uniqueness.", "authors": [ { "name": "The-Anh Vu-Le" } ], "classification": { "arxiv": [ "math.MG", "math.OC" ], "msc2020": [ "49Q10", "52A10", "52A38", "52A40", "68V20" ] }, "formalization": { "theorem_names": [ "Baek.gerver_params_exists", "Baek.gerver_params_unique", "Baek.gerver_sofa_area", "Baek.gerver_sofa_optimal", "Baek.gerver_sofa_unique", "Baek.gerver_sofa_stable", "Baek.gerver_sofa_angle_stable", "Baek.gerver_sofa_stability_exponent", "Bridge.isMovingSofa_iff", "Bridge.sofaConstant_eq", "Bridge.gerversSofa_eq", "FormalConjectures.MovingSofa.GerversSofa.AB\u03c6\u03b8Spec.existsUnique", "FormalConjectures.MovingSofa.isMovingSofa_gerversSofa", "FormalConjectures.MovingSofa.sofaConstant_eq_volume_gerversSofa", "FormalConjectures.MovingSofa.volume_eq_sofaConstant_iff_congruent_gerversSofa", "Certificate.coercive_certificate", "Certificate.gerver_triple" ] }, "id": "PALOMAR-2026-10-02-000008", "path": "entries/PALOMAR-2026-10-02-000008-v5.json", "preservation": { "repositories": [ { "commit": "1174c5c25c1c66b807e368446d6682e8d0276759", "fork_repository": "PalomarArchive/vltanh--lean4-moving-sofa--dfc797b6e6b6", "source_repository": "vltanh/lean4-moving-sofa" } ] }, "published_at": "2026-10-08T06:43:06Z", "source": { "commit": "1174c5c25c1c66b807e368446d6682e8d0276759", "project_path": null, "repository": "vltanh/lean4-moving-sofa" }, "status": "registered", "title": "Optimality, uniqueness and stability of Gerver's sofa from one coercive certificate, formalized in Lean 4", "trust": { "level": "high" }, "version": 5, "versions": 5 }, { "abstract": "A sorry-free Lean 4 formalization of improved running-time exponents for 3SUM, Exact Triangle, the (min,+)-product and all-pairs shortest paths (APSP), on the word RAM of the published formalization (anthropics/formal-math, 3sum-apsp) of the paper of Alman and Vassilevska Williams, \"Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs\" (arXiv:2610.06783). That paper gives deterministic algorithms for 3SUM in time O(n^1.9992) and for the (min,+)-product and APSP in time O(n^2.99942), through Exact Triangle in time O(n^(3 - 0.0017)). The project analyzes the same algorithm again and changes it in a few places: an exact count of the leaves of its recursion, a host that charges the solver at the true size of the middle part, and encodings restricted to the leaves that are read. The theorems submitted state, in exactly the sense of upstream's trusted file EndStatement.lean (programs of a word RAM with O(log n)-bit words, numbers of polynomial absolute value, every instance, every size), that 3SUM is solved in O(n^1.99896) steps and Exact Triangle in O(n^2.99791) steps, and, by carrying out the all-edges route that footnote 10 of the paper mentions, that the (min,+)-product and APSP are solved in O(n^2.99791) steps instead of O(n^2.99931) through the standard reductions. None of them has a hypothesis; every proof depends only on the axioms propext, Classical.choice and Quot.sound. The library also proves general theorems for all admissible parameters, a closed form for the optimal saving of the method at each ratio of parameters, and that the supremum of that saving over all ratios lies in (0.002095, 0.002096], so that no choice of parameters gives 3SUM an exponent below 1.998952 with this method. The pruned encoder and the two new hosts are written as programs of the machine and their running times are proved on it. All Lean, Python and LaTeX was written by the language model Claude Fable 5.1 (Anthropic) in Claude Code, under the direction of the maintainer, who wrote no Lean by hand; the move to Lean v4.35.0-rc2, with the port of the included part of upstream, was made by Claude Opus 5.5 (Anthropic) in the same way. An accompanying paper (papers/improved-exponents/) states every result with its Lean name.", "authors": [ { "name": "Jihoon Hyun" } ], "classification": { "arxiv": [ "cs.DS", "cs.LO" ], "msc2020": [ "68Q25", "68W40", "68V20" ] }, "formalization": { "theorem_names": [ "ImprovedExponents.threeSum_pruned", "ImprovedExponents.exactTriangle_pruned", "ImprovedExponents.minPlus_allEdges_pruned", "ImprovedExponents.apsp_allEdges_pruned" ] }, "id": "PALOMAR-2026-10-08-000003", "path": "entries/PALOMAR-2026-10-08-000003-v1.json", "preservation": { "repositories": [ { "commit": "0725f2477587d67212936777b8a264c9793ac7bb", "fork_repository": "PalomarArchive/qawbecrdtey--ImprovedExponents--333412a9c407", "source_repository": "qawbecrdtey/ImprovedExponents" } ] }, "published_at": "2026-10-08T05:35:09Z", "source": { "commit": "0725f2477587d67212936777b8a264c9793ac7bb", "project_path": null, "repository": "qawbecrdtey/ImprovedExponents" }, "status": "registered", "title": "Improved exponents for 3SUM and APSP from triangles in sparse lopsided graphs", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "There exists a locally finite carrier \u039b in the real line whose whole complex strongly tempered Meyer space has Hamel dimension \u2135\u2080. Both a tempered distribution and its distributional Fourier transform have local atomic formulas on \u039b, with coefficient families whose absolute values are summable after division by some polynomial weight (1 + |x|)^N. The same carrier supports the entire space; finite-dimensional exponent layers prove the rank upper bound and independent original parity modes prove the lower bound. The radius schedule is selected classically. This is an existence theorem for one carrier, not an effective construction or a classification of all carriers.", "authors": [ { "name": "Jamie Martin" } ], "classification": { "arxiv": [ "math.CA", "math.FA" ], "msc2020": [ "42A38", "46F12", "52C23" ] }, "formalization": { "theorem_names": [ "CountablyInfiniteStrongCrystallineMeasures.strongCardinalClaim" ] }, "id": "PALOMAR-2026-10-08-000002", "path": "entries/PALOMAR-2026-10-08-000002-v1.json", "preservation": { "repositories": [ { "commit": "4d3163a0ce1dd93fa48b7d4677fe78db15c0af9c", "fork_repository": "PalomarArchive/balancedscorpion--countably-infinite-dimension-strong-meyer-spaces--8319e9fa53b3", "source_repository": "balancedscorpion/countably-infinite-dimension-strong-meyer-spaces" } ] }, "published_at": "2026-10-08T03:47:50Z", "source": { "commit": "4d3163a0ce1dd93fa48b7d4677fe78db15c0af9c", "project_path": null, "repository": "balancedscorpion/countably-infinite-dimension-strong-meyer-spaces" }, "status": "registered", "title": "A strongly tempered Meyer space of countably infinite Hamel dimension", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 proof, checked against Mathlib, of a sharp quantitative propagation of chaos estimate for mean field diffusions dX_i = (a(X_i) + N^{-1} sum_j K(X_i, X_j)) dt + sqrt 2 dW_i with a globally Lipschitz one-body drift a and a bounded, globally Lipschitz pairwise interaction K. If every k-particle marginal of the (possibly correlated and singular) exchangeable initial law lies within squared quadratic Wasserstein distance C_0 (k/N)^2 of the corresponding power of the McKean-Vlasov initial law, then the same holds on every finite time interval [0, T] with an explicit constant C_T that depends neither on N nor on the dimension. No entropy bound, transport inequality or independence of the initial particles is assumed.", "authors": [ { "name": "Yuhao Lei" } ], "classification": { "arxiv": [ "math.PR" ], "msc2020": [ "60K35", "82C22", "60H10", "35Q84", "49Q22" ] }, "formalization": { "theorem_names": [ "SharpChaos.sharp_propagation_of_chaos" ] }, "id": "PALOMAR-2026-10-08-000001", "path": "entries/PALOMAR-2026-10-08-000001-v1.json", "preservation": { "repositories": [ { "commit": "84dbe881384ad2f0358857b025ef6fd824ebc9a1", "fork_repository": "PalomarArchive/yuhaolei-ctrl--Sharp-W2-to-W2-Propagation-of-Chaos--916e37ae9fb6", "source_repository": "yuhaolei-ctrl/Sharp-W2-to-W2-Propagation-of-Chaos" } ] }, "published_at": "2026-10-08T01:43:28Z", "source": { "commit": "84dbe881384ad2f0358857b025ef6fd824ebc9a1", "project_path": null, "repository": "yuhaolei-ctrl/Sharp-W2-to-W2-Propagation-of-Chaos" }, "status": "registered", "title": "Sharp Wasserstein propagation of chaos from correlated initial data", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 development around the tensor criterion of the classification of finite Euclidean Ramsey configurations (openai/math, family 172). With the criterion's definitions restated character for character from that project's challenge file: every set of six distinct points on a circle satisfies the criterion, by an explicit certificate (the product of the two bilinear forms through the first three and the last three points of a homogeneous parametrisation of the circle, scaled and conjugated by the parametrisation); the sharp dimension theorem: every spherical set of at most m + 4 distinct points, m the dimension of its affine span, satisfies the criterion in its own space and coordinate field (either the augmented vectors admit real isolating quadrics and the sphere lifts by the adjugate, or a dependence among them is supported on six concyclic points while every other point adds a dimension, and the six-point certificate is carried into the ambient space, symmetrised, adjusted to the common sphere, extended point by point and descended from R tensor R to F tensor F), so every spherical set of at most six points satisfies it in any dimension with any affine span; every spherical set with algebraic coordinates satisfies it, of any size, by the separability idempotent of F tensor F; the criterion always implies sphericity, so for algebraic coordinates it is exactly sphericity. Given the sufficiency half of the classification as an explicit hypothesis (quantified as the source states its Theorem 1.1), every spherical set of at most six points, every spherical set of at most m + 4 points and every spanning spherical set with algebraic coordinates is Ramsey; given both halves, a spanning set with algebraic coordinates is Ramsey if and only if it is spherical. The source proves the Ramsey property for at most five circle points; six is the first case its method of independent quadratic rows cannot reach. With the seven-point non-Ramsey circle set of A cyclic non-Ramsey heptagon (arXiv:2609.23327), the least size of a non-Ramsey spherical set is seven, given the classification, and, for m at least 2, m + 5 the least size of one with affine span of dimension m, since the heptagon with m - 2 further sphere points fails the criterion and is not Ramsey; the research note in note/ states this, verifies by an exact derivation identity that the heptagon fails the criterion, and proves the sharpness on paper (nothing in Lean says that a set fails the criterion). The classification is not proved here. The restated definitions and a few adapted lemmas derive from Apache-2.0 work of OpenAI (NOTICE, LICENSES/Apache-2.0.txt); the rest is MIT.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05D10", "52C10" ] }, "formalization": { "theorem_names": [ "SixConcyclic.six_concyclic_fieldCriterion", "SixConcyclic.at_most_six_concyclic_fieldCriterion", "SixConcyclic.at_most_six_spherical_spanning_fieldCriterion", "SixConcyclic.at_most_five_spherical_fieldCriterion", "SixConcyclic.algebraic_spherical_fieldCriterion", "SixConcyclic.fieldCriterion_spherical", "SixConcyclic.algebraic_fieldCriterion_iff_spherical", "SixConcyclic.at_most_six_spherical_ramsey_of_sufficiency", "SixConcyclic.at_most_six_circle_points_ramsey_of_sufficiency", "SixConcyclic.algebraic_spherical_ramsey_of_sufficiency", "SixConcyclic.algebraic_ramsey_iff_spherical_of_classification", "SixConcyclic.spherical_fieldCriterion_of_card_le_finrank_add_four", "SixConcyclic.at_most_six_spherical_fieldCriterion", "SixConcyclic.spherical_ramsey_of_sufficiency_of_card_le_finrank_add_four" ] }, "id": "PALOMAR-2026-10-07-000011", "path": "entries/PALOMAR-2026-10-07-000011-v1.json", "preservation": { "repositories": [ { "commit": "6bf3f4019bf0ced69f609abe3d517f9d7c31e55e", "fork_repository": "PalomarArchive/JD-Jones-ASES--six-concyclic-lean--c1d655f6e8fd", "source_repository": "JD-Jones-ASES/six-concyclic-lean" } ] }, "published_at": "2026-10-07T15:56:05Z", "source": { "commit": "6bf3f4019bf0ced69f609abe3d517f9d7c31e55e", "project_path": null, "repository": "JD-Jones-ASES/six-concyclic-lean" }, "status": "registered", "title": "Six concyclic points satisfy the Euclidean Ramsey tensor criterion, as does every spherical set of at most m + 4 points of affine dimension m and every algebraic spherical set; given the classification's sufficiency half, these sets are Ramsey (the algebraic ones when spanning)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of classical factorial-square constructions, exclusion of prime endpoints, existence of minimum representation cardinalities for composite endpoints, and the cutoff D_k = empty for k greater than six. The density question from Erd\u0151s problem 374 is defined but not proved. No mathematical novelty or solution of the density problem is claimed.", "authors": [ { "name": "alexander" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11D57", "11A41" ] }, "formalization": { "theorem_names": [ "Erdos374.rep_two", "Erdos374.square_inD_two", "Erdos374.adjacent_identity", "Erdos374.four_factor_identity", "Erdos374.six_factor_identity", "Erdos374.minimum_unique", "Erdos374.prime_no_rep", "Erdos374.product_separated_rep_six", "Erdos374.product_adjacent_rep_four", "Erdos374.product_equal_rep_two", "Erdos374.product_has_rep_le_six", "Erdos374.exists_minimum_le", "Erdos374.product_minimum_le_six", "Erdos374.rep_527_six", "Erdos374.rep_card_le", "Erdos374.composite_minimum_le_six", "Erdos374.D_empty_above_six", "Erdos374.countD_le" ] }, "id": "PALOMAR-2026-10-07-000010", "path": "entries/PALOMAR-2026-10-07-000010-v1.json", "preservation": { "repositories": [ { "commit": "40857297956c031d92b1bc30b007a46575c60c5d", "fork_repository": "PalomarArchive/AItoBit--erdos374-palomar--3718c5153338", "source_repository": "AItoBit/erdos374-palomar" } ] }, "published_at": "2026-10-07T15:34:20Z", "source": { "commit": "40857297956c031d92b1bc30b007a46575c60c5d", "project_path": null, "repository": "AItoBit/erdos374-palomar" }, "status": "registered", "title": "Erd\u0151s 374: classical square products of factorials", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean formalization of N\u03b6(\u03c3,T) \u226a\u03c3,\u03b5 T^(2(1\u2212\u03c3)+\u03b5), hence A(\u03c3) \u2264 2, for fixed 3/4 < \u03c3 < 1 and \u03b5 > 0; zeros with Re(\u03c1) \u2265 \u03c3 and 0 < Im(\u03c1) \u2264 T are counted with multiplicity.", "authors": [ { "name": "Mark Lewko" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26" ] }, "formalization": { "theorem_names": [ "DensityThreeQuarters.density_bound" ] }, "id": "PALOMAR-2026-10-07-000009", "path": "entries/PALOMAR-2026-10-07-000009-v1.json", "preservation": { "repositories": [ { "commit": "5dd0dd391c117ef4d8fd04a6b16b4b1ad5735d88", "fork_repository": "PalomarArchive/mrklwk--zeta-zero-density-lean--4326f648d928", "source_repository": "mrklwk/zeta-zero-density-lean" } ] }, "published_at": "2026-10-07T15:00:33Z", "source": { "commit": "5dd0dd391c117ef4d8fd04a6b16b4b1ad5735d88", "project_path": "lean", "repository": "mrklwk/zeta-zero-density-lean" }, "status": "registered", "title": "The density hypothesis for \u03c3 > 3/4", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Properly nested nonempty compact connected subsets A and B of a compact metric space are the endpoints of a continuous embedded interval of nonempty compact connected subsets, strictly increasing under inclusion. The ambient space need not be connected.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54F15", "54B20" ] }, "formalization": { "theorem_names": [ "OrderArcs.exists_order_arc" ] }, "id": "PALOMAR-2026-10-07-000008", "path": "entries/PALOMAR-2026-10-07-000008-v1.json", "preservation": { "repositories": [ { "commit": "8b802e2f62a31b08276a67021cd40f6941c5c256", "fork_repository": "PalomarArchive/Arthur742Ramos--Math-Formalization--4c88ddbef95a", "source_repository": "Arthur742Ramos/Math-Formalization" } ] }, "published_at": "2026-10-07T13:39:49Z", "source": { "commit": "8b802e2f62a31b08276a67021cd40f6941c5c256", "project_path": "topology/order-arcs", "repository": "Arthur742Ramos/Math-Formalization" }, "status": "registered", "title": "Order Arcs Between Subcontinua", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The matrix multiplication exponent \u03c9 is the least exponent such that n\u00d7n matrices can be multiplied in n^(\u03c9+o(1)) operations. This repository proves in Lean 4 / Mathlib that \u03c9 < 2.3710449 over every commutative ring, improving the previous bound \u03c9 < 2.371177 (Dupont et al. 2026). \u03c9 is defined via the exact tensor rank of n\u00d7n matrix multiplication. The new ingredient is shared extraction across recursion levels of the asymmetric laser method. The formalization covers the full argument, from the Coppersmith\u2013Winograd degeneration to an exact rational certificate checked in the Lean kernel. The construction and proof were produced largely by AI systems under the authors' direction.", "authors": [ { "name": "Oliver Proudfoot" }, { "name": "Andrew Perrault" } ], "classification": { "arxiv": [ "cs.CC", "math.CO" ], "msc2020": [ "68Q17", "68Q25", "15A69", "68V20" ] }, "formalization": { "theorem_names": [ "MatrixBounds.Numeric.SuppliedCertifiedPipeline.exponent_lt" ] }, "id": "PALOMAR-2026-10-07-000007", "path": "entries/PALOMAR-2026-10-07-000007-v1.json", "preservation": { "repositories": [ { "commit": "93492a4dd1ee90fd98cd7b339af77bb8bbc8be55", "fork_repository": "PalomarArchive/olliepro--LeanMMBound--bc7df2f4d6b7", "source_repository": "olliepro/LeanMMBound" } ] }, "published_at": "2026-10-07T13:29:49Z", "source": { "commit": "93492a4dd1ee90fd98cd7b339af77bb8bbc8be55", "project_path": null, "repository": "olliepro/LeanMMBound" }, "status": "registered", "title": "LeanMMBound: the matrix multiplication exponent is below 2.3710449", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We give an explicit compact tile, called CLARK (Chair with Local Asymmetric Registration Keys), in five-dimensional Euclidean space. It modifies a chair of 31 unit five-cubes with 256 rational pyramidal keys. Copies may be translated, rotated or reflected. The selected theorem proves tiling existence, no nonzero translation period in any physical tiling, and a finite full Euclidean symmetry group of cardinality at most 3840 for every such tiling. No registration, matching-rule or hierarchy assumption is imposed. A trivial symmetry group is not asserted.", "authors": [ { "name": "Conor Grogan" } ], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [] }, "formalization": { "theorem_names": [ "PalomarMonotiles.T5_strongAperiodicity" ] }, "id": "PALOMAR-2026-10-07-000006", "path": "entries/PALOMAR-2026-10-07-000006-v1.json", "preservation": { "repositories": [ { "commit": "5f5ceedee770b76c21e8f1105e5e4e3b84b8d676", "fork_repository": "PalomarArchive/jconorgrogan--palomar-chair-verification--3baebe3c8c75", "source_repository": "jconorgrogan/palomar-chair-verification" } ] }, "published_at": "2026-10-07T12:38:06Z", "source": { "commit": "5f5ceedee770b76c21e8f1105e5e4e3b84b8d676", "project_path": null, "repository": "jconorgrogan/palomar-chair-verification" }, "status": "registered", "title": "The CLARK Tile: An Aperiodic Monotile in Five Dimensions", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization, against Mathlib, of the theorem that the first Betti number (minimal number of generators of the toric ideal) of a numerical semigroup minimally generated by four elements n0 < n1 < n2 < n3 is at most 2W, where W = (n3 - n0) / gcd(n1 - n0, n2 - n0, n3 - n0); in particular at most twice the width n3 - n0. This replaces the conjectured quadratic bound (w+1 choose 2) of Herzog\u2013Stamate (2014) and Caviglia\u2013Moscariello\u2013Sammartano (2024) in embedding dimension four; the bound 2w is attained for every width w >= 4 (an informal companion result; sharpness is not formalized).", "authors": [ { "name": "Ben Keene" } ], "classification": { "arxiv": [ "math.AC", "math.CO" ], "msc2020": [ "13D02", "20M14", "13F65", "05E40" ] }, "formalization": { "theorem_names": [ "SemigroupBetti.beta1_le_two_mul_normalizedWidth", "SemigroupBetti.beta1_le_two_mul_width" ] }, "id": "PALOMAR-2026-10-07-000005", "path": "entries/PALOMAR-2026-10-07-000005-v1.json", "preservation": { "repositories": [ { "commit": "c94da4f5cb07e2ccf62c9ae18887aedd73e95bf2", "fork_repository": "PalomarArchive/benkeene--semigroup-betti--1ce9f720e04c", "source_repository": "BenKeene/semigroup-betti" } ] }, "published_at": "2026-10-07T07:06:12Z", "source": { "commit": "c94da4f5cb07e2ccf62c9ae18887aedd73e95bf2", "project_path": null, "repository": "BenKeene/semigroup-betti" }, "status": "registered", "title": "Linear upper bounds for the first Betti number of four-generated numerical semigroups", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We formalize four explicit Szemer\u00e9di\u2013Trotter incidence bounds for m distinct points and n distinct affine lines in a field plane. In characteristic zero the bound is 3(mn)^(2/3) + m + n. In positive characteristic p, an arbitrary field admits the additional term 2mn/p. This term is absent when mn \u2264 p\u00b3. Over the prime field ZMod p the additional term improves to mn/p without a size restriction. The development adapts the author's polynomial-method argument, uses generic coordinates for the improved linear term, and proves a finite-plane counting estimate for the prime-field refinement.", "authors": [ { "name": "Mark Lewko" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "52C10", "05B25" ] }, "formalization": { "theorem_names": [ "IncidenceBounds.arbitrary_charZero", "IncidenceBounds.arbitrary_charP", "IncidenceBounds.subcritical", "IncidenceBounds.prime_field" ] }, "id": "PALOMAR-2026-10-07-000004", "path": "entries/PALOMAR-2026-10-07-000004-v1.json", "preservation": { "repositories": [ { "commit": "844228bc6c042e59373888d1b69f4e215d1c00e4", "fork_repository": "PalomarArchive/mrklwk--szemeredi-trotter-lean--f1954e3e06af", "source_repository": "mrklwk/szemeredi-trotter-lean" } ] }, "published_at": "2026-10-07T06:46:45Z", "source": { "commit": "844228bc6c042e59373888d1b69f4e215d1c00e4", "project_path": null, "repository": "mrklwk/szemeredi-trotter-lean" }, "status": "registered", "title": "Explicit Szemer\u00e9di\u2013Trotter bounds over arbitrary fields", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Todorov\u2013Valov Theorem 3.3: for two disjoint closed sets with nonempty interiors in a metric indecomposable continuum, one open cover obstructs every separator map onto every nonempty compact connected Hausdorff target. The classical composant properties remain as foundations.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54F15", "54D05", "54D30" ] }, "formalization": { "theorem_names": [ "Composants.mem_composant_self", "Composants.dense_composant", "Composants.isConnected_composant", "Composants.composants_eq_or_disjoint", "Composants.iUnion_composant", "Composants.composant_countable_union", "Composants.isMeagre_composant", "Composants.uncountably_many_composants", "Composants.alexandroff_continua" ] }, "id": "PALOMAR-2026-10-07-000003", "path": "entries/PALOMAR-2026-10-07-000003-v1.json", "preservation": { "repositories": [ { "commit": "53be25d26551d18b006d3a625fc5b145cc73cebd", "fork_repository": "PalomarArchive/Arthur742Ramos--Math-Formalization--4c88ddbef95a", "source_repository": "Arthur742Ramos/Math-Formalization" } ] }, "published_at": "2026-10-07T02:31:35Z", "source": { "commit": "53be25d26551d18b006d3a625fc5b145cc73cebd", "project_path": "topology/composants", "repository": "Arthur742Ramos/Math-Formalization" }, "status": "registered", "title": "Alexandroff Property of Metric Indecomposable Continua", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A candidate Lean formalization of fifteen selected statements from Yongxi Lin's table of weak type (1,1) upper bounds for singular integrals and maximal operators. Bounds include full-vector complex-input Riesz, complex Beurling, full and traceless Frobenius Hessians, both projections, centred intervals and balls, and all six Poisson and heat maximal bounds. The strict planar square bound below 3.616 is excluded from the submitted scope at the human author's request.", "authors": [ { "name": "Yongxi Lin" } ], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "42B25", "42B20", "31B15" ] }, "formalization": { "theorem_names": [ "PartialBalayage.interval_weakTypeConstant_le_two", "PartialBalayage.ball_weakTypeConstant_two_le_exp", "PartialBalayage.ball_weakTypeConstant_le_rpow", "PartialBalayage.heat_weakTypeConstant_le_formula", "PartialBalayage.poisson_weakTypeConstant_le_formula", "PartialBalayage.heat_weakTypeConstant_two_le_exact", "PartialBalayage.poisson_weakTypeConstant_two_le_exact", "PartialBalayage.heat_weakTypeConstant_one_le_exact", "PartialBalayage.poisson_weakTypeConstant_one_le_exact", "PartialBalayage.hessian_weakTypeConstant_le_formula", "PartialBalayage.hessian_weakTypeConstant_two_le_exact", "PartialBalayage.beurling_weakTypeConstant_le_two", "PartialBalayage.tracelessHessian_weakTypeConstant_le", "PartialBalayage.projections_weakTypeConstants_le_exact", "PartialBalayage.riesz_weakTypeConstant_le_two" ] }, "id": "PALOMAR-2026-10-07-000002", "path": "entries/PALOMAR-2026-10-07-000002-v1.json", "preservation": { "repositories": [ { "commit": "f0777f541fd38a1be42a87e6c69806c8ac488015", "fork_repository": "PalomarArchive/CoolRmal--partial-balayage-lean--d01ff6ab4734", "source_repository": "CoolRmal/partial-balayage-lean" } ] }, "published_at": "2026-10-07T01:45:50Z", "source": { "commit": "f0777f541fd38a1be42a87e6c69806c8ac488015", "project_path": null, "repository": "CoolRmal/partial-balayage-lean" }, "status": "registered", "title": "Two partial balayage principles for weak-type estimates", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "There is a counterexample to Vaught's conjecture for L_{omega_1,omega}. This formalization is based on an unpublished draft by Robin Knight. It constructs a sentence in a countable relational language with exactly aleph-one isomorphism classes of countable models and no nonempty perfect set of pairwise nonisomorphic model codes. This is not a counterexample to the first-order Vaught conjecture.", "authors": [ { "name": "Nathanael Ackerman" }, { "name": "Cameron Freer" }, { "name": "Robin Knight" } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03C15", "03C75", "03E15" ] }, "formalization": { "theorem_names": [ "PalomarChallenge.independent_challenge" ] }, "id": "PALOMAR-2026-10-07-000001", "path": "entries/PALOMAR-2026-10-07-000001-v1.json", "preservation": { "repositories": [ { "commit": "032ccb7a25b0ff6227128aa0aeba549c5901ea9a", "fork_repository": "PalomarArchive/cameronfreer--vaught-conjecture-palomar--e07ed0df01d9", "source_repository": "cameronfreer/vaught-conjecture-palomar" } ] }, "published_at": "2026-10-07T01:31:06Z", "source": { "commit": "032ccb7a25b0ff6227128aa0aeba549c5901ea9a", "project_path": null, "repository": "cameronfreer/vaught-conjecture-palomar" }, "status": "registered", "title": "A counterexample to Vaught's conjecture for infinitary logic", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Block-diagonal matrix factors interleaved with fixed coordinate permutations define low-parametric matrix families. This formalization proves orthogonal-factor sufficiency for two and three real layers, its complex unitary analogue, and explicit four-layer counterexamples over both fields. It establishes an exact two-factor cell-rank criterion and real and unitary cosine-sine exhaustion for the (p,2,p) shuffle family. Two-layer images with complete within-layer block sharing, arbitrary overlap and arbitrary fixed carriers have a smooth atlas in the inherited matrix topology, an exact tangent image, a smooth Frobenius projector, a whole-tangent-bundle Cayley retraction and smooth intrinsic gradient sections. Shared orthogonal block exponential curves have normal ambient acceleration and constant Frobenius speed under the no-off-diagonal-overlap condition. Three-layer (2,q,2) families have genuine topological singularities for every real q >= 3 and for odd unitary q >= 3. The full supported proof library is included; experimental special-unitary drafts and the explicitly listed remaining problems are not claimed.", "authors": [ { "name": "Ali Aliev" } ], "classification": { "arxiv": [ "math.DG", "math.NA", "math.RA" ], "msc2020": [ "15A23", "15B10", "53B21", "57N35" ] }, "formalization": { "theorem_names": [ "GSPalomar.two_factor", "GSPalomar.three_factor", "GSPalomar.four_factor_counterexample", "GSPalomar.unitary_two_factor", "GSPalomar.unitary_three_factor", "GSPalomar.unitary_four_factor_counterexample", "GSPalomar.real_cs_exhaustion", "GSPalomar.unitary_cs_exhaustion", "GSPalomar.rank_one_cone_singularity", "GSPalomar.real_three_factor_singularity", "GSPalomar.real_singular_point", "GSPalomar.unitary_singular_point", "GSPalomar.orthogonal_shared_geometry", "GSPalomar.unitary_shared_geometry", "GSPalomar.two_factor_rank_criterion", "GSPalomar.orthogonal_shared_block_geodesic" ] }, "id": "PALOMAR-2026-10-06-000011", "path": "entries/PALOMAR-2026-10-06-000011-v1.json", "preservation": { "repositories": [ { "commit": "1270ba6cf3050f53a4beb98f0d65ab7459d4494b", "fork_repository": "PalomarArchive/alialiev78--gs_Lean_formalization--4e789898d047", "source_repository": "alialiev78/gs_Lean_formalization" } ] }, "published_at": "2026-10-06T17:21:05Z", "source": { "commit": "1270ba6cf3050f53a4beb98f0d65ab7459d4494b", "project_path": null, "repository": "alialiev78/gs_Lean_formalization" }, "status": "registered", "title": "Structured orthogonal and unitary matrices: factorization, geometry and singularities", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Every continuous surjection between compact Hausdorff spaces factors through a compact Hausdorff quotient that collapses exactly the connected components of each original fiber. The first map has connected fibers and the second has totally disconnected fibers. Any two such factorizations have a unique commuting homeomorphism. Empty spaces are permitted.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54C10", "54D05", "54D30", "54F05" ] }, "formalization": { "theorem_names": [ "MonotoneLight.isClosed_fiberRel", "MonotoneLight.exists_monotone_light_factorization", "MonotoneLight.unique_monotone_light_factorization" ] }, "id": "PALOMAR-2026-10-06-000010", "path": "entries/PALOMAR-2026-10-06-000010-v1.json", "preservation": { "repositories": [ { "commit": "759906c53ae14a660ab0e473dcf075d859c5a5a9", "fork_repository": "PalomarArchive/Arthur742Ramos--monotone-light-factorization-lean--a4e3fc9d4555", "source_repository": "Arthur742Ramos/monotone-light-factorization-lean" } ] }, "published_at": "2026-10-06T11:07:21Z", "source": { "commit": "759906c53ae14a660ab0e473dcf075d859c5a5a9", "project_path": null, "repository": "Arthur742Ramos/monotone-light-factorization-lean" }, "status": "registered", "title": "Monotone-Light Factorization of Compact Hausdorff Surjections", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 formalization of Catalan's conjecture, proved by Preda Mih\u0103ilescu: 8 and 9 are the only consecutive positive proper perfect powers. The formal proof follows the cyclotomic proof in Yuri Bilu's expositions. Cassels' relations, Stickelberger annihilation, and height and counting estimates exclude the case in which one exponent is congruent to 1 modulo the other. Thaine's theorem on circular units, together with a Runge-type approximation argument, settles the remaining case. The class-field-theoretic input comes from a fixed, vendored subset of the ClassFieldTheory library. The registered results are four theorems: Catalan's conjecture over the natural numbers, in the form stated in Google DeepMind's Formal Conjectures; the same equation for integer bases greater than one; the classification of solutions of x^p \u2212 y^q = 1 with nonzero integer bases of either sign; and Mih\u0103ilescu's theorem that x^p = y^q + 1 has no solution in nonzero integers when p and q are odd primes. All four depend only on the axioms propext, Classical.choice and Quot.sound.", "authors": [ { "name": "Yao Xu" } ], "classification": { "arxiv": [ "math.NT", "cs.LO" ], "msc2020": [ "11D61", "11R18", "11R29", "11R37", "68V20" ] }, "formalization": { "theorem_names": [ "PalomarCatalan.catalans_conjecture", "PalomarCatalan.catalan_int", "PalomarCatalan.catalan_int_signed", "PalomarCatalan.mihailescu_odd_primes" ] }, "id": "PALOMAR-2026-10-06-000009", "path": "entries/PALOMAR-2026-10-06-000009-v1.json", "preservation": { "repositories": [ { "commit": "597469e8eba93909efc94892e6f7df5ea4951fe3", "fork_repository": "PalomarArchive/cadamcat--catalan-lean4--fdc727faf3c1", "source_repository": "cadamcat/catalan-lean4" } ] }, "published_at": "2026-10-06T08:38:52Z", "source": { "commit": "597469e8eba93909efc94892e6f7df5ea4951fe3", "project_path": null, "repository": "cadamcat/catalan-lean4" }, "status": "registered", "title": "A Lean 4 formalization of Catalan's conjecture (Mih\u0103ilescu's theorem)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For a Borel subset E of the Euclidean plane with Hausdorff dimension d in (1, 5/4], a strict upper bound on its packing dimension by the explicit piecewise curve B_H(d) ensures a point y in E whose pinned distance set has positive one-dimensional Lebesgue measure. The three branches are 2d-1, 1+r_2(d-1), and 1/(3-2d), with the transitions and algebraic root defined independently in Challenge.lean. This is a sufficient criterion with an additional packing-dimension hypothesis, not a resolution of the general planar Falconer conjecture.", "authors": [ { "name": "Yongxi Lin" } ], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "28A80", "42B10" ] }, "formalization": { "theorem_names": [ "FalconerPacking.exists_pin_volume_pinnedDistances_pos" ] }, "id": "PALOMAR-2026-10-06-000008", "path": "entries/PALOMAR-2026-10-06-000008-v1.json", "preservation": { "repositories": [ { "commit": "70140ccedfb6de71342299523a21b1550df69ab9", "fork_repository": "PalomarArchive/CoolRmal--falconer-packing--054762f09ea9", "source_repository": "CoolRmal/falconer-packing" } ] }, "published_at": "2026-10-06T08:09:27Z", "source": { "commit": "70140ccedfb6de71342299523a21b1550df69ab9", "project_path": null, "repository": "CoolRmal/falconer-packing" }, "status": "registered", "title": "A Hausdorff\u2013packing criterion for self-pinned distance sets", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization, with Mathlib, of the main results of the paper \"The Schur degree of block sums: L(4) = 16 and L(5) \u2265 49\" (paper/schur-degree-block-sums.md in this repository). Eliahou and Revuelta (Discrete Math. 344 (2021) 112332) defined a number L(n) through the Schur degree of the set of block sums of a sequence of positive integers, proved S(n\u22121) + 1 \u2264 L(n) \u2264 R_{n\u22121}(3) \u2212 1 and S(n) \u2264 n\u00b7L(n), and conjectured L(n) = S(n\u22121) + 1 (their Conjecture 5.6), that is, L(4) = 14 and L(5) = 45. The formalization proves L(4) = 16 (erL_four), 49 \u2264 L(5) \u2264 65 (erL_five_bounds), the lift that turns a cover of the nonzero elements of ZMod m\u2081 \u00d7 ZMod m\u2082 by q sets that are sumfree in the group into a sequence whose block sums have Schur degree at most q (lift_lemma), its corollary m\u2081m\u2082 \u2264 L(n) (le_erL_of_groupPartition), and Theorem 4.1 and the upper bound of Proposition 5.3 of Eliahou and Revuelta, with the pigeonhole bound ramseyBound k in place of R_k(3) (le_sdeg_blockSums, erL_le). So Conjecture 5.6 is false for n = 4 and n = 5. The Lean upper bound for L(5) is 65, not the bound 61 of the paper: the formalization uses the pigeonhole bound R\u2084(3) \u2264 66 and not the bound R\u2084(3) \u2264 62 of Fettes, Kramer and Radziszowski. The finite checks use the Lean kernel (decide), with no native code. The sequence W, the lift, the proofs, the programs and the Lean formalization were produced with the AI system Claude (Anthropic; model Claude Opus 5.5) under the direction of the author.", "authors": [ { "name": "Adam McKenna" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D10", "11B75", "05C55" ] }, "formalization": { "theorem_names": [ "ClassicalSchurClaims.erL_four", "ClassicalSchurClaims.erL_five_bounds", "ClassicalSchurClaims.le_erL_of_groupPartition", "ClassicalSchurClaims.lift_lemma", "ClassicalSchurClaims.le_sdeg_blockSums", "ClassicalSchurClaims.erL_le" ] }, "id": "PALOMAR-2026-10-06-000007", "path": "entries/PALOMAR-2026-10-06-000007-v1.json", "preservation": { "repositories": [ { "commit": "933000c135141347835300ca13b3a15ff050d03a", "fork_repository": "PalomarArchive/mysticflounder--schur-degree-block-sums--b45c786b5e30", "source_repository": "mysticflounder/schur-degree-block-sums" } ] }, "published_at": "2026-10-06T05:54:20Z", "source": { "commit": "933000c135141347835300ca13b3a15ff050d03a", "project_path": "lean", "repository": "mysticflounder/schur-degree-block-sums" }, "status": "registered", "title": "The Schur degree of block sums: L(4) = 16 and L(5) \u2265 49", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Hausdorff nullity for all finite products of squared successive logarithms and an upper parabolic box dimension bound of 25/23 for interior singular sets of unforced three-dimensional suitable weak Navier\u2013Stokes solutions.", "authors": [ { "name": "Yongxi (Aaron) Lin" } ], "classification": { "arxiv": [ "math.AP", "math.MG" ], "msc2020": [ "35Q30", "28A78", "28A80" ] }, "formalization": { "theorem_names": [ "FluidSingularSets.singularSet_iteratedLogHausdorffMeasure_zero", "FluidSingularSets.singularSet_logSquaredHausdorffMeasure_zero", "FluidSingularSets.singularSet_upperBoxDimension_le" ] }, "id": "PALOMAR-2026-10-06-000006", "path": "entries/PALOMAR-2026-10-06-000006-v1.json", "preservation": { "repositories": [ { "commit": "dd3ae61ce421a634818f52a6e11a8b78dd1bfc97", "fork_repository": "PalomarArchive/CoolRmal--FluidSingularSets--a936c36bd90d", "source_repository": "CoolRmal/FluidSingularSets" } ] }, "published_at": "2026-10-06T04:18:16Z", "source": { "commit": "dd3ae61ce421a634818f52a6e11a8b78dd1bfc97", "project_path": null, "repository": "CoolRmal/FluidSingularSets" }, "status": "registered", "title": "FluidSingularSets", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Every nontrivial compact connected subset of a Hausdorff space contains two distinct points whose individual removals leave connected sets. A compact connected subset containing all non-cut points of a compact connected set equals that set.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54D05", "54D30", "54F15" ] }, "formalization": { "theorem_names": [ "NonCutPoints.exists_two_noncut_points", "NonCutPoints.eq_of_contains_noncut_points" ] }, "id": "PALOMAR-2026-10-06-000005", "path": "entries/PALOMAR-2026-10-06-000005-v1.json", "preservation": { "repositories": [ { "commit": "b1d0ed0338d098a4c84622828390cc84027df8dc", "fork_repository": "PalomarArchive/Arthur742Ramos--non-cut-points-lean--51757670d602", "source_repository": "Arthur742Ramos/non-cut-points-lean" } ] }, "published_at": "2026-10-06T02:45:48Z", "source": { "commit": "b1d0ed0338d098a4c84622828390cc84027df8dc", "project_path": null, "repository": "Arthur742Ramos/non-cut-points-lean" }, "status": "registered", "title": "Non-Cut Points of Hausdorff Continua", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 disproof of Erd\u0151s problem 575 in both its unrestricted and cycle-containing formulations: a finite family F of forbidden graphs containing a bipartite member need not contain a bipartite H \u2208 F with ex(n; H) = O(ex(n; F)). Proves the unrestricted case from scratch via the bipartite forest family F_0 = {K_{1,2}, 2K_2} (where ex(n; F_0) \u2264 1 while ex(n; K_{1,2}) \u2265 \u230an/2\u230b and ex(n; 2K_2) \u2265 n - 1), and resolves the cycle-containing case by adapting OpenAI's all-bipartite cyclic counterexample family from Erd\u0151s problem 180 (where ex(n; F) = O(n^(21/16)) and ex(n; H) = \u03a9(n^(4/3)) for every H \u2208 F).", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C35" ] }, "formalization": { "theorem_names": [ "Erdos575.not_erdos_575_corrected", "Erdos575.not_erdos_575", "Erdos575.not_erdos_575_all_bipartite", "Erdos575.correctedCounterexample", "Erdos575.quantitativeCounterexample", "Erdos575.ForestCounterexample.forestCounterexample", "Erdos575.ForestCounterexample.forestFamily_not_isBipartiteCompact", "Erdos575.erdos_575", "Erdos575.erdos_575_unrestricted" ] }, "id": "PALOMAR-2026-10-06-000004", "path": "entries/PALOMAR-2026-10-06-000004-v1.json", "preservation": { "repositories": [ { "commit": "b135fd8f6e5ee25129c4855e06c93739dd80651d", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-06T02:18:29Z", "source": { "commit": "b135fd8f6e5ee25129c4855e06c93739dd80651d", "project_path": "erdos-575", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 575: bipartite extremal-graph compactness", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization, with Mathlib, of an upper bound for the Schur numbers in terms of the triangle Ramsey numbers, and of the structure of a colouring at the limit of that bound. If every colouring with k colours of the pairs of r points has a monochromatic triangle (R_k(3) \u2264 r), then the interval [1, 2((k + 1)\u230a(r \u2212 1)/2\u230b + 1)] is not covered by k + 1 sumfree sets, so S(k + 1) \u2264 2(k + 1)\u230a(r \u2212 1)/2\u230b + 1. With R_2(3) \u2264 6 this gives S(3) \u2264 13, which is exact. Under the hypothesis R_4(3) \u2264 61, a computer-assisted claim of M. Tatarevic that this project does not prove, the pigeonhole step gives R_5(3) \u2264 302, and the bound then gives S(6) \u2264 1801; the hypothesis is an explicit argument (TriangleRamsey 4 61) of each theorem that uses it. The frontier theorems describe a Schur colouring of the largest interval that the bound allows. Let R_k(3) \u2264 u + 1, (k + 1)u = 2t and m = (k + 2)t, and let c be a Schur colouring of [1, 2m + 1] with k + 2 colours. Then each colour occurs exactly t times in [1, m]; with q = c(m + 1), the neighbourhood of the centre m in the colour q is regular in every other colour; and c(m + 1 \u2212 d) = c(m + 1 + d) for every d in [1, m] with c(d) = q. For six colours under R_4(3) \u2264 61, a Schur colouring of [1, 1801] gives five sets of 60 points, each closed under x \u21a6 1800 \u2212 x with no fixed point and with differences in at most four colours. These theorems do not exclude such a colouring and do not prove S(6) \u2264 1800. The arguments for the bound and for the frontier were first proposed by AI agents based on ChatGPT (OpenAI) in a project discussion. A Claude agent (Anthropic) audited the argument for the bound; the AI system Claude checked each step of the frontier argument, restated it with explicit hypotheses, and wrote all Lean proofs, under the direction of the author. The results have not been peer reviewed.", "authors": [ { "name": "Adam McKenna" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D10", "11B75", "05C55" ] }, "formalization": { "theorem_names": [ "ClassicalSchur.triangleRamsey_succ", "ClassicalSchur.not_coveredBySumFree_Icc_of_triangleRamsey", "ClassicalSchur.not_coveredBySumFree_Icc_fourteen_three", "ClassicalSchur.not_coveredBySumFree_Icc_six_of_triangleRamsey_four_sixtyOne", "ClassicalSchur.coveredBySumFree_of_schurColoring", "ClassicalSchur.exists_schurColoring_of_coveredBySumFree", "ClassicalSchur.card_filter_Icc_eq_of_frontier", "ClassicalSchur.card_centralNbhd_of_frontier", "ClassicalSchur.card_colorNbhd_centralNbhd_of_frontier", "ClassicalSchur.color_eq_of_card_filter_eq", "ClassicalSchur.color_reflect_of_frontier", "ClassicalSchur.endpointNbhd_of_frontier", "ClassicalSchur.even_of_frontier", "ClassicalSchur.schur_six_frontier_structure" ] }, "id": "PALOMAR-2026-10-06-000003", "path": "entries/PALOMAR-2026-10-06-000003-v1.json", "preservation": { "repositories": [ { "commit": "6b1c69732d73428042efce8d563f67604320cad4", "fork_repository": "PalomarArchive/mysticflounder--schur-centred-bound--746e036536c8", "source_repository": "mysticflounder/schur-centred-bound" } ] }, "published_at": "2026-10-06T01:33:00Z", "source": { "commit": "6b1c69732d73428042efce8d563f67604320cad4", "project_path": null, "repository": "mysticflounder/schur-centred-bound" }, "status": "registered", "title": "S(6) \u2264 1801 if R\u2084(3) \u2264 61: a centred Schur bound and the structure at the frontier", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A nonempty compact connected Hausdorff space covered by countably many pairwise disjoint closed sets has one piece equal to the whole space. Includes arbitrary countable index types, empty pieces, finite partitions, and the continuous-choice corollary.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54D05", "54D30", "54F15" ] }, "formalization": { "theorem_names": [ "Sierpinski.exists_eq_univ_of_nat_closed_partition", "Sierpinski.exists_eq_univ_of_countable_closed_partition", "Sierpinski.continuous_choice_eq_one" ] }, "id": "PALOMAR-2026-10-06-000002", "path": "entries/PALOMAR-2026-10-06-000002-v1.json", "preservation": { "repositories": [ { "commit": "406069edac48c4845bbf303ecc83ee9ef000b99a", "fork_repository": "PalomarArchive/Arthur742Ramos--sierpinski-closed-partition-lean--fd6de382b003", "source_repository": "Arthur742Ramos/sierpinski-closed-partition-lean" } ] }, "published_at": "2026-10-06T00:47:23Z", "source": { "commit": "406069edac48c4845bbf303ecc83ee9ef000b99a", "project_path": null, "repository": "Arthur742Ramos/sierpinski-closed-partition-lean" }, "status": "registered", "title": "Sierpinski Closed Partitions of Continua", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of unconditional bounds and conditional reductions for Erd\u0151s problem 1139 on consecutive P_2 almost-primes (\u03a9(u_k) \u2264 2). Extending Chojecki's (2026) Erd\u0151s 689 double-covering construction via protected-reserve omission, it proves unconditionally that limsup (u_{k+1} - u_k) / log(k + 1) \u2265 A > 1 and that every large height X contains an almost-prime-free interval of length \u2265 c log X (c > 1), alongside the mixed prime/prime-square CRT bridge, PNT primorial asymptotics, and squared-core deficiency classification. Using a squared-prime core and scale-adaptive dyadic prime shells, it also derives the full infinite-limsup conjecture conditionally from the Green\u2013Tao\u2013Ziegler (2010\u20132012) linear-forms asymptotic and proves its reverse implication to arbitrary-length prime progressions.", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11N05", "11N36", "11B25" ] }, "formalization": { "theorem_names": [ "Erdos1139.Palomar.uniform_maximal_gap", "Erdos1139.Palomar.limsup_strictly_gt_one", "Erdos1139.Palomar.cover_crt_interval", "Erdos1139.Palomar.seven_square_crt", "Erdos1139.Palomar.primorial_log_limit", "Erdos1139.Palomar.limsup_top_of_sparse", "Erdos1139.Palomar.core_classification" ] }, "id": "PALOMAR-2026-10-06-000001", "path": "entries/PALOMAR-2026-10-06-000001-v1.json", "preservation": { "repositories": [ { "commit": "38ad7ea8de1efe526f665c2b5204465e5ed879fe", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-06T00:36:59Z", "source": { "commit": "38ad7ea8de1efe526f665c2b5204465e5ed879fe", "project_path": "erdos-1139", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 1139: large gaps between almost-primes", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Every component of a proper closed subset of a compact connected Hausdorff space meets its frontier. Through each point of the closed subset there is a compact connected subset of it meeting that frontier.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.GN" ], "msc2020": [ "54D05", "54D30" ] }, "formalization": { "theorem_names": [ "BoundaryBumping.closed_component_meets_frontier", "BoundaryBumping.exists_subcontinuum_meeting_frontier" ] }, "id": "PALOMAR-2026-10-05-000009", "path": "entries/PALOMAR-2026-10-05-000009-v1.json", "preservation": { "repositories": [ { "commit": "58831416a12f73312ac01cdaec19c060a6d4b142", "fork_repository": "PalomarArchive/Arthur742Ramos--boundary-bumping-lean--12cab9337e49", "source_repository": "Arthur742Ramos/boundary-bumping-lean" } ] }, "published_at": "2026-10-05T22:35:11Z", "source": { "commit": "58831416a12f73312ac01cdaec19c060a6d4b142", "project_path": null, "repository": "Arthur742Ramos/boundary-bumping-lean" }, "status": "registered", "title": "Boundary Bumping for Closed Subsets of Continua", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Over any characteristic-zero field, two elements P and Q of the first Weyl algebra satisfying QP-PQ=1 generate the algebra when P has at most six nonzero homogeneous components for deg X=1 and deg Y=-1. The degree, order and mass of Q are unrestricted. The proof combines sparse-polynomial rigidity with internally proved Newton and Weyl-algebra structural results.", "authors": [ { "name": "Idris Ali Shaik" } ], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "16S32", "16W20", "16W50" ] }, "formalization": { "theorem_names": [ "Dixmier.Palomar.massSixGeneration" ] }, "id": "PALOMAR-2026-10-05-000006", "path": "entries/PALOMAR-2026-10-05-000006-v2.json", "preservation": { "repositories": [ { "commit": "61783d52b6ae44cd2d8d20ad6cb798e7bbbce3ff", "fork_repository": "PalomarArchive/shaikidris--DixmierMassSix-Palomar--ea6fbe79e423", "source_repository": "shaikidris/DixmierMassSix-Palomar" } ] }, "published_at": "2026-10-05T18:57:39Z", "source": { "commit": "61783d52b6ae44cd2d8d20ad6cb798e7bbbce3ff", "project_path": null, "repository": "shaikidris/DixmierMassSix-Palomar" }, "status": "registered", "title": "The rank-one Dixmier conjecture for elements of mass at most six", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A Lean formalization of the superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift (Armstrong, Bou-Rabee and Kuusi): the quenched superdiffusive invariance principle (Theorem A), quantitative homogenization of the Dirichlet problem (Theorem B), the large-scale C^{0,gamma} and C^{1,gamma} regularity theory (Theorems C and D) and the sharp asymptotics of the renormalized diffusivities (Theorem 5.1), with every result of Sections 2 to 8 of the paper that their proofs use. It is built on the public CoarseGraining homogenization library and the public MarkovProcess library.", "authors": [ { "name": "Scott Armstrong" } ], "classification": { "arxiv": [ "math.PR", "math-ph", "math.AP" ], "msc2020": [ "60K37", "35B27", "60J60", "60F17" ] }, "formalization": { "theorem_names": [ "SuperdiffusionCLT.StatementAudit.TheoremA.theoremA" ] }, "id": "PALOMAR-2026-10-05-000008", "path": "entries/PALOMAR-2026-10-05-000008-v1.json", "preservation": { "repositories": [ { "commit": "b407c97da6af47b63f8067aaf531e2e255bfc78d", "fork_repository": "PalomarArchive/scottnarmstrong--SuperdiffusionCLT--db2e44683f08", "source_repository": "scottnarmstrong/SuperdiffusionCLT" } ] }, "published_at": "2026-10-05T18:45:33Z", "source": { "commit": "b407c97da6af47b63f8067aaf531e2e255bfc78d", "project_path": null, "repository": "scottnarmstrong/SuperdiffusionCLT" }, "status": "registered", "title": "Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A conditional Lean 4 formalization of E. Markman, \"Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds\" (arXiv:2502.03415v2). The paper proves that the Hodge-Weil classes of polarized abelian sixfolds of Weil type with discriminant -1 are algebraic (Theorem 1.5.1) and deduces the Hodge conjecture for abelian fourfolds (Corollary 1.6.1).\n\nEvery result of the paper about cohomology classes, Clifford algebras, spin representations, Hodge structures and period domains is proved in a linear-algebra model of the cohomology of abelian varieties (abelian varieties up to isogeny as polarizable rational Hodge structures of weight one), by the paper's arguments except the departures listed under fidelity (notably Proposition 6.1.2 and Lemma 4.0.2, proved by other arguments with the owner's approval): rational secant lines to the spinor variety and the abelian 2n-folds of Weil type they give (Sections 2-4), the Spin(V)-equivariance of Orlov's equivalence (Section 6, Proposition 6.1.2), Theorem 1.4.1(3) and (4) (the characteristic class of the secant-square sheaf stays of Hodge type on every Weil-type deformation of X x X^, and its K-translates with h^3 span Q h^3 + HW), Corollary 1.3.2, the cohomological results of Section 8, and Section 10. The results the paper cites from Chevalley, Igusa (Lemmas 1-2, Section 2, Proposition 3 except the orbits over subfields in which -d is not a square), Golyshev-Lunts-Orlov, Orlov, Huybrechts, Trautman, van Geemen (Def. 4.9, and Th. 5.2(3) through Landherr's theorem and Meyer's theorem), Markman (2023) and Schoen (Prop. 10, from push-forward of cycles, by Voisin's argument) are proved in the case the paper uses.\n\nTheorem 1.5.1 and Corollary 1.6.1 are proved assuming, as named hypotheses in their statements, the paper's own sheaf-theoretic Sections 7-9 (in the form: the class kappa_3(E) is algebraic near X x X^ wherever kappa(E) is of Hodge type) and results from algebraic geometry not yet available in Lean: about algebraic cycles, pullbacks, push-forwards along projections and products of algebraic classes, the Lefschetz (1,1) theorem, Voisin's algebraicity loci and a theorem of Ramon-Mari; about Hodge structures, two theorems of Moonen-Zarhin. Remark 10.1.2(1) assumes part of Igusa's Proposition 3 when -d is not a square. Eleven statements of the paper are false or misprinted as printed and are formalized in corrected form (the basis of Example 10.2.2 is also misprinted, with no effect on the value computed there); REPORT.md audits the paper.", "authors": [ { "name": "The-Anh Vu-Le" } ], "classification": { "arxiv": [ "math.AG" ], "msc2020": [ "14C30", "14C25", "14K22", "14K12", "15A66", "14F08" ] }, "formalization": { "theorem_names": [ "WeilClasses.Challenge.Kd_Nm", "WeilClasses.Challenge.fX_mul_self", "WeilClasses.Challenge.JX_mem_weilDomain", "WeilClasses.Challenge.discIs_XXhat", "WeilClasses.Challenge.rank_chE", "WeilClasses.Challenge.theorem1_4_1_3", "WeilClasses.Challenge.theorem1_4_1_4", "WeilClasses.Challenge.theorem1_5_1", "WeilClasses.Challenge.corollary1_6_1" ] }, "id": "PALOMAR-2026-10-05-000007", "path": "entries/PALOMAR-2026-10-05-000007-v1.json", "preservation": { "repositories": [ { "commit": "4be865ee7242c8ac4e7b7f92b192b31f9d1c3338", "fork_repository": "PalomarArchive/vltanh--lean4-markman25-hodge-abelian-fourfolds--041dfda52aff", "source_repository": "vltanh/lean4-markman25-hodge-abelian-fourfolds" } ] }, "published_at": "2026-10-05T17:07:39Z", "source": { "commit": "4be865ee7242c8ac4e7b7f92b192b31f9d1c3338", "project_path": null, "repository": "vltanh/lean4-markman25-hodge-abelian-fourfolds" }, "status": "registered", "title": "Weil classes on abelian sixfolds and the Hodge conjecture for abelian fourfolds (Markman)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A formalization and provably correct implementation of a linear-time algorithm for triangulating a simple polygon. The implementation is a concrete program for an explicit exact real RAM. Lean proves that on every simple polygon with n vertices the program halts within C(n+1) steps, with natural-number words at most K(n+1), and outputs a conforming triangulation with exactly n - 2 triangles on the input vertices. No general-position assumption is made. The proofs use exactly Lean's three standard axioms (propext, Classical.choice, Quot.sound), with no sorry and no native_decide. They are checked by the Lean kernel and replayed by independent kernels. The algorithm is a variant of Chazelle's (B. Chazelle, Triangulating a simple polygon in linear time, Discrete & Computational Geometry 6 (1991), 485-524), and the main theorem is the triangulation conclusion of his Theorem 4.3. The program follows the structure of Chazelle's algorithm: visibility maps of polygonal chains, merging with conformality restoration and granularity enforcement, ray-shooting and arc-cutting oracles built with planar separators, the up phase and the down phase, and the final stage from the visibility map to the triangulation. It departs from the paper's algorithm in documented ways; in particular its down phase recurses only for n >= 2^30 + 2. The running-time bound is proved for this program, not for the algorithm as the paper describes it. The program is also given by name with explicit (very large) constants, and a merge-level refinement theorem shows that its run follows an abstract model of the algorithm written from the paper.", "authors": [ { "name": "Robert D. Barish" } ], "classification": { "arxiv": [ "cs.CG" ], "msc2020": [ "68U05", "68Q25", "68W40", "52B55" ] }, "formalization": { "theorem_names": [ "Chazelle.linear_time_triangulation" ] }, "id": "PALOMAR-2026-10-05-000003", "path": "entries/PALOMAR-2026-10-05-000003-v2.json", "preservation": { "repositories": [ { "commit": "92cc7745aa026aad3e80e7a72fe88cfbd909614a", "fork_repository": "PalomarArchive/RBarish-UTokyo--LinearTimeTriangulation-ChazelleAlgorithm-Lean4--0d58a8483c03", "source_repository": "RBarish-UTokyo/LinearTimeTriangulation-ChazelleAlgorithm-Lean4" } ] }, "published_at": "2026-10-05T14:21:34Z", "source": { "commit": "92cc7745aa026aad3e80e7a72fe88cfbd909614a", "project_path": null, "repository": "RBarish-UTokyo/LinearTimeTriangulation-ChazelleAlgorithm-Lean4" }, "status": "registered", "title": "Linear-Time Triangulation of Simple Polygons (Chazelle's Algorithm) in Lean 4", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A complete Lean 4 proof of all three parts of Erd\u0151s problem 1054. For the least integer f(n) whose k smallest divisors sum to n, we prove that f(n) is not o(n), is not o(n) on any density-one set, and satisfies limsup f(n)/n = \u221e on every density-one subtype. Building on the qualitative divisor-prefix sieve and almost-all binary Goldbach proof from Principia-Math-Solutions, this formalization proves the Formal Conjectures statements along with a sharp second-moment lower-density bound c / [A^3 (1 + log A)^4] for odd n with f(n) > A n, the Tao\u2013Kova\u010d small-ratio upper bound #{n \u2264 X : 0 < f(n) \u2264 \u03b4 n} \u2264 C \u03b4^3 X, liminf_{f(n)>0} f(n)/n = 0, and the asymptotic density equivalence between {s(2d)} and even aliquot values.", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11A25", "11N36", "11P32" ] }, "formalization": { "theorem_names": [ "Erdos1054.Palomar.answer_i", "Erdos1054.Palomar.answer_ii", "Erdos1054.Palomar.answer_iii", "Erdos1054.Palomar.official_answer_i", "Erdos1054.Palomar.official_answer_ii", "Erdos1054.Palomar.official_answer_iii", "Erdos1054.Palomar.limsup_on_every_density_one", "Erdos1054.Palomar.odd_subtype_limsup", "Erdos1054.Palomar.f_undefined_at_2", "Erdos1054.Palomar.f_undefined_at_5", "Erdos1054.Palomar.quantitative_odd_sharp_second_moment_logarithmic_endpoint", "Erdos1054.Palomar.quantitative_odd_sharp_second_moment_logarithmic_endpoint_eventual_count", "Erdos1054.Palomar.quantitative_odd_almost_full_three_plus_epsilon", "Erdos1054.Palomar.quantitative_odd_pure_power_six", "Erdos1054.Palomar.small_ratio_count_le_cubic", "Erdos1054.Palomar.small_ratio_upper_density_le_cubic", "Erdos1054.Palomar.littleO_on_subtype_imp_represented_density_zero", "Erdos1054.Palomar.f_sigma_le", "Erdos1054.Palomar.frequently_represented_small_ratio", "Erdos1054.Palomar.cofactor_two_lower_density_eq_even_aliquot", "Erdos1054.Palomar.cofactor_two_upper_density_eq_even_aliquot" ] }, "id": "PALOMAR-2026-10-05-000005", "path": "entries/PALOMAR-2026-10-05-000005-v1.json", "preservation": { "repositories": [ { "commit": "0917cd01090b52088aa82b09c8c84689047b8171", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-05T07:51:18Z", "source": { "commit": "0917cd01090b52088aa82b09c8c84689047b8171", "project_path": "erdos-1054", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 1054: sums of smallest divisors", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 proof of the negative answer to Erd\u0151s problem 958: for every n \u2265 4, there exists an n-point configuration in the Euclidean plane determining n - 1 distinct interpoint distances with multiplicities {1, 2, ..., n - 1} that is neither equidistant on a line nor equidistant on a circle. Formalizing the May 2025 construction of Clemen, Dumitrescu, and Liu, the proof places n - 1 equally spaced points on a short arc of the unit circle together with the center (0, 0), uses strict monotonicity of chord lengths on [0, \u03c0/3) to compute the exact distance-multiplicity profile, and eliminates collinear and concyclic progressions algebraically. It also proves the positive baseline results that every equidistant collinear configuration of size n \u2265 2 has n - 1 distinct distances with multiplicities {1, ..., n - 1}, and that for every n \u2265 2 there exists an n-point circular equidistant configuration with that distance-multiplicity profile.", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "52C10", "05D99" ] }, "formalization": { "theorem_names": [ "Erdos958.Palomar.erdos_958", "Erdos958.Palomar.not_erdos_958", "Erdos958.Palomar.clemen_dumitrescu_liu", "Erdos958.Palomar.equidistantOnLine_has_profile", "Erdos958.Palomar.equidistantOnCircle_exists_has_profile" ] }, "id": "PALOMAR-2026-10-05-000004", "path": "entries/PALOMAR-2026-10-05-000004-v1.json", "preservation": { "repositories": [ { "commit": "08152671ab394392fd941b18654daba734c26518", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-05T07:11:27Z", "source": { "commit": "08152671ab394392fd941b18654daba734c26518", "project_path": "erdos-958", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 958: planar point sets with distance multiplicities {1, ..., n - 1}", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 proof of the negative answer to Erd\u0151s problem 2 (Hough 2015; Balister\u2013Bollob\u00e1s\u2013Morris\u2013Sahasrabudhe\u2013Tiba 2022): there is an absolute constant B such that every finite covering of the integers by distinct moduli > 1 contains a modulus \u2264 B. For any finite set D of moduli above B with N = lcm(D), the proof builds a probability measure on ZMod N prime by prime via the BBMST half-cap fiber update, preserving the progression bound w(z \u2261 a mod d) \u2264 2^\u03c9(d)/d while bounding each prime-p loss by C (log p)^6 / (p - 1)^2 via Euler factorization and Mertens' third theorem. Combining this summable tail with Euler's product bound on A-smooth reciprocals keeps the covered mass below 1/2, proving the Formal Conjectures ideal statement (erdos_2), its direct negation (not_arbitrarilyLarge_ideal_coverings), the uniform minimum-modulus bound (minimum_modulus_bound), and finite-set noncoverage (finite_noncoverage_bound).", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11A07", "11B25" ] }, "formalization": { "theorem_names": [ "Erdos2.Standalone.erdos_2", "Erdos2.Standalone.not_arbitrarilyLarge_ideal_coverings", "Erdos2.Standalone.minimum_modulus_bound", "Erdos2.Standalone.finite_noncoverage_bound" ] }, "id": "PALOMAR-2026-10-05-000002", "path": "entries/PALOMAR-2026-10-05-000002-v1.json", "preservation": { "repositories": [ { "commit": "7caa4144741e10f96cf79d6f7313a0efd9a18b67", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-05T05:09:05Z", "source": { "commit": "7caa4144741e10f96cf79d6f7313a0efd9a18b67", "project_path": "erdos-2", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 2: minimum modulus of distinct covering systems", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Higher-cell attachment invariance for ordinary topological fundamental groups. For any path-connected topological space X, any basepoint, an index type in an independent universe, independently varying cell dimensions at least three, and arbitrary continuous unit-sphere attaching maps, the genuine adjunction quotient of X and the coproduct of closed Euclidean unit balls is constructed with its quotient topology. The actual inclusion-induced fundamental-group homomorphism is bijective and is the underlying homomorphism of the displayed multiplicative equivalence. Mixed dimensions and the empty family are included. No finiteness, countability, CW, separation or local-connectivity hypotheses are added, and no retraction, simple-connectivity or kernel oracle is assumed.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.AT", "math.CT" ], "msc2020": [ "55Q05", "18B40" ] }, "formalization": { "theorem_names": [ "HigherCellAttachment.higher_cell_attachment" ] }, "id": "PALOMAR-2026-10-05-000001", "path": "entries/PALOMAR-2026-10-05-000001-v1.json", "preservation": { "repositories": [ { "commit": "bcf788e349b4969f4d8379ee9cb4df7a28352af1", "fork_repository": "PalomarArchive/Arthur742Ramos--higher-cell-attachment-lean--839dc8f39578", "source_repository": "Arthur742Ramos/higher-cell-attachment-lean" } ] }, "published_at": "2026-10-05T02:16:29Z", "source": { "commit": "bcf788e349b4969f4d8379ee9cb4df7a28352af1", "project_path": null, "repository": "Arthur742Ramos/higher-cell-attachment-lean" }, "status": "registered", "title": "Fundamental Groups of Arbitrary Higher-Cell Attachments", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The full arbitrary-family two-cell attachment theorem for ordinary topological fundamental groups. For any path-connected topological space X, any basepoint, and any indexed family of continuous circle attaching maps, a genuine adjunction quotient is constructed from X and the coproduct of closed complex unit disks. The actual inclusion-induced fundamental-group homomorphism is surjective and its kernel is precisely the normal closure of the once-around attaching loops, transported along the specified basepoint paths. A quotient group isomorphism is proved compatible with the actual inclusion map. No finiteness, CW, separation, local-connectivity or semilocal-simple-connectivity hypotheses are added.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "math.AT", "math.CT" ], "msc2020": [ "55Q05", "18B40" ] }, "formalization": { "theorem_names": [ "CellAttachment.two_cell_attachment" ] }, "id": "PALOMAR-2026-10-04-000011", "path": "entries/PALOMAR-2026-10-04-000011-v1.json", "preservation": { "repositories": [ { "commit": "1c671038e320cecc58ca1d81dca11bbe2e2a507c", "fork_repository": "PalomarArchive/Arthur742Ramos--cell-attachment-lean--38d25cf83333", "source_repository": "Arthur742Ramos/cell-attachment-lean" } ] }, "published_at": "2026-10-04T20:55:47Z", "source": { "commit": "1c671038e320cecc58ca1d81dca11bbe2e2a507c", "project_path": null, "repository": "Arthur742Ramos/cell-attachment-lean" }, "status": "registered", "title": "Fundamental Groups of Arbitrary Two-Cell Attachments", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 proof of the affirmative answer to Erd\u0151s problem 546 (Sudakov's theorem): there exists an absolute constant C > 0 such that every finite simple graph G with m edges and no isolated vertices has diagonal Ramsey number R(G) \u2264 2^(C \u221am). Formalizing Sudakov's 2011 proof\u2014including high-degree vertex deletion, greedy bounded-degree embedding and hereditary sparse cuts, low-density monochromatic-pair extraction, and iterative clique-reservoir amplification. This proves the uniform explicit bound R(G) \u2264 2^(4000 \u221am) across all finite simple graphs without isolated vertices (including m = 0), together with the exact Formal Conjectures statement.", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C55", "05C35", "05D10" ] }, "formalization": { "theorem_names": [ "erdos_546_original_statement", "Erdos546.erdos_546", "Erdos546.sudakov_sparse_bound", "Erdos546.sparse_graph_ramsey_witness", "Erdos546.boundedDegree_sparse_cut", "Erdos546.exists_monoPair_of_low_edgeDensity", "Erdos546.quantitative_monoPair_amplification" ] }, "id": "PALOMAR-2026-10-04-000010", "path": "entries/PALOMAR-2026-10-04-000010-v1.json", "preservation": { "repositories": [ { "commit": "7caa4144741e10f96cf79d6f7313a0efd9a18b67", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-04T20:31:23Z", "source": { "commit": "7caa4144741e10f96cf79d6f7313a0efd9a18b67", "project_path": "erdos-546", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 546: sparse graph Ramsey numbers", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A fully-dynamic Maximum-Cardinality-Search (MCS) ordering algorithm that, on a single-edge insertion or deletion into a loop-free undirected graph, scans the affected window between the two endpoints for the first step at which the current ordering's choice stops being a legal MCS pick, and only then regreedies the suffix (probe-then-repair). Correctness - every order produced stays a valid MCS ordering of the updated graph, and in fact equals the from-scratch canonical ordering - is machine-checked in Lean 4 / Mathlib, together with locality (the first position whose choice can change lies inside the window between the endpoints; after a break the suffix is regreedyed, so positions past the window may also change), an executable boolean-adjacency implementation proved to refine the abstract update, and per-update cost in a unit-cost query model (counting legality checks and pick steps along the real executable control flow): at most n units for an insertion and n + 1 for a deletion, so the worst case matches static recomputation. A stability-boundary result is also proved: if the neighbour count of every vertex in one graph exceeds that in the other by at most one along every chosen prefix, the two graphs admit a common MCS ordering; in particular inserting or deleting a matching, or flipping a single edge, always preserves a common MCS ordering.", "authors": [ { "name": "Md Thoriqul Islam Thonmay" }, { "name": "Gregory Morse" } ], "classification": { "arxiv": [ "cs.DS", "math.CO" ], "msc2020": [ "05C85", "68Q25" ] }, "formalization": { "theorem_names": [ "Challenge.init_valid", "Challenge.insert_update_valid", "Challenge.delete_update_valid", "Challenge.greedySuffix_full", "Challenge.update_from_prefix", "Challenge.setEdge_preserves_symm", "Challenge.setEdge_preserves_loopless", "Challenge.mcsOrder_eq_greedySuffix", "Challenge.exec_insert_recompute_valid", "Challenge.exec_delete_recompute_valid", "Challenge.IsMCSOrdering_unique", "Challenge.insert_update_eq_initOrder", "Challenge.delete_update_eq_initOrder", "Challenge.legal_upto_earlierPos", "Challenge.legal_after_laterPos", "Challenge.delete_legal_between", "Challenge.insert_break_iff", "Challenge.insert_update_eq_self_iff", "Challenge.delete_update_eq_self_iff", "Challenge.exec_insert_update_eq", "Challenge.exec_delete_update_eq", "Challenge.exec_insert_cost_le", "Challenge.exec_delete_cost_le", "Challenge.exec_insert_cost_of_no_break", "Challenge.exec_delete_cost_of_no_break", "Challenge.exec_insert_cost_of_break", "Challenge.exec_delete_cost_of_break", "Challenge.exec_insert_update_c_fst", "Challenge.exec_delete_update_c_fst", "Challenge.exists_common_ordering", "Challenge.common_order_extends_prefix", "Challenge.insert_matching_common_order", "Challenge.delete_matching_common_order", "Challenge.flip_common_order", "Challenge.triangle_no_common_order", "Challenge.p4_no_common_order", "Challenge.mixed_matching_no_common_order" ] }, "id": "PALOMAR-2026-10-04-000009", "path": "entries/PALOMAR-2026-10-04-000009-v1.json", "preservation": { "repositories": [ { "commit": "a872fddca7025ae0ecedd654238fa0c06729dc2e", "fork_repository": "PalomarArchive/thonmay--thesis-lean--090f8db89fc3", "source_repository": "thonmay/thesis-lean" } ] }, "published_at": "2026-10-04T18:32:42Z", "source": { "commit": "a872fddca7025ae0ecedd654238fa0c06729dc2e", "project_path": null, "repository": "thonmay/thesis-lean" }, "status": "registered", "title": "Dynamic MCS Ordering Maintenance", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A complete Lean 4 proof of the affirmative answer to Erd\u0151s problem 956: for all sufficiently large n, there exist n pairwise-disjoint translates of a compact convex planar set with more than n^(1+c) unordered pairs at Euclidean set-distance 1 (for c = 1/4). Adapting the parabolic construction announced by Valtr (2005) and detailed in Chojecki's April 2026 manuscript, the formalization proves the supporting-hyperplane theorem for a centrally symmetric signed parabolic cap and constructs a four-layer parabolic grid of disjoint translates. This establishes the explicit lower bound h(n) > (1/26) n^(4/3) for all n \u2265 80 and the sharper eventual bound h(n) > (2/5) n^(4/3).", "authors": [ { "name": "Linmiao Xu" } ], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52C10", "52A10", "05D99" ] }, "formalization": { "theorem_names": [ "Erdos956.Palomar.erdos_956_superlinear", "Erdos956.Palomar.erdos_956", "Erdos956.Palomar.erdos_956_omega_four_thirds", "Erdos956.Palomar.erdos_956_four_layer_polynomial", "Erdos956.Palomar.erdos_956_eventual_two_fifths" ] }, "id": "PALOMAR-2026-10-04-000008", "path": "entries/PALOMAR-2026-10-04-000008-v1.json", "preservation": { "repositories": [ { "commit": "2afe25fe500b036dfabef8e00c369a434bfff763", "fork_repository": "PalomarArchive/linrock--math-proofs--14cf37780a40", "source_repository": "linrock/math-proofs" } ] }, "published_at": "2026-10-04T16:32:39Z", "source": { "commit": "2afe25fe500b036dfabef8e00c369a434bfff763", "project_path": "erdos-956", "repository": "linrock/math-proofs" }, "status": "registered", "title": "Erd\u0151s 956: unit distances between disjoint convex translates", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The critical-window laws for autocatalytic emergence in reversible binary-polymer networks lead to survival profiles for independent open reaction channels. For both the ordered split-position and commuting-factor quotient models, with all binary words of length at most two as food, explicit total algorithms enclose each survival probability at every rational openness in [0,1] to any prescribed positive rational accuracy. Computable seed lengths and finite-record contraction estimates provide uniform truncation bounds on intervals bounded away from zero. The associated intensity profiles S(1 \u2212 exp(\u2212\u03bb)) are smaller than every fixed power of \u03bb as \u03bb tends to zero from above. Rational interval bounds also compare these profiles with finite-network RAF-existence probabilities under explicit catalyst-pool and input-probability conditions. These results extend the split and quotient critical-window laws with certified approximation and quantitative control.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.PR", "q-bio.MN" ], "msc2020": [ "60K35", "92E20", "05C80", "03D78", "68V20" ] }, "formalization": { "theorem_names": [ "RAF.Concrete.reaction_length_add", "RAF.Concrete.reaction_left_pos", "RAF.Concrete.reaction_right_pos", "RAF.Concrete.reaction_product_le", "RAF.Concrete.reaction_pow_split", "RAF.Concrete.molLength_reactionLeft", "RAF.Concrete.molLength_reactionRight", "RAFCriticalWindowQuantitative.seedTest_exists", "HordijkSteelThreshold.measurable_currySplitField", "RAFReactionQuotient.precedes_total", "RAFCriticalWindowQuantitative.geometricTest_exists", "RAFCriticalWindowQuantitative.repair_ratio", "RAFCriticalWindowQuantitative.quantitative_critical_window_resolution" ] }, "id": "PALOMAR-2026-10-04-000007", "path": "entries/PALOMAR-2026-10-04-000007-v1.json", "preservation": { "repositories": [ { "commit": "b46f8d0747b6dfb4c00a50d386f3df4f330e861e", "fork_repository": "PalomarArchive/mmislan--palomar-p046-critical-window-shape-law--e3ad53e8d156", "source_repository": "mmislan/palomar-p046-critical-window-shape-law" } ] }, "published_at": "2026-10-04T15:07:36Z", "source": { "commit": "b46f8d0747b6dfb4c00a50d386f3df4f330e861e", "project_path": null, "repository": "mmislan/palomar-p046-critical-window-shape-law" }, "status": "registered", "title": "Effective approximation and low-intensity flatness of RAF critical-window profiles", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Every connected finite simple graph admits an edge partition into at most ceil(|V(G)|/2) simple paths when at most two vertices of its induced even-degree graph have degree greater than three. One positive designated even vertex can be exposed twice at the ceiling budget. Two designated even vertices can be exposed twice simultaneously within floor(|V(G)|/2)+1 paths, giving simultaneous ceiling exposure at odd order. Exceptional and ordinary degrees are unrestricted. Simultaneous even-order ceiling exposure and the unrestricted Gallai conjecture are not claimed.", "authors": [ { "name": "Idris Ali Shaik" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C38", "05C70" ] }, "formalization": { "theorem_names": [ "Gallai.Submission.prescribed_endpoint", "Gallai.Submission.ceiling_bound", "Gallai.Submission.simultaneous_floor_add_one", "Gallai.Submission.odd_order_simultaneous" ] }, "id": "PALOMAR-2026-10-04-000006", "path": "entries/PALOMAR-2026-10-04-000006-v1.json", "preservation": { "repositories": [ { "commit": "5f4130a2e14b836cfaa86cad6ae59beb9768b9ea", "fork_repository": "PalomarArchive/shaikidris--gallai-two-exception--3391d32f463e", "source_repository": "shaikidris/gallai-two-exception" } ] }, "published_at": "2026-10-04T13:47:26Z", "source": { "commit": "5f4130a2e14b836cfaa86cad6ae59beb9768b9ea", "project_path": null, "repository": "shaikidris/gallai-two-exception" }, "status": "registered", "title": "Gallai's conjecture with two exceptional even vertices", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A formalization of E. S. Allman, J. H. Degnan and J. A. Rhodes, \"Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent\", J. Math. Biol. 62 (2011) 833-862. The multispecies coalescent is defined from Kingman's coalescent on forests of gene lineages (its generator, the matrix exponential, and a recursion over the species tree).\n\nProved: the distribution of unrooted topological gene trees, with one lineage sampled per taxon, determines the rooted species tree and its internal branch lengths when there are at least five taxa (Theorem 9); on four taxa it determines exactly the unrooted metric species tree (Proposition 3); with several lineages per taxon it also determines the corresponding pendant branch lengths (Corollary 10); and all of this holds for nonbinary species trees (Proposition 11). Every other result of the paper is proved too, with the explicit gene tree distributions, linear invariants and inequalities for five taxa, and the cited results that the proofs use: Tavare's formula, the reconstruction of trees from quartets, and Cayley's count of tree shapes.", "authors": [ { "name": "The-Anh Vu-Le" } ], "classification": { "arxiv": [ "q-bio.PE", "math.PR" ], "msc2020": [ "92D15", "60J90", "60J27", "05C05", "62P10" ] }, "formalization": { "theorem_names": [ "ADR11.Challenge.unrootedDist_nonneg", "ADR11.Challenge.unrootedDist_sum", "ADR11.Challenge.fourTaxa_balanced", "ADR11.Challenge.fourTaxa_caterpillar", "ADR11.Challenge.theorem9", "ADR11.Challenge.theorem9_four", "ADR11.Challenge.proposition3", "ADR11.Challenge.corollary10", "ADR11.Challenge.proposition11_theorem9", "ADR11.Challenge.proposition11_theorem9_four", "ADR11.Challenge.proposition11_corollary10" ] }, "id": "PALOMAR-2026-10-04-000005", "path": "entries/PALOMAR-2026-10-04-000005-v1.json", "preservation": { "repositories": [ { "commit": "2db7f81359758bc49f52bc13c51f2640f8f2a7fd", "fork_repository": "PalomarArchive/vltanh--lean4-adr11--aeb74929e5e0", "source_repository": "vltanh/lean4-adr11" } ] }, "published_at": "2026-10-04T12:48:36Z", "source": { "commit": "2db7f81359758bc49f52bc13c51f2640f8f2a7fd", "project_path": null, "repository": "vltanh/lean4-adr11" }, "status": "registered", "title": "Identifying the rooted species tree from unrooted gene trees under the coalescent", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib formalization of Theorems A and B of arXiv:2609.35632 on the central configurations of the planar Newtonian four-body problem. For every choice of four positive masses and every cyclic order of the bodies there is exactly one strictly convex central configuration with that cyclic order, up to similarity, so there are exactly three convex central configurations up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos (2012). The main step of the paper is Theorem B: at every strictly convex central configuration of positive masses, the Hessian form Q of U + (\u03bb/2)I is at least one quarter of its radial part K. It follows that every convex central configuration is a nondegenerate local minimum of U I^(1/2), whose Hessian vanishes exactly on the tangent space of the orbit of the similarities. The computer-assisted step, a branch and bound with interval arithmetic over the three-dimensional set of normalized convex central configurations in the coordinates of Corbera, Cors and Roberts, proves tr S < 3/4 and is checked by Lean's kernel with `decide +kernel` in 7307 certificate theorems. No compiled code is trusted, and the proofs use only Lean's three standard axioms.", "authors": [ { "name": "Tejasvi Singh Tomar" } ], "classification": { "arxiv": [ "math.DS", "math-ph" ], "msc2020": [ "70F10", "70F15", "58K05", "65G40" ] }, "formalization": { "theorem_names": [ "C4.theoremA_slice", "C4.theoremA", "C4.theoremA_cyclic", "C4.theoremB", "C4.nondegenerate", "C4.convex_count" ] }, "id": "PALOMAR-2026-10-02-000011", "path": "entries/PALOMAR-2026-10-02-000011-v2.json", "preservation": { "repositories": [ { "commit": "353653eceec402c993183f240beacaa20e6f39ad", "fork_repository": "PalomarArchive/tejasvi--lean4body-uniqueness--ceab4c756231", "source_repository": "tejasvi/lean4body-uniqueness" } ] }, "published_at": "2026-10-04T11:13:07Z", "source": { "commit": "353653eceec402c993183f240beacaa20e6f39ad", "project_path": null, "repository": "tejasvi/lean4body-uniqueness" }, "status": "registered", "title": "Uniqueness of four-body convex central configurations for all masses", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A Lean 4 formalization of the quantum Pauli basis test from MIP*=RE. The comparator-configured results give a value-one symmetric projective consistent commuting strategy, quantum soundness of the simultaneous low-degree test, soundness of the Pauli basis test up to local isometries with the explicit deltaQld error, and the corresponding qubit-coordinate form.", "authors": [ { "name": "Ruixuan Deng" } ], "classification": { "arxiv": [ "quant-ph" ], "msc2020": [ "81P45" ] }, "formalization": { "theorem_names": [ "MIPStarRE.QPBT.Palomar.exists_spcc_value_one", "MIPStarRE.QPBT.Palomar.exists_ld_soundness", "MIPStarRE.QPBT.Palomar.pauli_soundness", "MIPStarRE.QPBT.Palomar.pauli_soundness_qubit" ] }, "id": "PALOMAR-2026-10-04-000004", "path": "entries/PALOMAR-2026-10-04-000004-v1.json", "preservation": { "repositories": [ { "commit": "179a37893d2a037cce0587ea932f5da819c7d78c", "fork_repository": "PalomarArchive/Dengnifer--MIPStarRE-QPBT--18006c01b8b8", "source_repository": "Dengnifer/MIPStarRE-QPBT" } ] }, "published_at": "2026-10-04T09:50:13Z", "source": { "commit": "179a37893d2a037cce0587ea932f5da819c7d78c", "project_path": null, "repository": "Dengnifer/MIPStarRE-QPBT" }, "status": "registered", "title": "Quantum Pauli basis test", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean formalization of Stafford's Conjecture 3.8 for every characteristic-zero field and every Weyl-algebra rank, proving that each nonzero d admits F, R, and S with 1 = d*R + F*d*S. The development also proves an exact Bernstein-degree fixed-source strengthening and derived localization, adjoint, and evolutionary results.", "authors": [ { "name": "Christopher Albert" } ], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "16S32" ] }, "formalization": { "theorem_names": [ "Stafford38Challenge.universalStatement" ] }, "id": "PALOMAR-2026-09-05-000007", "path": "entries/PALOMAR-2026-09-05-000007-v3.json", "preservation": { "repositories": [ { "commit": "0bb3aa929b931bf5d82d90f62d7508d3ae1dccc1", "fork_repository": "PalomarArchive/itpplasma--stafford38-formal--e4b6caf6307d", "source_repository": "itpplasma/stafford38-formal" } ] }, "published_at": "2026-10-04T06:15:21Z", "source": { "commit": "0bb3aa929b931bf5d82d90f62d7508d3ae1dccc1", "project_path": null, "repository": "itpplasma/stafford38-formal" }, "status": "registered", "title": "Stafford 3.8", "trust": { "level": "high" }, "version": 3, "versions": 3 }, { "abstract": "A Lean-checked proof of FKS Problem 1 for graphs on at most 2d vertices, for every d \u2265 4: every finite simple graph of maximum degree at most d with no connected component isomorphic to K_{d+1} has an injective placement in Euclidean d-space, with every vertex of norm 1/\u221a2 and every edge at distance exactly 1. Non-edges are unconstrained and may also have length 1. The unrestricted question is false at d = 4: Mishra and Senthilkumar (2026) showed that the square of the nine-cycle, on 2d + 1 = 9 vertices, has no such placement. The bound proved here is therefore sharp at d = 4. Neither result settles the question restricted to d \u2265 5. The exception is interpreted componentwise: a K_{d+1} component already precludes such a placement even when other components are present. The research audience is discrete geometers and extremal graph theorists studying spherical unit-distance representations, graph dimension and bounded-degree embedding problems.", "authors": [ { "name": "Dishant Shah" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C62", "52C10" ] }, "formalization": { "theorem_names": [ "FKSProblem1.Palomar.target" ] }, "id": "PALOMAR-2026-09-30-000028", "path": "entries/PALOMAR-2026-09-30-000028-v2.json", "preservation": { "repositories": [ { "commit": "b0d32cb58e8f1d6e4b6e5993d6243fc46a88dd6e", "fork_repository": "PalomarArchive/Dishah3241--FKSProblem1--ff00f0d7257c", "source_repository": "Dishah3241/FKSProblem1" } ] }, "published_at": "2026-10-04T04:16:55Z", "source": { "commit": "b0d32cb58e8f1d6e4b6e5993d6243fc46a88dd6e", "project_path": null, "repository": "Dishah3241/FKSProblem1" }, "status": "registered", "title": "FKS Problem 1 on at most twice the dimension vertices", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A Lean proof that every family of 14-element subsets of a 24-element set covering all 4-element subsets has at least 20 members, so the covering number C(24,14,4) is at least 20, improving the previously tabulated lower bound of 19. Point derivation (the Sch\u00f6nheim bound) then gives C(25,15,5) >= 34, improving the tabulated 32. The proof excludes a hypothetical 19-block covering using degree and codegree counting, a Gram-matrix rank argument that forces balanced bipartite components, and a structural analysis of the two resulting cases.", "authors": [ { "name": "James Chang" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B40" ] }, "formalization": { "theorem_names": [ "Covering.Palomar.covering_24_14_4_lower_bound", "Covering.Palomar.covering_25_15_5_lower_bound" ] }, "id": "PALOMAR-2026-10-04-000003", "path": "entries/PALOMAR-2026-10-04-000003-v1.json", "preservation": { "repositories": [ { "commit": "17bca0067a6bfa8c6c25669d97fca1190d20c8a3", "fork_repository": "PalomarArchive/jamesyc--covering--2d6067e8871e", "source_repository": "jamesyc/covering" } ] }, "published_at": "2026-10-04T01:55:46Z", "source": { "commit": "17bca0067a6bfa8c6c25669d97fca1190d20c8a3", "project_path": null, "repository": "jamesyc/covering" }, "status": "registered", "title": "Lower bounds for the covering numbers C(24,14,4) and C(25,15,5)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The full cycle characterization for arbitrary finite player types and arbitrary nonempty player-dependent action spaces. A real-valued exact potential exists if and only if deviators' payoff differences sum to zero around every finite closed path, every finite simple closed path, every simple four-cycle, or every two-player rectangle. The construction uses finite-subset induction and imposes no finiteness restriction on actions. Exact potentials are unique up to an additive constant. With finite actions, the cycle condition yields a pure Nash equilibrium. Adversarial examples distinguish matching pennies, infinite actions, unequal dependent actions, and the finite-action boundary.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91A10", "68V20" ] }, "formalization": { "theorem_names": [ "Potential.monderer_shapley_characterization", "Potential.exactPotential_unique", "Potential.fourCycle_exists_pureNash" ] }, "id": "PALOMAR-2026-10-04-000002", "path": "entries/PALOMAR-2026-10-04-000002-v1.json", "preservation": { "repositories": [ { "commit": "bce2e989bc8243f938a34ac0f027b57063c71380", "fork_repository": "PalomarArchive/Arthur742Ramos--monderer-shapley-potential-lean--5a16fe135330", "source_repository": "Arthur742Ramos/monderer-shapley-potential-lean" } ] }, "published_at": "2026-10-04T01:12:22Z", "source": { "commit": "bce2e989bc8243f938a34ac0f027b57063c71380", "project_path": null, "repository": "Arthur742Ramos/monderer-shapley-potential-lean" }, "status": "registered", "title": "Monderer\u2013Shapley Exact-Potential Characterization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Substantive Lean development for labelled distance-shell observations of anonymous integer populations. The focused 14-statement interface covers six result groups. Square grids have an exact four-corner interval-cut characterization of recovery of all populations of mass at most two after one erased report, an equivalent distance-sum compression contract, and exact corner-constrained optimum ceil(3(n+1)/2) for every n at least 3. For every positive q, the actual-distance equalizer cardinality of Q_(4q) lies between 2^(4q-1)+2q-1 and 2^(4q-1)+2q; explicit prefixes also give the infinite contradiction to the sourced exponential formula. Further selected results prove the long-rectangle optimum, ordinary 2-domination on every positive even bishop board, two exact Petersen double-total-domination values, and a connected real fractional-face counterexample. The complete library is included, not an external or assumption-based wrapper. Supporting algebra and geometric lemmas remain proved but unselected. The public README explains mathematical scope, antecedents, and limitations without treating known ingredients or all supporting lemmas as separate original results.", "authors": [ { "name": "Lennart Rudolph" } ], "classification": { "arxiv": [ "math.CO", "cs.IT" ], "msc2020": [ "05C12", "05C69" ] }, "formalization": { "theorem_names": [ "PalomarResults.Grid.gridRobust_iff_intervalCuts", "PalomarResults.Grid.gridMomentRobust_iff_gridRobust", "PalomarResults.Grid.grid_robust_sensor_lower_bound", "PalomarResults.Grid.exact_corner_grid_robust_minimum_all", "PalomarResults.Hypercube.infinite_counterexample_family", "PalomarResults.Hypercube.odd_union_equalizer_iff_antipodal_cover", "PalomarResults.Hypercube.equidistant_dimension_within_one", "PalomarResults.Bishop.exact_two_domination", "PalomarResults.Petersen11.exact_domination", "PalomarResults.Petersen16.exact_domination", "PalomarResults.Fractional.graph_connected", "PalomarResults.Fractional.counterexample", "PalomarResults.Rectangle.gridRobust_iff_intervalCuts", "PalomarResults.Rectangle.exact_long_rectangle_robust_minimum" ] }, "id": "PALOMAR-2026-10-04-000001", "path": "entries/PALOMAR-2026-10-04-000001-v1.json", "preservation": { "repositories": [ { "commit": "636768718796b33b36d344d04b8e5f2f840633dd", "fork_repository": "PalomarArchive/lennrt--palomar-formalizations--21c40838d48e", "source_repository": "lennrt/palomar-formalizations" } ] }, "published_at": "2026-10-04T00:36:46Z", "source": { "commit": "636768718796b33b36d344d04b8e5f2f840633dd", "project_path": "zenodo-23125523/palomar", "repository": "lennrt/palomar-formalizations" }, "status": "registered", "title": "Distance-Shell Tomography: Geometric Certificates for Fault-Tolerant Sensing, Moment Compression, and Hypercube Equalization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A full constructive formalization of the Falmagne\u2013Block\u2013Marschak theorem for any finite nonempty alternative set, without a cardinality bound. A normalized stochastic choice rule on all nonempty menus is generated by a single probability distribution over complete strict rankings if and only if all Block\u2013Marschak alternating-sum coefficients are nonnegative. The proof derives Boolean-lattice flow conservation, constructs a finite ranking mixture by positive-path subtraction, and proves exact lower-contour probability identities by M\u00f6bius inversion. Compiled adversarial examples show that regularity alone is insufficient.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "math.CO" ], "msc2020": [ "91B08", "91B16" ] }, "formalization": { "theorem_names": [ "Falmagne.falmagne_blockMarschak" ] }, "id": "PALOMAR-2026-10-03-000008", "path": "entries/PALOMAR-2026-10-03-000008-v1.json", "preservation": { "repositories": [ { "commit": "e6078b9b11af9010386207890fb79b792617e65e", "fork_repository": "PalomarArchive/Arthur742Ramos--falmagne-block-marschak-lean--337131130ac4", "source_repository": "Arthur742Ramos/falmagne-block-marschak-lean" } ] }, "published_at": "2026-10-03T17:37:41Z", "source": { "commit": "e6078b9b11af9010386207890fb79b792617e65e", "project_path": null, "repository": "Arthur742Ramos/falmagne-block-marschak-lean" }, "status": "registered", "title": "Falmagne\u2013Block\u2013Marschak Random-Utility Characterization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Milgrom and Shannon (1994), Theorem 4, in both directions: for an arbitrary lattice X, partially ordered parameter space T, and real-valued objective, strong-set monotonicity of constrained maximizers over every pair of strongly ordered feasible subsets is equivalent to quasisupermodularity of every section and both weak and strict preference-preservation clauses of single crossing. The theorem imposes no finiteness, completeness, topology, continuity, compactness, nonemptiness, or optimizer existence assumptions. Supporting results and counterexamples distinguish crossing clauses and fixed-domain from varying-feasible-set monotonicity. Earlier Lean sufficiency results in Econlib and nonmathlib4 are explicitly cited.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "math.OC" ], "msc2020": [ "91B02", "06B05", "03B35" ] }, "formalization": { "theorem_names": [ "MS1994.monotonicity_iff" ] }, "id": "PALOMAR-2026-10-03-000007", "path": "entries/PALOMAR-2026-10-03-000007-v1.json", "preservation": { "repositories": [ { "commit": "50a69ed78003a84ea849ed72dd24e0146829c4a6", "fork_repository": "PalomarArchive/Arthur742Ramos--milgrom-shannon-monotonicity-lean--e48b5a4260cf", "source_repository": "Arthur742Ramos/milgrom-shannon-monotonicity-lean" } ] }, "published_at": "2026-10-03T14:13:52Z", "source": { "commit": "50a69ed78003a84ea849ed72dd24e0146829c4a6", "project_path": null, "repository": "Arthur742Ramos/milgrom-shannon-monotonicity-lean" }, "status": "registered", "title": "Milgrom\u2013Shannon's Monotonicity Theorem: the Full Characterization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of all the numbered statements of F. Marbach, \"Continuity of solutions to abstract linear control systems\" (2026), https://hexagonmath.org/2610.00022. For abstract linear control systems in the sense of Weiss (a strongly continuous semigroup T on a Banach space X and bounded input maps Phi_t : L^p(R_+; U) -> X satisfying the concatenation identity), the main results are: for p = infinity, every system is zero-class, i.e. the operator norm of Phi_t tends to zero as t -> 0+ (Theorem 1.5), and t -> Phi_t u is continuous on [0, infinity) for every u in L^infinity (Corollary 1.6), a positive answer to Weiss' 1989 Problem 2.4. No integral representation of the input maps is assumed. The proof packs nearly extremal controls into one bounded operator on l^infinity and uses Phillips' lemma (Lemma 2.2), which is proved here. The development also covers the continuity of semigroup orbits (Proposition 1.2), the classical case p < infinity (Proposition 1.3), and the examples of Section 3: a non zero-class system for p < infinity (Proposition 3.1), an exotic invariant mean on periodic L^infinity functions (Lemma 3.2), and a scalar system with p = infinity without integral representation and not E_F-admissible for any Young function F (Proposition 3.3).", "authors": [ { "name": "Fr\u00e9d\u00e9ric Marbach" } ], "classification": { "arxiv": [ "math.OC", "math.FA" ], "msc2020": [ "93C25", "47D06", "93C05", "46B45" ] }, "formalization": { "theorem_names": [ "PalomarALCS.zero_class", "PalomarALCS.continuousOn_Phi" ] }, "id": "PALOMAR-2026-10-03-000001", "path": "entries/PALOMAR-2026-10-03-000001-v2.json", "preservation": { "repositories": [ { "commit": "bda56576f8b2730b73a55f273511c80fabbe56f4", "fork_repository": "PalomarArchive/frederic-marbach--alcs--85985fba62b5", "source_repository": "frederic-marbach/alcs" } ] }, "published_at": "2026-10-03T13:18:19Z", "source": { "commit": "bda56576f8b2730b73a55f273511c80fabbe56f4", "project_path": null, "repository": "frederic-marbach/alcs" }, "status": "registered", "title": "Continuity of solutions to abstract linear control systems", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A Lean formalization of two classical regularity theorems for the three-dimensional incompressible Navier\u2013Stokes equations. The Escauriaza\u2013Seregin\u2013\u0160ver\u00e1k theorem states that a Leray\u2013Hopf solution whose spatial L\u00b3 norm remains bounded in time is smooth at positive times and unique among Leray\u2013Hopf solutions with the same initial data. The Ladyzhenskaya\u2013Prodi\u2013Serrin theorem gives the corresponding conclusion under a family of mixed space-time integrability conditions. The development includes the local regularity theorem, L\u2075 space-time integrability under the endpoint L\u00b3 condition, and the supporting Carleman inequalities, unique continuation and backward uniqueness. It builds on the Caffarelli\u2013Kohn\u2013Nirenberg formalization and includes a manuscript presenting the proofs.", "authors": [ { "name": "Scott Armstrong" } ], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q30", "76D05", "35B65", "35D30", "35B60" ] }, "formalization": { "theorem_names": [ "ESSChallenge.essLocal", "ESSChallenge.essGlobal", "ESSChallenge.essL5Unique", "ESSChallenge.ladyzhenskayaProdiSerrin", "ESSChallenge.essSmooth", "ESSChallenge.serrinCriterion" ] }, "id": "PALOMAR-2026-10-03-000006", "path": "entries/PALOMAR-2026-10-03-000006-v1.json", "preservation": { "repositories": [ { "commit": "ae1a0e46086b9b16179dea76b41da10b01845151", "fork_repository": "PalomarArchive/scottnarmstrong--EscauriazaSereginSverak--c1d5d82817d7", "source_repository": "scottnarmstrong/EscauriazaSereginSverak" } ] }, "published_at": "2026-10-03T12:58:38Z", "source": { "commit": "ae1a0e46086b9b16179dea76b41da10b01845151", "project_path": null, "repository": "scottnarmstrong/EscauriazaSereginSverak" }, "status": "registered", "title": "Escauriaza\u2013Seregin\u2013\u0160ver\u00e1k and Ladyzhenskaya\u2013Prodi\u2013Serrin regularity", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Kamenica and Gentzkow's finite-state persuasion theorem from economic primitives: nonempty compact metric actions, continuous sender and receiver utilities, and a full-support common prior. Receiver best replies and sender-preferred tie actions exist. The induced sender payoff is bounded and upper semicontinuous. Finite signals realize exactly finite Bayes-plausible posterior splittings, with prescribed posteriors recovered only at nonnull messages. An optimal finite signal with at most the number of states plus two messages exists and dominates every finite experiment. Its value is the least concave majorant of the induced payoff on the entire belief simplex; strict persuasion gain is exactly the concavification gap. This covers the signal/splitting part of Proposition 1 and Corollary 2, including attainment. The straightforward recommendation branch and compact-state extension are outside this scope. Related finite splitting and abstract attainment are already formalized in Daniel Lyng's Econlib; no first-formalization or novelty claim is made.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91A26", "91B44", "52A20" ] }, "formalization": { "theorem_names": [ "BayesianPersuasion.economic_payoff_regular", "BayesianPersuasion.finite_signal_splitting", "BayesianPersuasion.optimal_information_and_concavification", "BayesianPersuasion.persuasion_gain_iff" ] }, "id": "PALOMAR-2026-10-03-000005", "path": "entries/PALOMAR-2026-10-03-000005-v1.json", "preservation": { "repositories": [ { "commit": "6bb350283717128bcd625daf93da18fe199e13d9", "fork_repository": "PalomarArchive/Arthur742Ramos--bayesian-persuasion-lean--b0746beaf304", "source_repository": "Arthur742Ramos/bayesian-persuasion-lean" } ] }, "published_at": "2026-10-03T12:27:03Z", "source": { "commit": "6bb350283717128bcd625daf93da18fe199e13d9", "project_path": null, "repository": "Arthur742Ramos/bayesian-persuasion-lean" }, "status": "registered", "title": "Optimal Information and Concavification in Bayesian Persuasion", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the main results of Constantin, Ignatova and Vicol on the regularity of asymptotically axisymmetric solutions of the three-dimensional Navier-Stokes equations with analytic forcing (arXiv:2609.20803). A suitable weak solution on the unit parabolic cylinder, smooth before the blow-up time, whose force is bounded in C^2 up to the blow-up time and real analytic in space, whose angular mean satisfies anisotropic Type II bounds, and which is exactly axisymmetric on a ball around the axis at every time, is axisymmetric and regular at the putative singular point (Theorem 1.1); the axisymmetric theorem (Theorem 1.3), the meridional smallness proposition, the consequences for the force (Corollary 2.3) and the lemmas of Sections 3 and 5 are formalized with it. The interior analyticity theorem of Kahane, the interior estimates of Serrin and the regularity criterion of Gustafson, Kang and Tsai, in the forms the paper uses, are proved rather than assumed. The thirteen results use only the standard axioms and build on the Lean formalization of the Caffarelli-Kohn-Nirenberg theorem; five of them are restated in a Mathlib-only comparator challenge and proved from the library theorems.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Vlad Vicol" } ], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q30", "35B44", "35Q31" ] }, "formalization": { "theorem_names": [ "CIVChallenge.mainTheorem", "CIVChallenge.axisymmetricTheorem", "CIVChallenge.meridionalSmallness", "CIVChallenge.interiorAnalyticity", "CIVChallenge.forceUnderCore" ] }, "id": "PALOMAR-2026-10-03-000004", "path": "entries/PALOMAR-2026-10-03-000004-v1.json", "preservation": { "repositories": [ { "commit": "9f539dc7bf7ddd414d3ef81df539e1bef229f681", "fork_repository": "PalomarArchive/scottnarmstrong--CIVAxisymmetric--a71f9ac656b8", "source_repository": "scottnarmstrong/CIVAxisymmetric" } ] }, "published_at": "2026-10-03T10:24:29Z", "source": { "commit": "9f539dc7bf7ddd414d3ef81df539e1bef229f681", "project_path": null, "repository": "scottnarmstrong/CIVAxisymmetric" }, "status": "registered", "title": "Regularity of asymptotically axisymmetric Navier\u2013Stokes solutions with analytic forcing", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalisation of a hitting time theorem of Wanfang Chen, Jared Le\u00f3n and Xizhi Liu. For every fixed integer r \u2265 3, assume all r-element subsets of n vertices are exposed in a uniformly random order. As n tends to infinity through multiples of r \u2212 1, with probability tending to one, a loose Hamilton cycle first emerges exactly when the last isolated vertex disappears.", "authors": [ { "name": "Jared Le\u00f3n" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C65", "05C80" ] }, "formalization": { "theorem_names": [ "HittingTimeLooseHamilton.main_result" ] }, "id": "PALOMAR-2026-10-03-000003", "path": "entries/PALOMAR-2026-10-03-000003-v1.json", "preservation": { "repositories": [ { "commit": "594001afb0fb7d80c581d0e2998a98355e87143c", "fork_repository": "PalomarArchive/WozMit--hitting-time-loose-hamilton--76aa0aa185d8", "source_repository": "WozMit/hitting-time-loose-hamilton" } ] }, "published_at": "2026-10-03T07:27:23Z", "source": { "commit": "594001afb0fb7d80c581d0e2998a98355e87143c", "project_path": null, "repository": "WozMit/hitting-time-loose-hamilton" }, "status": "registered", "title": "The hitting time for loose Hamilton cycles in r-graphs", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A general two-alternative implementation characterization on own-type domains containing an open ball: affine private and social valuation differences, nonzero social linear form, one active welfare cut, and a fixed independent opposing probability law force nonnegative coefficient alignment. Arbitrary measurable efficient lotteries have integrable Bayesian incentive compatible transfers exactly under alignment, with explicit ex-post truthful-opponent transfers and a measurable efficient allocation witness. Arbitrary distributions including atoms are covered. A genuine independent uniform-square auction model rules out efficiency even under only agent 0 incentive compatibility for every a,b>0. The separately proved finite sharp welfare/error frontier and original example corollary are retained. This does not formalize arbitrary-alternative Theorem 4.3 or genericity, and makes no novelty claim.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "JM.Quantitative.welfare_incentive_bound", "JM.Sharp.welfare_incentive_frontier", "JM.Sharp.frontier_attained", "JM.Sharp.feasible_iff", "JM.Sharp.exact_bic_loss", "JM.Sharp.exact_bic_attained", "JM.jehiel_moldovanu_impossibility", "JM.Binary.utility_integrable", "JM.Binary.congruence", "JM.Binary.aligned_ex_post", "JM.Binary.implementation_iff", "JM.Binary.efficient_bic_exists_iff", "JM.ContinuousAuction.uniform_probability", "JM.ContinuousAuction.prior_probability", "JM.ContinuousAuction.joint_probability", "JM.ContinuousAuction.welfare_difference", "JM.ContinuousAuction.utility_integrable", "JM.ContinuousAuction.impossibility" ] }, "id": "PALOMAR-2026-10-03-000002", "path": "entries/PALOMAR-2026-10-03-000002-v1.json", "preservation": { "repositories": [ { "commit": "4ab48f369106f8e7304ea62d6d9264a736ebf7b6", "fork_repository": "PalomarArchive/Arthur742Ramos--jehiel-moldovanu-lean--f6da463ab7ef", "source_repository": "Arthur742Ramos/jehiel-moldovanu-lean" } ] }, "published_at": "2026-10-03T01:40:47Z", "source": { "commit": "4ab48f369106f8e7304ea62d6d9264a736ebf7b6", "project_path": null, "repository": "Arthur742Ramos/jehiel-moldovanu-lean" }, "status": "registered", "title": "Continuous binary implementation congruence and sharp finite welfare frontier", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalisation of p-adic valuation laws for alternating cofactors C_d(p^a) = (p^(ad)+1)/(p^a+1) with odd d. It proves finite-window coefficient scaling and an exact pre-extremal restricted-binomial-gcd formula, identifies the sparse endpoint, proves the least extremal row T_p(C_d(p^a)) = p^(ad)+1 for odd d >= 5, and gives the full cubic classification for p^(2a)-p^a+1. The Palomar Challenge records coefficient scaling, the exact bounded gcd formula, the odd-degree least-row theorem, and the cubic classification.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11B65" ] }, "formalization": { "theorem_names": [ "PascalCofactorsPalomar.coefficient_scaling", "PascalCofactorsPalomar.restricted_gcd_valuation_bounded", "PascalCofactorsPalomar.targetA_d", "PascalCofactorsPalomar.targetC" ] }, "id": "PALOMAR-2026-10-02-000014", "path": "entries/PALOMAR-2026-10-02-000014-v1.json", "preservation": { "repositories": [ { "commit": "02a52e71a0a5ab1e77824490d0c47650d4a45691", "fork_repository": "PalomarArchive/jfairfaxball-348--pascal-cofactors--9d6b3a0fe1db", "source_repository": "jfairfaxball-348/pascal-cofactors" } ] }, "published_at": "2026-10-02T20:32:00Z", "source": { "commit": "02a52e71a0a5ab1e77824490d0c47650d4a45691", "project_path": null, "repository": "jfairfaxball-348/pascal-cofactors" }, "status": "registered", "title": "Pascal Cofactors", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 proof of the Cook\u2013Levin theorem and of the NP-completeness of several variants of satisfiability, for an explicit machine model, using only Lean's core library. The compared theorems state that the following languages are NP-complete: SAT, the set of binary codes of satisfiable CNF formulas (Complexity.sat_np_complete); 3-SAT, with at most three literals per clause (threeSAT_np_complete); exactly-3-SAT, with exactly three literals per clause on three distinct variables (exactThreeSAT_np_complete); (\u22641,\u22642)-SAT, with two or three literals per clause and every variable occurring at most once positively and at most twice negatively (leOneLeTwoSAT_np_complete); SAT with variable indices written in binary (binarySAT_np_complete); planar 3-SAT, whose variable\u2013clause incidence graph is planar (planarThreeSAT_np_complete); and Lichtenstein's planar 3-SAT, whose incidence graph stays planar with the cycle through the variables added (cyclePlanarThreeSAT_np_complete). Planarity is Gonthier's from the formal proof of the Four Color Theorem: a graph is planar when a hypermap on its half-edges embeds it with genus zero, the genus being computed from the Euler formula. NP-complete means: in NP, and every language in NP reduces to it by a polynomial-time many-one (Karp) reduction. A further compared theorem, inNP_iff_nondeterministicPolyTime, proves that NP, defined by verifiers, is the class of languages accepted in polynomial time by nondeterministic machines with two transition tables. Languages are sets of binary words. Computation is by deterministic Turing machines with a single two-way infinite tape over a four-symbol alphabet, and time is the number of machine steps. NP is defined by polynomial-time verifiers and polynomially bounded certificates, the standard certificate characterization; a reduction is a function computed by such a machine within a polynomial number of steps. Formulas are coded with unary variable indices and list lengths (binary indices for the binary variant); words that code no formula belong to none of the languages.\nFor an arbitrary NP verifier, the hardness proof builds a CNF formula describing the computation tableau of the verifier on the input and an unknown certificate; it is satisfiable exactly when some certificate makes the verifier accept. The reduction is itself implemented as a machine: it is written in a small language of loops over unary counters, compiled through finite programs on Boolean stacks to the single-tape model, and its running time is proved by counting the executed steps. SAT is reduced to 3-SAT by replacing each clause with a chain of clauses of at most three literals linked by fresh variables, and the reductions are composed as machines. Exactly-3-SAT and (\u22641,\u22642)-SAT are reached by transforming the Cook\u2013Levin formula generator itself: every literal occurrence gets its own copy variable, a cycle of implications links the copies of each variable, long clauses are split by chains, and short clauses are padded. The binary variant is reached by a machine converting unary indices to binary. For planar 3-SAT, the Cook\u2013Levin formula is turned into a grid formula in the style of Lichtenstein (one column of crossover gadgets per clause, copies of each variable joined by nested two-literal clauses), proved equisatisfiable and planar with its variable cycle through a general lemma on formulas drawn as combs; the plain version subdivides the cycle edges. Membership in NP is proved with explicit verifier machines for every language; for the planar versions the certificate also describes a planar embedding. For the nondeterministic characterization, a deterministic verifier simulates a nondeterministic machine with its choices read from the certificate, and conversely a nondeterministic machine guesses a certificate in a unary budget and runs the verifier. The development was written by AI under human direction: GPT-6 Astra (OpenAI) produced the proof, and Claude Opus 5.5 (Anthropic) ran its full verification, revised the statement for auditability and prepared it for Palomar, with the variants and the nondeterministic characterization proved by Claude agents. The proof uses only the axioms propext, Classical.choice and Quot.sound.", "authors": [ { "name": "Robert D. Barish" } ], "classification": { "arxiv": [ "cs.CC", "cs.LO" ], "msc2020": [ "68Q17", "68Q15", "03D15", "68Q04", "68V20" ] }, "formalization": { "theorem_names": [ "Complexity.sat_np_complete", "Complexity.threeSAT_np_complete", "Complexity.exactThreeSAT_np_complete", "Complexity.leOneLeTwoSAT_np_complete", "Complexity.binarySAT_np_complete", "Complexity.planarThreeSAT_np_complete", "Complexity.cyclePlanarThreeSAT_np_complete", "Complexity.inNP_iff_nondeterministicPolyTime" ] }, "id": "PALOMAR-2026-10-02-000013", "path": "entries/PALOMAR-2026-10-02-000013-v1.json", "preservation": { "repositories": [ { "commit": "3e603e795b44fedcf7d6dd65b7d1e30547ead925", "fork_repository": "PalomarArchive/RBarish-UTokyo--NPCompleteness-Lean4--67a5f2be4177", "source_repository": "RBarish-UTokyo/NPCompleteness-Lean4" } ] }, "published_at": "2026-10-02T19:47:25Z", "source": { "commit": "3e603e795b44fedcf7d6dd65b7d1e30547ead925", "project_path": null, "repository": "RBarish-UTokyo/NPCompleteness-Lean4" }, "status": "registered", "title": "NP-completeness of SAT, 3-SAT and their variants (the Cook\u2013Levin theorem) in Lean 4", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Hordijk and Steel (https://arxiv.org/abs/1605.03919, Section 7, Eq. (21)) conjectured a zero-one threshold for RAF emergence at linear mean catalysis. An interior limiting law is established for the reversible binary-polymer model with monomer/dimer food and reaction channels obtained by identifying the split descriptions u + v and v + u precisely when uv = vu. Each molecule independently catalyzes each channel, with one catalysis coordinate shared by both directions. For every real sequence f(n)/n \u2192 \u03bb > 0, the clamped channel parameter eventually equals f(n)/|J_n|, and the probability of a reflexively autocatalytic and food-generated (RAF) set converges to a limit L(\u03bb) depending only on \u03bb. The function L is nondecreasing and satisfies 0 < L(\u03bb) \u2264 1 \u2212 exp(\u221234\u03bb) < 1 at every positive finite intensity. Thus a zero-one step threshold is excluded in this channel model.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.PR", "q-bio.MN" ], "msc2020": [ "60K35", "92E20", "05C80" ] }, "formalization": { "theorem_names": [ "RAFReactionQuotient.palomar_quotient_critical_window" ] }, "id": "PALOMAR-2026-10-02-000012", "path": "entries/PALOMAR-2026-10-02-000012-v1.json", "preservation": { "repositories": [ { "commit": "12b505dcc1ccc5ee2d06b32692af201d452a2a7c", "fork_repository": "PalomarArchive/mmislan--palomar-p036-quotient-channel-raf-limit--e02bae3b1dd7", "source_repository": "mmislan/palomar-p036-quotient-channel-raf-limit" } ] }, "published_at": "2026-10-02T19:32:01Z", "source": { "commit": "12b505dcc1ccc5ee2d06b32692af201d452a2a7c", "project_path": null, "repository": "mmislan/palomar-p036-quotient-channel-raf-limit" }, "status": "registered", "title": "Critical-window emergence of autocatalytic sets under reaction-channel quotienting", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Finite Gershkov\u2013Goeree\u2013Kushnir\u2013Moldovanu\u2013Shi (2013) Theorem 2 for arbitrary finite agents and alternatives, heterogeneous ordered finite scalar type supports, arbitrary independent positive support masses, and linear private values with nonnegative slopes and arbitrary constants. The library constructs a feasible dominant-strategy mechanism preserving every support type's interim expected utility and ex ante social surplus. Weighted monotone lifting is proved by a compact feasible quadratic minimizer and an improving perturbation. Adjacent transfer bounds imply full-report IC by telescoping. Selected results are finite equivalence, weighted monotone lifting and optimizer monotonicity; scalar IC and support-restriction facts remain in the library as supporting lemmas. Validation evidence is recorded separately for the exact candidate.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91A27", "91A10", "91B44", "03B35" ] }, "formalization": { "theorem_names": [ "GGKMS.finite_bayesian_dominant_equivalence", "GGKMS.weighted_monotone_lifting", "GGKMS.minimizer_monotone" ] }, "id": "PALOMAR-2026-10-02-000010", "path": "entries/PALOMAR-2026-10-02-000010-v1.json", "preservation": { "repositories": [ { "commit": "0e1bed171ab66b194d5ee28ac02818b9a7bc5073", "fork_repository": "PalomarArchive/Arthur742Ramos--gershkov-bic-dic-lean--d98725fc9369", "source_repository": "Arthur742Ramos/gershkov-bic-dic-lean" } ] }, "published_at": "2026-10-02T18:37:38Z", "source": { "commit": "0e1bed171ab66b194d5ee28ac02818b9a7bc5073", "project_path": null, "repository": "Arthur742Ramos/gershkov-bic-dic-lean" }, "status": "registered", "title": "Finite Bayesian-to-Dominant-Strategy Equivalence", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of S. Armstrong and V. Vicol, \"Anomalous diffusion by fractal homogenization\" (arXiv:2305.05048). For every alpha in (0, 1/3) it constructs one divergence-free, Z^2-periodic drift b in C^0_t C^{0,alpha}_x intersect C^{0,alpha}_t C^0_x on [0,1] x T^2, and proves three properties of the advection-diffusion equation with this drift: anomalous dissipation of scalar variance for every mean-zero H^1 datum (Theorem 1.1 of the paper); a uniform C^{0,mu}([0,1]; L^2) bound by C times the H^1 norm of the datum along a set of diffusivities accumulating at zero (Remark r.LeBron.2); and non-uniqueness of vanishing-diffusivity limits for a class of data containing cos(2 pi n x_1) for infinitely many n (a corrected form of the paper's no-selection principle). The three properties are combined in one theorem, restated in a Mathlib-only comparator challenge. The formalization also proves existence and uniqueness of weak solutions and the classical facts the paper uses without proof.", "authors": [ { "name": "Scott Armstrong" }, { "name": "Vlad Vicol" } ], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35K15", "35B27", "76F25", "35Q35" ] }, "formalization": { "theorem_names": [ "AVChallenge.anomalous_dissipation_full" ] }, "id": "PALOMAR-2026-10-02-000009", "path": "entries/PALOMAR-2026-10-02-000009-v1.json", "preservation": { "repositories": [ { "commit": "c5b0e710c14cf41eca33dc2ae5d7aeceff829f99", "fork_repository": "PalomarArchive/scottnarmstrong--AVAnomalousDiffusion--6a23398d8017", "source_repository": "scottnarmstrong/AVAnomalousDiffusion" } ] }, "published_at": "2026-10-02T17:44:56Z", "source": { "commit": "c5b0e710c14cf41eca33dc2ae5d7aeceff829f99", "project_path": null, "repository": "scottnarmstrong/AVAnomalousDiffusion" }, "status": "registered", "title": "Anomalous diffusion by fractal homogenization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Bergemann\u2013Morris Theorem 1 for arbitrary finite players and states with heterogeneous action and original-signal spaces: a normalized decision rule is obedient if and only if a finite marginal-preserving information expansion supports a valid mixed Bayes\u2013Nash strategy profile inducing its joint law. The recommendation expansion and the converse for arbitrary extra signals and independent behavioral randomization are proved. Null fibers are handled without conditioning, and exact induction is equivalent to the paper's positive-support quotient. Zero prior masses are permitted, extending the paper's full-support assumption. The one-player decision-problem specialization and explicit examples are included. Theorem 2 proves that individual sufficiency is equivalent to BCE outcome inclusion across all finite basic games with arbitrary heterogeneous actions and utilities and full-support common priors, on a nonempty finite state space. Its converse constructs actual couplings from finite quadratic-scoring games and weighted compactness.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91A27", "91A10", "91B44", "03B35" ] }, "formalization": { "theorem_names": [ "BayesCorrelated.finite_characterization", "BayesCorrelated.mixedBNE_induces_BCE", "BayesCorrelated.expansion_normalized", "BayesCorrelated.expansion_null_fiber", "BayesCorrelated.strategyProduct_isProbability", "BayesCorrelated.induces_iff_positive_support", "BayesCorrelated.onePlayer_obedienceGain", "BayesCorrelated.onePlayer_characterization", "BayesCorrelated.information_order_characterization", "BayesCorrelated.individuallySufficient_outcome", "BayesCorrelated.moreIncentiveConstrained_outcome_any_universe", "BayesCorrelated.onePlayer_individual_sufficiency", "BayesCorrelated.onePlayer_information_order" ] }, "id": "PALOMAR-2026-10-02-000007", "path": "entries/PALOMAR-2026-10-02-000007-v1.json", "preservation": { "repositories": [ { "commit": "1f2d307e49d5dbcc33984709f9c7bf8691554fbc", "fork_repository": "PalomarArchive/Arthur742Ramos--bayes-correlated-equilibrium-lean--cecbf64a937e", "source_repository": "Arthur742Ramos/bayes-correlated-equilibrium-lean" } ] }, "published_at": "2026-10-02T15:22:28Z", "source": { "commit": "1f2d307e49d5dbcc33984709f9c7bf8691554fbc", "project_path": null, "repository": "Arthur742Ramos/bayes-correlated-equilibrium-lean" }, "status": "registered", "title": "Finite Bayes Correlated Equilibrium, Information Expansions and Information Order", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalization of the AGV (d'Aspremont-G\u00e9rard-Varet-Arrow) expected-externality mechanism: with independent types, ex post efficiency, Bayesian incentive compatibility, and budget balance are jointly attainable. Agents are indexed by Fin n with 2 \u2264 n; each agent has a finite nonempty type space with an independent PMF prior; alternatives form a finite nonempty type; values are private. The efficient rule xmax maximizes utilitarian welfare, and agvTransfer pays each agent its expected externality minus an equal share of the other agents' receipts. Interim individual rationality is not claimed.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "AGV.agv_efficient", "AGV.agv_bic", "AGV.agv_budget_balanced", "AGV.agv_theorem" ] }, "id": "PALOMAR-2026-10-02-000006", "path": "entries/PALOMAR-2026-10-02-000006-v1.json", "preservation": { "repositories": [ { "commit": "4e1f6b369836e6ec0a5443ea57424b5ed71b97d6", "fork_repository": "PalomarArchive/Arthur742Ramos--agv-mechanism-lean--2dd26d94a5d0", "source_repository": "Arthur742Ramos/agv-mechanism-lean" } ] }, "published_at": "2026-10-02T14:11:36Z", "source": { "commit": "4e1f6b369836e6ec0a5443ea57424b5ed71b97d6", "project_path": null, "repository": "Arthur742Ramos/agv-mechanism-lean" }, "status": "registered", "title": "agv-mechanism-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalization of Maskin's theorem on Nash implementation: necessity of Maskin monotonicity and sufficiency of monotonicity plus no-veto power via the canonical mechanism. Alternatives form a finite nonempty type, agents are indexed by Fin n, preferences are complete preorders, and the social choice correspondence is nonempty-valued. Sufficiency requires at least three agents; implementation is in pure strategies.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "Maskin.maskin_necessity", "Maskin.maskin_sufficiency" ] }, "id": "PALOMAR-2026-10-02-000005", "path": "entries/PALOMAR-2026-10-02-000005-v1.json", "preservation": { "repositories": [ { "commit": "9c041756dcee8e572b73f4c12b62e6c95322db09", "fork_repository": "PalomarArchive/Arthur742Ramos--maskin-nash-implementation-lean--e0bbd25de94a", "source_repository": "Arthur742Ramos/maskin-nash-implementation-lean" } ] }, "published_at": "2026-10-02T12:54:40Z", "source": { "commit": "9c041756dcee8e572b73f4c12b62e6c95322db09", "project_path": null, "repository": "Arthur742Ramos/maskin-nash-implementation-lean" }, "status": "registered", "title": "maskin-nash-implementation-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of the Herzog\u2013Sch\u00f6nheim conjecture (the index form, which implies the cardinality form asked in Erd\u0151s problem 274) for every solvable group, finite or infinite: if a solvable group is partitioned into finitely many, and at least two, left cosets of subgroups, then two of the subgroups have the same index (hence, in the cardinality form, the same cardinality). The conjecture for arbitrary groups is open and is not claimed. The infinite case reduces to a finite solvable quotient by the common core of the parts; the finite case is proved for a least counterexample by considering its largest prime divisor p, using the subgroup generated by the p-prime-part elements, divisibility restrictions on local indices obtained from fixed-point counts in a primitive solvable quotient, and an extra closure argument at p = 7. The statements use the same exact covering structure as the formal-conjectures file for Erd\u0151s problem 274.", "authors": [ { "name": "Murali Menon" } ], "classification": { "arxiv": [ "math.GR", "math.CO" ], "msc2020": [ "20D60", "20D10", "05B40" ] }, "formalization": { "theorem_names": [ "Erdos274.Palomar.herzog_schonheim_solvable", "Erdos274.Palomar.erdos_274_solvable" ] }, "id": "PALOMAR-2026-10-02-000004", "path": "entries/PALOMAR-2026-10-02-000004-v1.json", "preservation": { "repositories": [ { "commit": "d701decf37c225a3e5326fade4bd92121b1f93d4", "fork_repository": "PalomarArchive/Jostamon--erdos274-hs-solvable--2a7a53287c1d", "source_repository": "Jostamon/erdos274-hs-solvable" } ] }, "published_at": "2026-10-02T07:53:20Z", "source": { "commit": "d701decf37c225a3e5326fade4bd92121b1f93d4", "project_path": null, "repository": "Jostamon/erdos274-hs-solvable" }, "status": "registered", "title": "The Herzog\u2013Sch\u00f6nheim conjecture for solvable groups", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization of the Cremer-McLean full surplus extraction theorem (Econometrica, 1988): for a single indivisible object with finitely many bidders, finite type spaces, and a correlated joint prior with full support, if every bidder's family of conditional distributions over the others' types is convex independent (no conditional lies in the convex hull of the rest), then there exists a Bayesian incentive compatible, interim individually rational direct mechanism that allocates ex post efficiently and extracts the full surplus: every bidder's interim expected utility is zero, so expected seller revenue equals the expected maximum value. The proof hinges on constructing side-bet lotteries with zero own-conditional expectation and strictly positive cross-conditional expectation, obtained by strict separation of each conditional from the convex hull of the others, scaled to dominate any misreport gain.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "CremerMcLean.separating_lottery", "CremerMcLean.bic_of_good_lotteries", "CremerMcLean.full_extraction" ] }, "id": "PALOMAR-2026-10-02-000003", "path": "entries/PALOMAR-2026-10-02-000003-v1.json", "preservation": { "repositories": [ { "commit": "e0578ad905149bb579a2b4fe7930720055497c50", "fork_repository": "PalomarArchive/Arthur742Ramos--cremer-mclean-lean--b8c346df28ca", "source_repository": "Arthur742Ramos/cremer-mclean-lean" } ] }, "published_at": "2026-10-02T03:57:36Z", "source": { "commit": "e0578ad905149bb579a2b4fe7930720055497c50", "project_path": null, "repository": "Arthur742Ramos/cremer-mclean-lean" }, "status": "registered", "title": "Cremer-McLean Full Surplus Extraction Theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For n = 1, \u2026, 7: the least radius of a closed disk holding n unit squares with disjoint interiors, each placed and rotated freely, and every packing that attains it. The radii are \u221a2/2, \u221a5/2, 5\u221a17/16, \u221a2, \u221a(5/2), \u221aq* \u2248 1.68854 and \u221a13/2, where q* is a quartic irrational. Up to rotation about the disk centre and relabelling, the optimal packing is unique for n \u2264 6: the square, the 2 \u00d7 1 rectangle, the T, the 2 \u00d7 2 block, the plus, and for six squares a central square with four neighbours, two of them pushed along its sides, and a sixth square turned by \u03c0/4 between those two. For n = 7 the optimal packings form a three-parameter family: two columns of two squares and, between them, a column of three whose heights vary independently, at least 1 apart and within \u221a3 \u2212 1/2 of the disk centre. The proofs for three to six squares bound the arcs the squares occupy on a small circle about the disk centre. For six squares this finds the square that contains the disk centre; five fixed points label the others, and a weighted sum of separating inequalities, with estimates uniform in the angles of the squares, forces the optimal radius. For seven squares, each square avoiding the disk centre gets a direction from it, and in the optimal disk two disjoint such squares get directions at least \u03c0/3 apart. We found no earlier proof for five, six or seven squares, and no exact one for three, where a computer-assisted enclosure of the radius and the optimal arrangements was known. Four squares, uniqueness included, were settled by hand in 2026, by the same idea as here.", "authors": [ { "name": "The-Anh Vu-Le" } ], "classification": { "arxiv": [ "math.MG" ], "msc2020": [ "52C15" ] }, "formalization": { "theorem_names": [ "SquaresInCircles.optimal_radius", "SquaresInCircles.optimal_packings" ] }, "id": "PALOMAR-2026-10-02-000002", "path": "entries/PALOMAR-2026-10-02-000002-v1.json", "preservation": { "repositories": [ { "commit": "621f2340a12f661a51b613719107a75cf40ab03a", "fork_repository": "PalomarArchive/vltanh--lean4-squares-in-circles--0ba28607e811", "source_repository": "vltanh/lean4-squares-in-circles" } ] }, "published_at": "2026-10-02T02:40:53Z", "source": { "commit": "621f2340a12f661a51b613719107a75cf40ab03a", "project_path": null, "repository": "vltanh/lean4-squares-in-circles" }, "status": "registered", "title": "Optimal packings of one to seven unit squares in a disk", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of H. T. Pham and L. Sauermann, \"On Graham's rearrangement conjecture\" (arXiv:2602.15797v1). Graham conjectured in 1971 that for every prime p, every subset S of Z_p \\ {0} has an ordering whose partial sums are pairwise distinct. The formalization proves the paper's main theorem (Theorem 1.2): for every 0 < \u03b1 < 1 there is a constant C_\u03b1 > 0 such that, for every prime p, every S \u2286 Z_p \\ {0} with C_\u03b1 \u2264 |S| \u2264 p^(1-\u03b1) has such an ordering. It also proves the paper's anticoncentration bounds for the sum of a uniformly random m-element subset R of a set S \u2286 Z_p with |S| \u2265 2: the probability that the sum of R equals any given z is at most 1/p + C/(|S|\u221am), for an absolute constant C > 0, when C log|S| \u2264 m \u2264 10^-3 |S|/log|S| (Theorem 1.3), and, for every 0 < \u03b5 < 1, at most 1/p + C_\u03b5 \u221a(log|S|)/(|S|\u221am) for every positive integer m \u2264 (1-\u03b5)|S| (Corollary 1.4). Every result that the paper proves is formalized, following its argument, and so are the two hypergeometric tail bounds that its proofs take from the Chernoff bound of Janson, \u0141uczak and Ruci\u0144ski.", "authors": [ { "name": "The-Anh Vu-Le" } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B75", "05D40", "11B30", "60C05", "11P70", "05A05" ] }, "formalization": { "theorem_names": [ "PhamSauermann.theorem_1_2", "PhamSauermann.theorem_1_3", "PhamSauermann.corollary_1_4" ] }, "id": "PALOMAR-2026-10-02-000001", "path": "entries/PALOMAR-2026-10-02-000001-v1.json", "preservation": { "repositories": [ { "commit": "e4d1034e026688b6ef3966dde89437d823a4262e", "fork_repository": "PalomarArchive/vltanh--lean4-graham-rearrangement-conjecture--7f9d98b8f7e8", "source_repository": "vltanh/lean4-graham-rearrangement-conjecture" } ] }, "published_at": "2026-10-02T02:31:30Z", "source": { "commit": "e4d1034e026688b6ef3966dde89437d823a4262e", "project_path": null, "repository": "vltanh/lean4-graham-rearrangement-conjecture" }, "status": "registered", "title": "On Graham's rearrangement conjecture (Pham\u2013Sauermann)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Two components of the proof, in the paper \"Linnik's constant is at most 3.99\" (paper/linnik399.pdf in this repository), that every reduced residue class modulo q contains a prime below q^3.99 for all large q. That bound itself is NOT formally verified here: the paper's zero-location arguments, case analysis and exterior regimes have conventional proofs and computer checks only. (1) The graded near lemma: fix test data (a detector f satisfying Heath-Brown's Condition 1, a Gram test g satisfying his Conditions 1 and 2, a Selberg-sieve weight with heights h_k >= 0 on cells [t_k, t_{k+1}) with t_0 > 1/3 + 2 eps', finitely many nonnegative anchor offsets, a positive lower bound for the weight omega where f is nonzero, and I_B > 0) and eta > 0. For all large q, take a finite family of entries (chi_j, gamma_j, delta_j) of non-principal characters mod q with heights |gamma_j| <= T <= (log q)/3 and a safe anchor at height 2T+1, whose characters are pairwise distinct whenever some sieve height h_k is nonzero (condition (iii) of the paper's Theorem 8.4; without sieve weights a character may repeat, and repeated characters enter through the pair excess e_j). Then the prime-sum responses satisfy (sum_j a_j R_j)_+^2 <= (1+eta) I_B [sum_j (D(delta_j) - d_1 + e_j) a_j^2 + (d_1 + eta)(sum_j a_j)^2] for all a_j >= 0. This is a weighted version of Heath-Brown's Lemma 12.1 (1992), which Heath-Brown remarks (section 16, item 9) no neat way of achieving \"seems available\". Its zero form: the response lemma (a response is bounded below by the zeros it keeps, via the local explicit formula), the threshold form exists tau: sum_j (v_j - tau)_+^2 / D_j <= 1 - tau^2/d with features computed from kept zeros, provided every diagonal numerator D(delta_j) - d_1 + e_j is positive, the entry bounds F(hi - s_j) of the rows, the normalization used by the leaf builders, and the vanishing of the pair excess for separated same-character pairs. (2) The leaf LPs: an exact integer checker for the certificates of the leaf linear programs (bisection trees over threshold boxes, first-order and tangent relaxations with safe rounding, exclusions, weak duality with a free dual) is proved sound: if it accepts a certificate with value v, every feasible point of the LP has objective at most v < 10^16. A bridge lemma turns the threshold form into the LP's near rows. Seven certificates (six leaves of the 3.99 corpus and the program of the large first-zero range) are checked by the kernel (decide +kernel) in the repository (not part of the compared theorems). The analytic part is conditional on four published estimates, stated as explicit hypotheses in the Challenge, each no stronger than its source: Xylouris's Lemmas 3.1 and 3.2 (= Heath-Brown's Lemmas 5.3 and 5.2), Burgess's character-sum bound with k = 3 as printed for primitive characters (Heath-Brown's Lemma 2.1; the form for all non-principal characters is derived), and Graham's asymptotic for Selberg-sieve sums. Mertens' estimate, Abel and partial summation, the Selberg majorant, the decay of Laplace transforms, the finiteness and multiplicities of zeros in a disc, the equality of multiplicities of conjugate zeros for real characters, and all error bookkeeping are proved.", "authors": [ { "name": "Eric Naslund" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M20", "11M26", "11N13", "11N35", "11Y35", "68V15", "90C05" ] }, "formalization": { "theorem_names": [ "GradedNear.graded_near_lemma", "GradedNear.response_lemma", "GradedNear.graded_near_threshold", "GradedNear.keptBound_single", "GradedNear.keptBound_pair", "GradedNear.single_zero_threshold", "GradedNear.keptBound_double", "GradedNear.keptBound_pair_neg", "GradedNear.zeroTerm_conj", "GradedNear.builder_threshold", "GradedNear.pairExcess_eq_zero", "GradedNear.burgessBound_of_primitive", "GradedNear.zmult_conj", "GradedNear.keptBound_pair_neg_real", "GradedNear.Cert.near_row_of_bins", "GradedNear.Cert.checkLeaf_sound", "GradedNear.Cert.checkLeaf_lt", "GradedNear.Cert.certified" ] }, "id": "PALOMAR-2026-10-01-000020", "path": "entries/PALOMAR-2026-10-01-000020-v1.json", "preservation": { "repositories": [ { "commit": "ecdbe3e3e08ad499d2c749ff417cc461f57334c5", "fork_repository": "PalomarArchive/enaslund--linniks-constant-3.99--88c18998ca3f", "source_repository": "enaslund/linniks-constant-3.99" } ] }, "published_at": "2026-10-01T22:52:17Z", "source": { "commit": "ecdbe3e3e08ad499d2c749ff417cc461f57334c5", "project_path": null, "repository": "enaslund/linniks-constant-3.99" }, "status": "registered", "title": "Linnik's constant is at most 3.99: A formalization of a weighted version of Heath-Brown's near-density estimate, and of the proof's certificate checker", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For arbitrary finite bidders with heterogeneous finite type spaces, a nonnegative normalized real joint prior, and an interim allocation rule with values in [0,1], a randomized allocation of at most one item realizes the weighted interim masses if and only if every collection of bidder-specific type subsets satisfies Border's union-probability inequality. Weighted feasibility is proved equivalent to conditional feasibility on positive marginal support. Null-type interim values remain unrestricted within [0,1]. For a product of nonnegative normalized distributions, the independent-prior corollary has right side 1 minus the product of the complementary marginal probabilities. Full real-valued sufficiency is proved by finite threshold decomposition and compact convex separation; no sufficiency oracle is assumed.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91B26", "90C05", "03B35" ] }, "formalization": { "theorem_names": [ "Border.border_feasibility", "Border.feasible_iff_conditional", "Border.border_conditional_feasibility", "Border.independent_border_feasibility", "Border.independent_border_conditional_feasibility", "Border.allocation_le_one" ] }, "id": "PALOMAR-2026-10-01-000019", "path": "entries/PALOMAR-2026-10-01-000019-v1.json", "preservation": { "repositories": [ { "commit": "690e62ec02d68f1e993fd53e46963d79b7993a2b", "fork_repository": "PalomarArchive/Arthur742Ramos--border-auction-feasibility-lean--13f9c1532ffa", "source_repository": "Arthur742Ramos/border-auction-feasibility-lean" } ] }, "published_at": "2026-10-01T22:42:21Z", "source": { "commit": "690e62ec02d68f1e993fd53e46963d79b7993a2b", "project_path": null, "repository": "Arthur742Ramos/border-auction-feasibility-lean" }, "status": "registered", "title": "Border's Finite Auction Feasibility Theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Hordijk and Steel (https://arxiv.org/abs/1605.03919, Section 7, Eq. (21)) conjectured a zero-one threshold for RAF emergence at linear mean catalysis. In the uniform reversible binary-polymer model with split-position reaction identities and food consisting of monomers and dimers, a nontrivial limiting law is established. Each molecule independently catalyzes each reaction, with one catalysis coordinate shared by both directions. If the mean catalysis parameter f(n) satisfies f(n)/n \u2192 \u03bb > 0, the probability of a reflexively autocatalytic and food-generated (RAF) set converges to L(\u03bb). The function L is continuous and nondecreasing, with 0 < L(\u03bb) \u2264 1 \u2212 exp(\u221236\u03bb) < 1. Thus no finite positive threshold separates limiting RAF probability zero from one in this model.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.PR", "q-bio.MN" ], "msc2020": [ "60K35", "92E20", "05C80" ] }, "formalization": { "theorem_names": [ "HordijkSteelThreshold.palomar_critical_window_variable_intensity" ] }, "id": "PALOMAR-2026-10-01-000017", "path": "entries/PALOMAR-2026-10-01-000017-v1.json", "preservation": { "repositories": [ { "commit": "f90984f2b107aad62c6efcceb0d8cb9da0fe62a4", "fork_repository": "PalomarArchive/mmislan--palomar-p002-binary-polymer-critical-window--3e56abbe9c7e", "source_repository": "mmislan/palomar-p002-binary-polymer-critical-window" } ] }, "published_at": "2026-10-01T21:33:55Z", "source": { "commit": "f90984f2b107aad62c6efcceb0d8cb9da0fe62a4", "project_path": null, "repository": "mmislan/palomar-p002-binary-polymer-critical-window" }, "status": "registered", "title": "A nontrivial critical window for RAF emergence in the binary polymer model", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For any nonempty report type T, any outcome type X, any real-valued valuation v : T -> X -> Real and allocation f : T -> X, DSIC payments exist exactly when every finite closed walk has nonnegative sum of weights l(s,t) = v(t,f(t)) - v(t,f(s)). The proof constructs payments as the infimum of finite path weights from a fixed anchor. Direct edges establish nonemptiness; a return edge supplies an explicit real lower bound before using the infimum's order properties. The payments are zero at the anchor and satisfy every implementing difference inequality. The characterization also holds for arbitrary agents with heterogeneous nonempty report spaces and decidable agent equality, by fixing other reports and assembling anchored slice payments. A three-type example satisfies all two-cycle inequalities but has a negative three-cycle and admits no DSIC payment.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91B03", "91B16", "03B35" ] }, "formalization": { "theorem_names": [ "Rochet.pathWeights_nonempty", "Rochet.pathWeights_lower_bound", "Rochet.pathWeights_bddBelow", "Rochet.potential_anchor", "Rochet.potential_edge", "Rochet.potential_bounds", "Rochet.dsic_iff_edge", "Rochet.anchoredPayment_spec", "Rochet.rochet", "Rochet.payment_eq_of_same_outcome", "Rochet.multiPayment_spec", "Rochet.multi_rochet", "Rochet.example_two_cycles", "Rochet.example_three_cycle", "Rochet.example_not_implementable" ] }, "id": "PALOMAR-2026-10-01-000016", "path": "entries/PALOMAR-2026-10-01-000016-v1.json", "preservation": { "repositories": [ { "commit": "88171ba4b232466c510d45863a91626a1c546a87", "fork_repository": "PalomarArchive/Arthur742Ramos--rochet-cyclic-monotonicity-lean--61bec3b9b9da", "source_repository": "Arthur742Ramos/rochet-cyclic-monotonicity-lean" } ] }, "published_at": "2026-10-01T19:48:58Z", "source": { "commit": "88171ba4b232466c510d45863a91626a1c546a87", "project_path": null, "repository": "Arthur742Ramos/rochet-cyclic-monotonicity-lean" }, "status": "registered", "title": "Rochet's Cyclic-Monotonicity Characterization", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of Gale's top trading cycles algorithm (1974) for the Shapley-Scarf housing market. Finitely many agents (Fin n) each own exactly one house (the endowment is a bijection endow : Fin n \u2243 Fin n) and have strict preferences over houses, given most-preferred-first as per-agent permutations rank i : Fin n \u2243 Fin n; agent i strictly prefers house a over house b iff (rank i).symm a < (rank i).symm b. The comparator-checked results are TTC.Palomar.ttcTerminates: the TTC procedure (each remaining agent points to the owner of their favourite remaining house, each house points to its owner; each round removes the single directed cycle reachable from a chosen remaining agent, whose members trade and leave) removes every agent within n rounds, since each round deletes at least one nonempty directed cycle from the finite functional pointing graph. This one-cycle-at-a-time form computes the same allocation as the simultaneous version (see the remark in README.md); TTC.Palomar.ttcInCore: the TTC allocation is in the ordinary core, i.e. no coalition S can reallocate its own endowments (Set.BijOn y S (M.endow '' S)) so that every member strictly prefers the deviation; TTC.Palomar.ttcUniqueStrictCore: strict-core uniqueness, i.e. for (x : Fin n \u2192 Fin n) with (hxb : Function.Bijective x) and (hx : StrictCore M x) the conclusion is x = ttcAllocation M. This is uniqueness among bijective strict-core allocations (no coalition can weakly block), not a singleton claim about the ordinary core. The library also proves the TTC allocation Pareto optimal (TTC.ttcPareto), individually rational (TTC.ttcIR), in the strict core (TTC.ttcInStrictCore), and strategyproof (TTC.ttcStrategyproof, Roth 1982: no unilateral misreport is profitable); these four are library-only and are not on the comparator surface. The Palomar registry API was searched on 2026-10-01 for \"top trading cycles\", \"housing market\", \"shapley scarf\", \"TTC\", and \"trading cycles\" and returned no existing TTC formalization among 397 entries; this is a dated, limited observation, not a universal prior-art claim.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B68" ] }, "formalization": { "theorem_names": [ "TTC.Palomar.ttcTerminates", "TTC.Palomar.ttcInCore", "TTC.Palomar.ttcUniqueStrictCore" ] }, "id": "PALOMAR-2026-10-01-000015", "path": "entries/PALOMAR-2026-10-01-000015-v1.json", "preservation": { "repositories": [ { "commit": "9fc2a2130983b494831f95b831b9cf28620cf556", "fork_repository": "PalomarArchive/Arthur742Ramos--top-trading-cycles-lean--0ea4ea731bf7", "source_repository": "Arthur742Ramos/top-trading-cycles-lean" } ] }, "published_at": "2026-10-01T18:31:20Z", "source": { "commit": "9fc2a2130983b494831f95b831b9cf28620cf556", "project_path": null, "repository": "Arthur742Ramos/top-trading-cycles-lean" }, "status": "registered", "title": "Gale's Top Trading Cycles (Shapley-Scarf Housing Market)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For a permutation pattern \u03c4, let Av_n(\u03c4) be the set of permutations of length n that avoid \u03c4. This formalization proves, for every n, that four recurrences from the source paper compute |Av_n(1342)| and |Av_n(12453)|. For 1342 they are the one-threshold recurrence W and the scalar kernel recurrence G, and for 12453 the literal two-threshold recurrence H and the transfer-kernel recurrence G. These are the cases d = 1 and d = 2 of the paper's counting recurrences for the classes Av(12...d \u2295 231), the direct sum of an increasing pattern and 231, whose kernel form gives polynomial-time algorithms for every fixed d (the complexity bounds themselves are not formalized). The counting sequence of Av(1342) has been known since B\u00f3na (1997). That of Av(12453) was previously known only up to n = 38, from a program of Biers-Ariel that evaluates the recurrence H, and the paper uses its kernel algorithm to compute it up to n = 150 (that computation is not formalized).", "authors": [ { "name": "Henning Arn\u00f3r Skeggi \u00dalfarsson" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A05", "05A15", "68W40", "68V15" ] }, "formalization": { "theorem_names": [ "Av12453.count_1342_literal", "Av12453.count_1342_kernel", "Av12453.count_12453_literal", "Av12453.count_12453_kernel" ] }, "id": "PALOMAR-2026-10-01-000014", "path": "entries/PALOMAR-2026-10-01-000014-v1.json", "preservation": { "repositories": [ { "commit": "d6133070300db0fd1d229f3e4566067ef1fc6efa", "fork_repository": "PalomarArchive/ulfarsson--public-12453--a16eaf45e498", "source_repository": "ulfarsson/public-12453" } ] }, "published_at": "2026-10-01T15:46:21Z", "source": { "commit": "d6133070300db0fd1d229f3e4566067ef1fc6efa", "project_path": "formal/Av12453", "repository": "ulfarsson/public-12453" }, "status": "registered", "title": "Counting recurrences for the permutation classes Av(1342) and Av(12453)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "In their study of pathologies in recursion categories, Montagna (1989) and Di Paola\u2013Montagna (1991) introduce the algebraic systems S\u2032 and S\u2032_T, respectively, and claim that they are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension T of Peano arithmetic (PA), we exhibit two unary programs whose partial functions are provably equal in T, separately at each standard input. Composing each after a program that searches for a T-proof of contradiction and returns its code yields programs that are not equivalent in this sense. An alternative proof uses the productivity of the complement of the diagonal halting set. Montagna's S\u2032 is the case T = PA. More generally, for consistent T \u2287 PA, pointwise provable equality is a composition congruence exactly when T proves every true \u03a0\u2070\u2081 sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable T \u2287 PA by G\u00f6del's second incompleteness theorem. For every extension T \u2287 PA, the least composition congruence containing pointwise provable equality is extensional equality if T is \u03a3\u2070\u2081-sound and the universal relation otherwise.", "authors": [ { "name": "Florian Lengyel" } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03D75", "03F40", "08A30", "18B10" ] }, "formalization": { "theorem_names": [ "FailureOfComposition.Palomar.obstruction_four_properties", "FailureOfComposition.Palomar.no_quotient_composition_productive", "FailureOfComposition.Palomar.no_quotient_composition_godel", "FailureOfComposition.Palomar.pi_one_characterization", "FailureOfComposition.Palomar.generated_congruence_classification", "FailureOfComposition.Palomar.weak_totality_counterexample", "FailureOfComposition.Palomar.range_counterexample_godel", "FailureOfComposition.Palomar.range_counterexample_productive", "FailureOfComposition.Palomar.generated_quotient_partial_recursive" ] }, "id": "PALOMAR-2026-10-01-000013", "path": "entries/PALOMAR-2026-10-01-000013-v1.json", "preservation": { "repositories": [ { "commit": "fb32a19c4883baad83b286d4594cfbc6b1d2ad5c", "fork_repository": "PalomarArchive/flengyel--FailureOfComposition--79f0e34ed6ea", "source_repository": "flengyel/FailureOfComposition" } ] }, "published_at": "2026-10-01T14:11:39Z", "source": { "commit": "fb32a19c4883baad83b286d4594cfbc6b1d2ad5c", "project_path": "Lean", "repository": "flengyel/FailureOfComposition" }, "status": "registered", "title": "Pointwise provable equality and the failure of composition", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We formalise the proof of two major recent results on wandering domains of transcendental entire and meromorphic functions: every orbit in a wandering domain accumulates at infinity and at a non-isolated (derived) point of the spherical set of singular values.\nBoth results are obtained as a special case of a more general theorem, for locally defined functions on possibly disconnected complex one-manifolds. Let f be an open holomorphic function from an open subset O of such a manifold X into X. Different components of O may map to different components of X, even when those source components lie in the same ambient component. Suppose that U is a wandering component of the set of normality of f, and that z is in U. If the orbit of z is contained in a compact subset of X, then it accumulates at a non-isolated point s of the set of singular values. Moreover, this accumulation must occur close to singularities for infinitely many singular values. That is, there exists a sequence (s_n) of distinct singular values converging to s, a sequence of embedded discs D_n containing s and a sequence U_n of components of f^{-1}(D_n), with the following properties. (We say that the orbit of z has \"singular encounters\"; see HasSingularEncounterSequence in the formalisation.) * the discs D_n shrink to s; that is, every neighbourhood of s contains D_n for sufficiently large n; * each component U_n contains a point of the orbit of z; * s_n is a singular value of the restriction f:U_n\\to D_n. In particular, the orbit of z is not contained in any compact subset of O. In the special case of a transcendental entire or meromorphic function, we can take X to be the Riemann sphere, and hence compact, so that the hypotheses are satisfied for any wandering Fatou component.\nWe also prove a corresponding theorem for measurable wandering subsets of the complement of the domain of normality on which all iterates are defined and injective; here the conclusion is that almost every point of the set (in any chart) behaves as above: ambient escape along a subsequence, or singular encounters.\nAs corollaries we recover Epstein's absence of wandering domains for finite-type maps on arbitrary open subsets of compact, possibly disconnected complex one-manifolds, and the classical no-wandering results for rational maps, compact-surface self-maps, and class S entire and meromorphic functions.\nThis is an update from the previous registered submission on Palomar, https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000001&version=3 . The change in this update is to allow functions in the local result to take values in a possibly disconnected complex 1-manifold, rather than a Riemann surface. This aligns the definition with Epstein's original discussion of finite-type analytic maps, which likewise allow (compact) complex 1-manifolds.", "authors": [ { "name": "Nikolai Prochorov" }, { "name": "Lasse Rempe" }, { "name": "James Waterman" } ], "classification": { "arxiv": [ "math.DS", "math.CV" ], "msc2020": [ "37F10", "30D05", "30F45" ] }, "formalization": { "theorem_names": [ "BoundedWanderingDomains.wandering_domain_has_locally_uniform_escaping_subsequence", "MeromorphicDynamics.wandering_domain_has_locally_uniform_escaping_subsequence", "SurfaceDynamics.wandering_component_orbit_not_compactly_contained", "SurfaceDynamics.positive_area_wandering_saturation_not_compactly_contained", "BoundedWanderingDomains.wandering_orbit_accumulates_on_derived_singular_values", "SurfaceDynamics.wandering_orbit_has_escaping_or_derived_singular_subsequence", "BoundedWanderingDomains.no_bounded_wandering_domains_transcendental_entire", "BoundedWanderingDomains.no_wandering_domains_transcendental_entire_finite_singularValues", "SurfaceDynamics.no_wandering_domains_compact", "SurfaceDynamics.no_wandering_domains_rational", "SurfaceDynamics.compact_wandering_orbit_accumulates_on_derived_singular_values", "SurfaceDynamics.almost_every_wandering_point_has_source_escaping_subsequence", "SurfaceDynamics.positive_area_wandering_saturation_not_compactly_contained_away_from_derived", "SurfaceDynamics.almost_every_wandering_point_has_escaping_or_derived_singular_subsequence", "SurfaceDynamics.no_wandering_domains_finite_type", "MeromorphicDynamics.no_wandering_domains_transcendental_meromorphic_finite_singularValues", "SurfaceDynamics.wandering_domain_has_escaping_or_singular_encounter_sequence", "SurfaceDynamics.almost_every_wandering_point_has_escaping_or_singular_encounter_sequence", "MeromorphicDynamics.wandering_orbit_accumulates_on_derived_singular_values" ] }, "id": "PALOMAR-2026-09-25-000001", "path": "entries/PALOMAR-2026-09-25-000001-v4.json", "preservation": { "repositories": [ { "commit": "6e306deb8169e834e9cc6662160c70231a7e2a83", "fork_repository": "PalomarArchive/LR-UK--Bounded-orbit-wandering-domains--c47223f53157", "source_repository": "LR-UK/Bounded-orbit-wandering-domains" } ] }, "published_at": "2026-10-01T14:01:10Z", "source": { "commit": "6e306deb8169e834e9cc6662160c70231a7e2a83", "project_path": null, "repository": "LR-UK/Bounded-orbit-wandering-domains" }, "status": "registered", "title": "Absence of bounded-orbit wandering domains", "trust": { "level": "high" }, "version": 4, "versions": 4 }, { "abstract": "A Lean formalization of Myerson's 1981 optimal auction theorem, regular case, in a single-parameter independent-private-values model with real types. Finitely many bidders draw independent types from TypeDist priors: probability measures on \u211d with a continuous monotone CDF and a nonnegative Lebesgue density that vanishes for negative types, so each prior's mass lies on [0, \u221e). The formal results are MyersonOptimalAuction.Palomar.virtualSurplusIdentity: under BIC, the support hypothesis (hsupp : \u2200 t : \u211d, 0 < t \u2192 0 < D.F t, i.e. the prior puts positive mass in (0, t] for every t > 0 so zero is in the support), and five integrability hypotheses, the expected payment equals the expected virtual surplus minus the interim utility at the reference type (type zero), expectedPayment p D = expectedVirtualSurplus x D - interimUtility x p 0; and MyersonOptimalAuction.Palomar.optimalAuction: for regular priors (hreg : Regular D, i.e. each bidder's virtual value \u03c8(t) = t - (1 - F t) / f t is nondecreasing on nonnegative types), the auction that allocates the item to the least-index bidder attaining the highest virtual value whenever that value is strictly positive (no sale otherwise), with envelope payments normalized to zero utility at type zero, is Bayesian incentive compatible and interim individually rational, and raises at least as much expected revenue as any feasible BIC and interim-IR mechanism. Independence is the product measure over type profiles; feasibility is pointwise (\u2211 i, X i t \u2264 1 for every profile t). Types range over all of \u211d; zero is only the reference/normalization type, never a lowest type. The Bochner integrals defining expectations are totalized, so every expectation claim carries the integrability hypotheses that make it non-vacuous. Ironing for irregular distributions is out of scope. The Palomar registry API searches on 2026-10-01 for \"optimal auction\", \"virtual surplus\", and \"virtual value\" returned no existing Myerson optimal auction formalization (\"optimal auction\" matched only PALOMAR-2026-10-01-000007, Bulow-Klemperer, a different theorem). These are dated, limited observations, not universal prior-art claims.", "authors": [ { "name": "Arthur Freitas Ramos" }, { "name": "David Barros Hulak" }, { "name": "Ruy Jose Guerra Barretto de Queiroz" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "MyersonOptimalAuction.Palomar.virtualSurplusIdentity", "MyersonOptimalAuction.Palomar.optimalAuction" ] }, "id": "PALOMAR-2026-10-01-000012", "path": "entries/PALOMAR-2026-10-01-000012-v1.json", "preservation": { "repositories": [ { "commit": "f78aa142d4c34ed70262f44250fa66eb1f95ffb4", "fork_repository": "PalomarArchive/Arthur742Ramos--myerson-optimal-auction-lean--3701856fdb67", "source_repository": "Arthur742Ramos/myerson-optimal-auction-lean" } ] }, "published_at": "2026-10-01T13:50:39Z", "source": { "commit": "f78aa142d4c34ed70262f44250fa66eb1f95ffb4", "project_path": null, "repository": "Arthur742Ramos/myerson-optimal-auction-lean" }, "status": "registered", "title": "Myerson's Optimal Auction Theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of Myerson's 1981 revenue equivalence theorem in a single-parameter Bayesian mechanism-design model with continuous real types. The formal result is RevenueEquivalence.Palomar.revenueEquivalence (X P1 P2 : \u211d \u2192 \u211d) (h1 : BIC X P1) (h2 : BIC X P2) (hU : interimUtility X P1 0 = interimUtility X P2 0) (t : \u211d) (ht : 0 \u2264 t) : P1 t = P2 t: two Bayesian incentive-compatible mechanisms with the same interim allocation rule and the same interim utility at the reference type (type zero) charge the same interim payments at every nonnegative type. Types range over all of \u211d; there is no lowest type, and zero is only the normalization point for the payment identity. The envelope lemma RevenueEquivalence.Palomar.envelope_integral and the expected-revenue corollary RevenueEquivalence.Palomar.expectedRevenueEqual are proved alongside; the corollary requires both interim payment rules to be integrable with respect to the prior ((hInt1 : Integrable P1 \u03bc) (hInt2 : Integrable P2 \u03bc)), so its conclusion is an equality of finite expected revenues rather than a vacuous totalized-integral identity. The Palomar registry API search on 2026-10-01 for \"revenue equivalence\" returned zero entries; the queries \"Myerson\" and \"auction\" returned only different theorems (see related_formalizations note). These are dated, limited observations, not universal prior-art claims.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "RevenueEquivalence.Palomar.envelope_integral", "RevenueEquivalence.Palomar.revenueEquivalence", "RevenueEquivalence.Palomar.expectedRevenueEqual" ] }, "id": "PALOMAR-2026-10-01-000011", "path": "entries/PALOMAR-2026-10-01-000011-v1.json", "preservation": { "repositories": [ { "commit": "b07bab2482b6ff98a71d650d5b4d4ac327e240b0", "fork_repository": "PalomarArchive/Arthur742Ramos--revenue-equivalence-lean--27517e4ee616", "source_repository": "Arthur742Ramos/revenue-equivalence-lean" } ] }, "published_at": "2026-10-01T10:34:02Z", "source": { "commit": "b07bab2482b6ff98a71d650d5b4d4ac327e240b0", "project_path": null, "repository": "Arthur742Ramos/revenue-equivalence-lean" }, "status": "registered", "title": "revenue-equivalence-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A machine-checked proof of Gleason's theorem in finite dimensions. On a complex Hilbert space of dimension at least 3, every frame function on the orthogonal projections has the form P \u21a6 Re Tr(\u03c1P) for exactly one density operator \u03c1. The statement is Kim Morrison's lean-eval problem gleason_theorem_finite, unchanged. The proof follows Cooke, Keane and Moran's elementary proof of Gleason's core lemma for matrices on \u2102\u1d3a, then moves the result to operators by taking matrices in an orthonormal basis.", "authors": [ { "name": "Zayn Blore" } ], "classification": { "arxiv": [ "math-ph", "quant-ph" ], "msc2020": [ "81P10", "81P16", "46C05", "46N50" ] }, "formalization": { "theorem_names": [ "gleason_theorem_finite" ] }, "id": "PALOMAR-2026-10-01-000010", "path": "entries/PALOMAR-2026-10-01-000010-v1.json", "preservation": { "repositories": [ { "commit": "3a9449ce0e3e9f9355f15827ce5d99a4460efa0c", "fork_repository": "PalomarArchive/zblore--gleason-finite-lean--7a8ddf8ab084", "source_repository": "zblore/gleason-finite-lean" } ] }, "published_at": "2026-10-01T08:58:30Z", "source": { "commit": "3a9449ce0e3e9f9355f15827ce5d99a4460efa0c", "project_path": null, "repository": "zblore/gleason-finite-lean" }, "status": "registered", "title": "gleason-finite-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the Green-Laffont characterization: for finite, nonempty agent and alternative types, efficiency and dominant-strategy incentive compatibility imply that payments have Groves form. The project also verifies four standard properties of the finite VCG mechanism with Clarke pivot payments.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "VCG.vcg_greenLaffont", "VCG.vcg_efficient", "VCG.vcg_truthful", "VCG.vcg_individualRational", "VCG.vcg_noDeficit" ] }, "id": "PALOMAR-2026-10-01-000009", "path": "entries/PALOMAR-2026-10-01-000009-v1.json", "preservation": { "repositories": [ { "commit": "aad0c41b99053cf4657d91f7446279557f87ae18", "fork_repository": "PalomarArchive/Arthur742Ramos--vcg-mechanism-lean--b451825d01a8", "source_repository": "Arthur742Ramos/vcg-mechanism-lean" } ] }, "published_at": "2026-10-01T04:02:50Z", "source": { "commit": "aad0c41b99053cf4657d91f7446279557f87ae18", "project_path": null, "repository": "Arthur742Ramos/vcg-mechanism-lean" }, "status": "registered", "title": "vcg-mechanism-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of Roberts theorem: for finite nonempty agent and alternative types with at least three alternatives, every onto dominant-strategy incentive compatible choice function is an affine maximizer. The proof follows the modular route: the taxation principle and weak monotonicity, reduction to strong monotonicity by tie-breaking, no-veto-power range analysis, the two-agent affine price representation, and induction on the number of agents.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B14" ] }, "formalization": { "theorem_names": [ "Roberts.roberts_theorem" ] }, "id": "PALOMAR-2026-10-01-000008", "path": "entries/PALOMAR-2026-10-01-000008-v1.json", "preservation": { "repositories": [ { "commit": "0c176c66d2afa62301297f443b1e40eda2387ca2", "fork_repository": "PalomarArchive/Arthur742Ramos--roberts-theorem-lean--051d38120180", "source_repository": "Arthur742Ramos/roberts-theorem-lean" } ] }, "published_at": "2026-10-01T02:31:47Z", "source": { "commit": "0c176c66d2afa62301297f443b1e40eda2387ca2", "project_path": null, "repository": "Arthur742Ramos/roberts-theorem-lean" }, "status": "registered", "title": "roberts-theorem-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean proofs of Yongxi Lin's original weak type (1,1) bounds: c_2 >= 1.6855 and c_2 <= 3.879 for cubes, c_2 <= e for Euclidean discs, and c_n <= (n/2)^(n/(n-2)) for balls in n >= 3. Also proves the classical cube bound c_d <= 2^d.", "authors": [ { "name": "Yongxi Lin" } ], "classification": { "arxiv": [ "math.CA", "math.FA" ], "msc2020": [ "42B25", "26D15", "35R11" ] }, "formalization": { "theorem_names": [ "CenteredMaximal.weakTypeConstant_le_two_pow", "CenteredMaximal.weakTypeConstant_two_le_upper", "CenteredMaximal.ofReal_phi_le_weakTypeConstant_two", "CenteredMaximal.lt_phi", "CenteredMaximal.phi_lt", "CenteredMaximal.ballWeakTypeConstant_two_le_exp", "CenteredMaximal.ballWeakTypeConstant_le_rpow" ] }, "id": "PALOMAR-2026-09-19-000002", "path": "entries/PALOMAR-2026-09-19-000002-v3.json", "preservation": { "repositories": [ { "commit": "72c022ba09032b522c878cca2f9b6c68217d77ea", "fork_repository": "PalomarArchive/CoolRmal--centered-maximal-constant--975187d5196c", "source_repository": "CoolRmal/centered-maximal-constant" } ] }, "published_at": "2026-10-01T02:09:29Z", "source": { "commit": "72c022ba09032b522c878cca2f9b6c68217d77ea", "project_path": null, "repository": "CoolRmal/centered-maximal-constant" }, "status": "registered", "title": "Centred maximal constants for cubes and Euclidean balls", "trust": { "level": "high" }, "version": 3, "versions": 3 }, { "abstract": "A Lean proof that a no-reserve second-price auction with one additional independent bidder has at least the expected revenue of the optimal normalized BIC auction under a regular common value distribution.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "cs.GT", "econ.TH" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "BulowKlemperer.Palomar.optimalAuction_expectedRevenue", "BulowKlemperer.Palomar.secondPriceAuction_expectedRevenue", "BulowKlemperer.Palomar.bulowKlemperer" ] }, "id": "PALOMAR-2026-10-01-000007", "path": "entries/PALOMAR-2026-10-01-000007-v1.json", "preservation": { "repositories": [ { "commit": "b6b7068f5846f63eb18baeb753fcdbd0acf3f99f", "fork_repository": "PalomarArchive/Arthur742Ramos--bulow-klemperer-lean--ede6ea4ee589", "source_repository": "Arthur742Ramos/bulow-klemperer-lean" } ] }, "published_at": "2026-10-01T01:59:43Z", "source": { "commit": "b6b7068f5846f63eb18baeb753fcdbd0acf3f99f", "project_path": null, "repository": "Arthur742Ramos/bulow-klemperer-lean" }, "status": "registered", "title": "bulow-klemperer-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Asymptotic correctness of the discriminative Kalman filter for a stationary linear Gaussian state model with nonlinear and non-Gaussian observations. Formalizes Appendix B, Theorem 2, Lemma 1, and Remarks 1\u20134 of Burkhart et al. (2020): weak convergence, eventual Gaussian well-definedness, L1 consistency, the exact Gaussian parameter update, the robust update, and the extension to random input distributions with convergence in probability.", "authors": [ { "name": "Michael C. Burkhart" } ], "classification": { "arxiv": [ "math.PR", "stat.TH" ], "msc2020": [ "60G35", "62F15", "03B35" ] }, "formalization": { "theorem_names": [ "DiscriminativeKalmanFilter.theorem2", "DiscriminativeKalmanFilter.lemma1", "DiscriminativeKalmanFilter.remark1", "DiscriminativeKalmanFilter.remark2", "DiscriminativeKalmanFilter.remark3", "DiscriminativeKalmanFilter.remark4" ] }, "id": "PALOMAR-2026-10-01-000006", "path": "entries/PALOMAR-2026-10-01-000006-v1.json", "preservation": { "repositories": [ { "commit": "6b91b7f967bc4523a3de9024bcb49be1aca42918", "fork_repository": "PalomarArchive/burkh4rt--Discriminative-Kalman-Filter-Formalized--ee0b4e8e744b", "source_repository": "burkh4rt/Discriminative-Kalman-Filter-Formalized" } ] }, "published_at": "2026-10-01T01:40:36Z", "source": { "commit": "6b91b7f967bc4523a3de9024bcb49be1aca42918", "project_path": null, "repository": "burkh4rt/Discriminative-Kalman-Filter-Formalized" }, "status": "registered", "title": "Discriminative Kalman Filter Formalized", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalization of all eleven numbered results and both counterexamples in Local conjugacy in prosolvable groups, arXiv:2609.37678v1. Closed supplements of a pronilpotent normal subgroup are conjugate exactly when locally conjugate under the stated structural hypotheses. Continuous nonabelian cohomology has the stated primary restriction decomposition. Local inclusion and fixed-point results, the GL(2,3) quaternion counterexample, and the order-162 Heisenberg wreath-product counterexample are formalized.", "authors": [ { "name": "Michael C. Burkhart" } ], "classification": { "arxiv": [ "math.GR" ], "msc2020": [ "20E18", "20E45", "20J06", "05E18" ] }, "formalization": { "theorem_names": [ "LocalConjugacy.theorem_1_1", "LocalConjugacy.lemma_1_2", "LocalConjugacy.corollary_1_3", "LocalConjugacy.corollary_1_4", "LocalConjugacy.proposition_2_1", "LocalConjugacy.proposition_2_2", "LocalConjugacy.proposition_2_3", "LocalConjugacy.proposition_3_1", "LocalConjugacy.proposition_3_2", "LocalConjugacy.proposition_4_1", "LocalConjugacy.proposition_4_2", "LocalConjugacy.counterexample_quaternion", "LocalConjugacy.counterexample_heisenberg" ] }, "id": "PALOMAR-2026-10-01-000005", "path": "entries/PALOMAR-2026-10-01-000005-v1.json", "preservation": { "repositories": [ { "commit": "a6ccde0b11cbef7dfd8344cbd21f5c6fd8edc887", "fork_repository": "PalomarArchive/burkh4rt--Local-Conjugacy-Formalized--47d7400b9819", "source_repository": "burkh4rt/Local-Conjugacy-Formalized" } ] }, "published_at": "2026-10-01T01:22:30Z", "source": { "commit": "a6ccde0b11cbef7dfd8344cbd21f5c6fd8edc887", "project_path": null, "repository": "burkh4rt/Local-Conjugacy-Formalized" }, "status": "registered", "title": "Local conjugacy in prosolvable groups", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of von Neumann's 1928 finite two-player zero-sum minimax theorem, a foundation of modern game theory and a cornerstone of mathematical economics. Nash equilibrium existence builds directly on its mixed-strategy solution concept by extending equilibrium existence beyond the two-player zero-sum case; this is historical motivation, not a claimed Lean proof of Nash's theorem. The formal result is Minimax.minimax_theorem {A B : Type*} [Fintype A] [Fintype B] [Nonempty A] [Nonempty B] (u : A \u2192 B \u2192 \u211d) : maxminValue u = minmaxValue u. The reported Palomar registry API search on 2026-09-29 for \"minimax\" returned zero entries. The M1 Mathlib survey found no finite zero-sum game specialization in the pinned Mathlib, whose general Sion saddle-point theorem is reused here.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "cs.GT", "math.OC" ], "msc2020": [ "91A05" ] }, "formalization": { "theorem_names": [ "Minimax.Palomar.maxmin_le_minmax", "Minimax.Palomar.minmax_le_maxmin", "Minimax.Palomar.minimax_theorem" ] }, "id": "PALOMAR-2026-10-01-000004", "path": "entries/PALOMAR-2026-10-01-000004-v1.json", "preservation": { "repositories": [ { "commit": "0d5e3fd6ec3d56f5cafa34886019c46668fcf9ff", "fork_repository": "PalomarArchive/Arthur742Ramos--minimax-theorem-lean--2fd26e80c8d0", "source_repository": "Arthur742Ramos/minimax-theorem-lean" } ] }, "published_at": "2026-10-01T00:38:48Z", "source": { "commit": "0d5e3fd6ec3d56f5cafa34886019c46668fcf9ff", "project_path": null, "repository": "Arthur742Ramos/minimax-theorem-lean" }, "status": "registered", "title": "minimax-theorem-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of exact distributional results for the random-permutation greedy maximal-independent-set process on finite trees. For every finite nonempty tree, the complete greedy output law is uniform over maximal independent sets exactly for K1 and K2. For the mixed-spider family T_{k,l}, the development counts maximal independent sets, proves exact output probabilities, and for l=2^k-k proves positive total-variation bias with decay O(sqrt(k)/4^k), equivalently O(sqrt(log n_k)/n_k^2) along n_k=2^k+k+1. The Stage-4 novelty classification is \"plausibly new with bounded uncertainty\"; no worldwide-priority, global-optimality, matching lower-bound, all-n extremal, or other stronger claim is made.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.CO", "math.PR" ], "msc2020": [ "05C05", "05C69" ] }, "formalization": { "theorem_names": [ "GreedyUniformity.tree_greedyLawEqUniform_iff_isK1OrK2", "GreedyUniformity.tree_bias_eq_zero_iff_isK1OrK2", "GreedyUniformity.mixedSpider_maximalIndependentSets_card", "GreedyUniformity.greedyProb_mixedSpiderCenter", "GreedyUniformity.greedyProb_mixedSpiderNoncenter", "GreedyUniformity.bias_mixedSpider_tuned_eq_expectation", "GreedyUniformity.bias_mixedSpider_tuned_pos", "GreedyUniformity.bias_mixedSpider_tuned_isBigO", "GreedyUniformity.bias_mixedSpider_tuned_isBigO_sqrt_log_order_div_order_sq" ] }, "id": "PALOMAR-2026-10-01-000003", "path": "entries/PALOMAR-2026-10-01-000003-v1.json", "preservation": { "repositories": [ { "commit": "1733c29a5165148d71a2f0bd1ed2dcd7c35309ce", "fork_repository": "PalomarArchive/jfairfaxball-348--Greedy-Uniformity-on-Trees--0c04ff5c5faa", "source_repository": "jfairfaxball-348/Greedy-Uniformity-on-Trees" } ] }, "published_at": "2026-10-01T00:28:54Z", "source": { "commit": "1733c29a5165148d71a2f0bd1ed2dcd7c35309ce", "project_path": null, "repository": "jfairfaxball-348/Greedy-Uniformity-on-Trees" }, "status": "registered", "title": "Greedy Uniformity on Trees", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization in Lean 4 of the paper 'A Short Proof of Ball's Plank Theorem' by Arseniy Akopyan and Alexander Polyanskii (arXiv:2609.36163). The paper gives a short proof of Keith Ball's 1991 theorem: if a centrally symmetric convex body in R^d is covered by finitely many planks, the sum of their relative widths with respect to the body is at least 1. The proof is based on a simple proof of a core theorem about linear functionals on R^n and an orthonormal basis. It also yields a short proof of Davenport's problem for centrally symmetric convex bodies (interior-disjointness of a homothet K/(n+1) from n hyperplanes).", "authors": [ { "name": "Arseniy Akopyan" } ], "classification": { "arxiv": [ "math.MG" ], "msc2020": [ "52A20", "52C17", "52A38", "52A40" ] }, "formalization": { "theorem_names": [ "PlankProblem.ball_plank_theorem", "PlankProblem.core_theorem", "PlankProblem.davenport_conjecture" ] }, "id": "PALOMAR-2026-10-01-000002", "path": "entries/PALOMAR-2026-10-01-000002-v1.json", "preservation": { "repositories": [ { "commit": "e57a2d8d0129b2d7ce3ee45c2064f5080c8c65eb", "fork_repository": "PalomarArchive/akopjan--Palomar-Bang-Ball--93f2b4a952dd", "source_repository": "akopjan/Palomar-Bang-Ball" } ] }, "published_at": "2026-10-01T00:18:27Z", "source": { "commit": "e57a2d8d0129b2d7ce3ee45c2064f5080c8c65eb", "project_path": null, "repository": "akopjan/Palomar-Bang-Ball" }, "status": "registered", "title": "A Short Proof of Ball's Plank Theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Despons, De Decker and Lacoste (https://arxiv.org/abs/2306.02366, conjecture following Eq. (33)) proposed that the affinity at maximum production flux is bounded below by the logarithm of the gross amplification ratio in seed-dependent autocatalytic networks. A two-species reversible mass-action counterexample is given by X \u21cc Y with forward/reverse rates (3, 2) and 2Y \u21cc 2X + Y with rates (7, 6). On the positive Y-stationary set, the production current has its unique global maximum at (X, Y) = (1, 1). At this point the exponential of the overall affinity is 7/4, strictly below the gross amplification ratio 2. The network has positive rates, an invertible net stoichiometric matrix, unit production mode and a positive detailed-balance state. The formal theorem establishes the counterexample; the correspondence with the published conjecture is supplied by the witness\u2019s stationary branch and reactant matrix, together with the source\u2019s affinity and chemostat conventions.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "q-bio.MN", "math.DS" ], "msc2020": [ "92C42", "80A30", "92C45" ] }, "formalization": { "theorem_names": [ "OptimalAffinity.optimalAffinity_seedLowerBound_false" ] }, "id": "PALOMAR-2026-10-01-000001", "path": "entries/PALOMAR-2026-10-01-000001-v1.json", "preservation": { "repositories": [ { "commit": "50199f4bc2999fb43a0486dcb7571f9dfd333ff2", "fork_repository": "PalomarArchive/mmislan--palomar-p004-optimal-affinity-counterexample--b9ec3e1231c9", "source_repository": "mmislan/palomar-p004-optimal-affinity-counterexample" } ] }, "published_at": "2026-10-01T00:06:23Z", "source": { "commit": "50199f4bc2999fb43a0486dcb7571f9dfd333ff2", "project_path": null, "repository": "mmislan/palomar-p004-optimal-affinity-counterexample" }, "status": "registered", "title": "A two-reaction counterexample to the gross-amplification affinity bound", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization of 18 claims from manuscript Theorems A, B, C, E (extension direction), F, G, H (hierarchy inclusions and characteristic-zero strictness), I, and J. The claims include coalgebraic rounding, algebraic and coalgebraic amenability, Hopf-module quotients, cleft extensions, group and Lie applications, and augmentation-associated graded amenability. Every selected claim is proved without unformalized literature inputs or project-local axioms. Theorem D, the equivalence form of E, and arbitrary-characteristic strictness in H remain in the library but are excluded from the submission.", "authors": [ { "name": "Laurent Bartholdi" } ], "classification": { "arxiv": [ "math.RA", "math.GR" ], "msc2020": [ "16T05", "43A07" ] }, "formalization": { "theorem_names": [ "HopfAmenability.palomar_rounding", "HopfAmenability.palomar_amenability", "Palomar.theoremB", "Palomar.theoremC", "Palomar.theoremC_module", "Palomar.theoremE_extension", "Palomar.theoremF_permutation", "Palomar.theoremF", "Palomar.theoremG_growth", "Palomar.theoremG_subalgebra", "Palomar.theoremG_quotient", "Palomar.theoremG_extension", "Palomar.theoremG_union", "Palomar.theoremH_EL_SL", "Palomar.theoremH_SL_AL", "Palomar.theoremH_charZero", "Palomar.theoremI", "Palomar.theoremJ" ] }, "id": "PALOMAR-2026-09-30-000037", "path": "entries/PALOMAR-2026-09-30-000037-v1.json", "preservation": { "repositories": [ { "commit": "6399798618014ed9a89694008cceb5b75707c7e0", "fork_repository": "PalomarArchive/laurentbartholdi--hopfamenability--5ba5f4713a92", "source_repository": "laurentbartholdi/hopfamenability" } ] }, "published_at": "2026-09-30T18:00:00Z", "source": { "commit": "6399798618014ed9a89694008cceb5b75707c7e0", "project_path": null, "repository": "laurentbartholdi/hopfamenability" }, "status": "registered", "title": "Amenability of Lie, group and Hopf algebras", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "There exists a locally finite set \u039b in the real line such that the complete complex vector space of tempered distributions locally given by atomic measures on \u039b, whose distributional Fourier transforms are also locally atomic on \u039b, has Hamel dimension \u2135\u2080. A countable family gives a unique finite linear expansion of every admissible distribution, and every finite expansion is admissible. The theorem includes the equivalence between a Schwartz-vanishing-ideal definition and the local atomic formulas on both Fourier sides. Temperedness is distributional; no additional polynomial total-variation bound is assumed. This is an exhaustive occurrence result for one constructed carrier, not a classification of arbitrary carriers.", "authors": [ { "name": "Jamie Martin" }, { "name": "Li Shen" } ], "classification": { "arxiv": [ "math.CA", "math.FA" ], "msc2020": [ "42A38", "46F12", "52C23" ] }, "formalization": { "theorem_names": [ "CountablyInfiniteCrystallineMeasures.exhaustiveCardinalClaim" ] }, "id": "PALOMAR-2026-09-30-000036", "path": "entries/PALOMAR-2026-09-30-000036-v1.json", "preservation": { "repositories": [ { "commit": "0de61962a757b5e930dfe59dd94d191ae1f87922", "fork_repository": "PalomarArchive/balancedscorpion--countably-infinite-dimension-meyer-spaces--08b04ade2e72", "source_repository": "balancedscorpion/countably-infinite-dimension-meyer-spaces" } ] }, "published_at": "2026-09-30T17:49:58Z", "source": { "commit": "0de61962a757b5e930dfe59dd94d191ae1f87922", "project_path": null, "repository": "balancedscorpion/countably-infinite-dimension-meyer-spaces" }, "status": "registered", "title": "A complete crystalline-measure space of countably infinite Hamel dimension", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For the explicit bivector metric induced by diag(1,1,1,-1,1,-1) and the Pfaffian cubic on fifteen real coordinates, the joint infinitesimal stabilizer is exactly the uniquely parameterized fifteen-dimensional orthogonal adjoint span. Under the metric condition, ninety specified cubic derivative checks certify preservation at every point. Within the joint stabilizer, commutation with a displayed local Lorentz Hodge operator is equivalent to annihilating its complementary reference and yields a unique six-plus-one decomposition, of dimension seven. Within this fixed-Hodge space, exactly the complementary line acts trivially on all extracted four-plane fields; six specified scalar readings reconstruct the entire visible action. The contribution is the explicit constructive classification and recovery certificates, situated in classical Pfaffian-preserver theory. The metric, cubic and four-plane are supplied data. PDT motivates the construction; selecting a physical coupling is outside these claims.", "authors": [ { "name": "Stephanie Alexander" } ], "classification": { "arxiv": [ "math.RT" ], "msc2020": [ "15A72", "15A04" ] }, "formalization": { "theorem_names": [ "PDTGeometry.jointClassification", "PDTGeometry.geometricRecovery", "PDTGeometry.finiteCubicCertificate", "PDTGeometry.fixedReference", "PDTGeometry.fixedClassification", "PDTGeometry.stabilizerDimensions", "PDTGeometry.fieldKernel", "PDTGeometry.fieldRecovery", "PDTGeometry.complementStructure" ] }, "id": "PALOMAR-2026-09-30-000035", "path": "entries/PALOMAR-2026-09-30-000035-v1.json", "preservation": { "repositories": [ { "commit": "0e436b9e3468e0393641d42fd37dfc9e3520ffbf", "fork_repository": "PalomarArchive/stalex444--pdt-metric-pfaffian-hodge-stabilizers--4541a9a1b3db", "source_repository": "stalex444/pdt-metric-pfaffian-hodge-stabilizers" } ] }, "published_at": "2026-09-30T15:37:54Z", "source": { "commit": "0e436b9e3468e0393641d42fd37dfc9e3520ffbf", "project_path": null, "repository": "stalex444/pdt-metric-pfaffian-hodge-stabilizers" }, "status": "registered", "title": "Metric\u2013Pfaffian Stabilizers and Fixed-Hodge Field Recovery", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "This is a Lean 4 formalization of the main theorem of the papers arXiv:1907.00405 and arXiv:2107.14616 by Ben Krause and Joris Roos. The result concerns l^p boundedness for a discrete maximal singular oscillatory operator of Stein-Wainger type, also called a discrete Carleson operator.", "authors": [ { "name": "Joris Roos" } ], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "42B20", "42B25", "42B15" ] }, "formalization": { "theorem_names": [ "discrete_carleson_stein_wainger" ] }, "id": "PALOMAR-2026-09-30-000034", "path": "entries/PALOMAR-2026-09-30-000034-v1.json", "preservation": { "repositories": [ { "commit": "aba0306bbb65ada1e7b412eed4f689f12a207f6e", "fork_repository": "PalomarArchive/roos-j--lean-discretesw--a92788269d63", "source_repository": "roos-j/lean-discretesw" } ] }, "published_at": "2026-09-30T15:24:08Z", "source": { "commit": "aba0306bbb65ada1e7b412eed4f689f12a207f6e", "project_path": null, "repository": "roos-j/lean-discretesw" }, "status": "registered", "title": "Formalization of a discrete Carleson theorem of Stein-Wainger type", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalisation of sharp p-adic extremal behaviour for the restricted binomial gcd G(N;m) = gcd{C(N,k) : 0 < k < N, m divides k}. For every prime p and m >= 2 with p not dividing m, it proves that the maximum of v_p(G(N;m)) over admissible rows is the least r >= 1 with m < p^r, with constructive attainment proved before the least extremal row T is used. It then proves, for every prime p and a >= 2, the exact least-row formula T_p(p^a+1) = p^(3a)+1.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11B65" ] }, "formalization": { "theorem_names": [ "PascalExtremesPalomar.targetA_attainment", "PascalExtremesPalomar.targetA", "PascalExtremesPalomar.targetB" ] }, "id": "PALOMAR-2026-09-30-000033", "path": "entries/PALOMAR-2026-09-30-000033-v1.json", "preservation": { "repositories": [ { "commit": "a7e414a9e5bcf931c79b2bd4d4ffa81eeb6cfc22", "fork_repository": "PalomarArchive/jfairfaxball-348--Pascal-Extremes--6b93e2d91cdd", "source_repository": "jfairfaxball-348/Pascal-Extremes" } ] }, "published_at": "2026-09-30T15:06:01Z", "source": { "commit": "a7e414a9e5bcf931c79b2bd4d4ffa81eeb6cfc22", "project_path": null, "repository": "jfairfaxball-348/Pascal-Extremes" }, "status": "registered", "title": "Pascal Extremes", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of Theorem A: every countable group embeds in a group of type FP_n over the integers for each finite n >= 2. The proof uses a 23-index, 15-triple Fano-plane construction to supply the finite input to geometric, homological, and relative Leary arguments. It does not assert the existence of one FP_infty overgroup.", "authors": [ { "name": "Laurent Bartholdi" } ], "classification": { "arxiv": [ "math.GR" ], "msc2020": [ "20J05" ] }, "formalization": { "theorem_names": [ "TheoremA.palomarStatement" ] }, "id": "PALOMAR-2026-09-30-000032", "path": "entries/PALOMAR-2026-09-30-000032-v1.json", "preservation": { "repositories": [ { "commit": "8970db6b7cc8ec2567dae486e48f34cf9226fe58", "fork_repository": "PalomarArchive/laurentbartholdi--theorem-a-fp-groups--a09c222fd486", "source_repository": "laurentbartholdi/theorem-a-fp-groups" } ] }, "published_at": "2026-09-30T14:44:02Z", "source": { "commit": "8970db6b7cc8ec2567dae486e48f34cf9226fe58", "project_path": null, "repository": "laurentbartholdi/theorem-a-fp-groups" }, "status": "registered", "title": "Embedding countable groups into groups of type FP_n", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the whole of J.-H. Evertse, On sums of S-units and linear recurrences (Compositio Math. 53 (1984)), built on the repository's libraries ArithmeticHeights and DiophantineApproximation, whose p-adic Subspace Theorem (Schlickewei's, the paper's Theorem 4) it uses. The library Evertse1984 is sorry-free and within Lean's three standard axioms. It proves Theorem 2 (a lower bound for S-integral sums without vanishing subsums, at any set of places), Theorem 1 (finitely many admissible projective solutions of x\u2080 + \u22ef + x\u2099 = 0), Corollaries 1 and 2, Lemmas 1 and 2 (polynomial values and shifted polynomial quotients that are S-units), Theorem 3 (for a non-degenerate linear recurrence with at least two characteristic roots, the largest prime ideal norm of u_r / u_s tends to infinity), and Corollaries 3 and 4 (P(u_r) \u2192 \u221e; u_r = u_s has finitely many solutions r \u2260 s). It also compares the paper's cited Theorem 4, the repository's p-adic Subspace Theorem. Fourteen statements are compared.", "authors": [ { "name": "Ralf Stephan" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11D61", "11B37", "11J87", "11R27" ] }, "formalization": { "theorem_names": [ "NumberField.exists_finset_submodule_of_integer_of_affineProd_le", "NumberField.exists_pos_forall_placeProd_le", "NumberField.finite_setOf_admissible_sum_eq_zero", "evertseTheorem1", "NumberField.exists_pos_forall_placeProd_iInf_le", "Rat.finite_setOf_sum_eq_zero_of_gcd_eq_one", "NumberField.exists_pos_forall_placeProd_eval_intCast", "NumberField.finite_setOf_eval_div_eval_mem_unit", "NumberField.finite_setOf_powerSum_div_mem_unit", "NumberField.finite_setOf_div_mem_unit_of_eq_powerSum", "NumberField.finite_setOf_powerSum_mem_unit", "NumberField.finite_setOf_mem_unit_of_eq_powerSum", "NumberField.finite_setOf_powerSum_eq", "NumberField.finite_setOf_eq_of_eq_powerSum" ] }, "id": "PALOMAR-2026-09-30-000031", "path": "entries/PALOMAR-2026-09-30-000031-v1.json", "preservation": { "repositories": [ { "commit": "f10f915bc689c86ae0abe45d4164dcb7df91e7ea", "fork_repository": "PalomarArchive/rwst--Subspace-Theorems--9bd9bba330e8", "source_repository": "rwst/Subspace-Theorems" } ] }, "published_at": "2026-09-30T14:23:34Z", "source": { "commit": "f10f915bc689c86ae0abe45d4164dcb7df91e7ea", "project_path": null, "repository": "rwst/Subspace-Theorems" }, "status": "registered", "title": "Evertse 1984: sums of S-units and linear recurrences", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the linear independence of 1, \u03b6(2) and \u03b6(3) over \u211a. The proof is inspired by A. Fauzan's work on the irrationality of \u03b6(5) and builds on the Lean formalization of the irrationality of Li\u2082(1/2), registered in Palomar as PALOMAR-2026-09-30-000013. It shows that the method of that proof can be pushed further. The main difference from the Li\u2082(1/2) proof is that the rational functions appearing here are allowed to have poles of order two. The remaining details are similar, with some minor modifications. The compared theorems depend on propext, Classical.choice and Quot.sound and on nothing else, and there is no sorry outside the deliberate holes in Challenge.lean.", "authors": [ { "name": "Qian Tang" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11M06" ] }, "formalization": { "theorem_names": [ "one_zeta_two_zeta_three_linearIndependent", "one_zeta_two_zeta_three_linearIndependent_series" ] }, "id": "PALOMAR-2026-09-30-000030", "path": "entries/PALOMAR-2026-09-30-000030-v1.json", "preservation": { "repositories": [ { "commit": "669c92bf7a3728e0de9ad80c373bc29fb5ca3d58", "fork_repository": "PalomarArchive/dtq1997--zeta2-zeta3-linear-independence--4378dbf7497d", "source_repository": "dtq1997/zeta2-zeta3-linear-independence" } ] }, "published_at": "2026-09-30T14:02:39Z", "source": { "commit": "669c92bf7a3728e0de9ad80c373bc29fb5ca3d58", "project_path": null, "repository": "dtq1997/zeta2-zeta3-linear-independence" }, "status": "registered", "title": "Linear independence of 1, \u03b6(2), \u03b6(3) over \u211a", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 formalization proving that the Moh curve parametrized by x=t^6+t^31, y=t^8, z=t^10 is cut out set-theoretically by two explicit equations over every characteristic-zero field. The development computes the polynomial and formal-power-series parametrization kernels, proves the radical equality for the two-generated witness ideal, and proves that the curve ideal has height two and arithmetic rank two, both globally and after completion. It also verifies the exact Hilbert-Burch presentation given by the manuscript's explicit four-by-three matrix.", "authors": [ { "name": "Aakash Gurung" } ], "classification": { "arxiv": [ "math.AC", "math.AG" ], "msc2020": [ "13C40", "14M10", "13H05", "13P10" ] }, "formalization": { "theorem_names": [ "MohP3.Palomar.global_setTheoretic_completeIntersection", "MohP3.Palomar.formalLocal_setTheoretic_completeIntersection" ] }, "id": "PALOMAR-2026-09-07-000012", "path": "entries/PALOMAR-2026-09-07-000012-v2.json", "preservation": { "repositories": [ { "commit": "abbdac729c91a6c7784a3dd15ff6828f2ce6e7fc", "fork_repository": "PalomarArchive/GrgAakash--moh-p3-set-theoretic-complete-intersection--dfb91e23119e", "source_repository": "GrgAakash/moh-p3-set-theoretic-complete-intersection" } ] }, "published_at": "2026-09-30T13:23:58Z", "source": { "commit": "abbdac729c91a6c7784a3dd15ff6828f2ce6e7fc", "project_path": null, "repository": "GrgAakash/moh-p3-set-theoretic-complete-intersection" }, "status": "registered", "title": "Moh's P3 curve is a set-theoretic complete intersection", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "Formal proof of the time-zero Black-Scholes formula for a European call on any strong solution of the geometric Brownian motion SDE, with positive initial price, strike, volatility and maturity, and constant drift and interest rate. The change of measure uses the constant-drift case of the general predictable (dynamic) Girsanov theorem. The discounted stopped asset is a martingale under one fixed terminal measure, and the call-payoff expectation equals the closed-form Black-Scholes expression.", "authors": [ { "name": "Ezzeri Esa" } ], "classification": { "arxiv": [ "math.PR", "q-fin.MF" ], "msc2020": [ "91G20", "60H10", "60H05" ] }, "formalization": { "theorem_names": [ "PalomarBlackScholes.black_scholes" ] }, "id": "PALOMAR-2026-09-30-000029", "path": "entries/PALOMAR-2026-09-30-000029-v1.json", "preservation": { "repositories": [ { "commit": "bde839683ba3cf1bf149355a40e9de9522babe92", "fork_repository": "PalomarArchive/savarin--lean-stochastic-calculus--55dac193d9d9", "source_repository": "savarin/lean-stochastic-calculus" } ] }, "published_at": "2026-09-30T12:18:58Z", "source": { "commit": "bde839683ba3cf1bf149355a40e9de9522babe92", "project_path": null, "repository": "savarin/lean-stochastic-calculus" }, "status": "registered", "title": "Black-Scholes pricing via Girsanov change of measure", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalisation of the automorphism-matching theorems and the quasi-isometry classification of Galton\u2013Watson trees from the paper \"The quasi-isometry classes of Galton\u2013Watson trees\" by Jayadev S. Athreya and Sascha Troscheit (arXiv:2609.23882). The audited statements are: the i.i.d. matching theorem on the binary tree (Theorem 5.1), the general Markov matching theorem for typed label processes on the binary tree, in its finite-type and zero-compatible forms (Theorem 6.1), the complete classification, up to quasi-isometry, of Galton\u2013Watson trees with finitely supported offspring law conditioned on infinite diameter (Theorem 1.2), the mutual embeddability theorem (Theorem 1.3), and two-value universality with its quantitative rate (Theorem 1.4). The statements are made in Challenge.lean over Mathlib alone and discharged by Solution.lean from four library targets.", "authors": [ { "name": "Jayadev S. Athreya" }, { "name": "Sascha Troscheit" } ], "classification": { "arxiv": [ "math.PR", "math.GR", "math.MG" ], "msc2020": [ "60J80", "51F30", "30L10", "28A80", "60K35", "68V20" ] }, "formalization": { "theorem_names": [ "Challenge.audit_graph_leaf_matching_bound", "Challenge.audit_graph_full_matching_bound", "Challenge.audit_exists_infinite_tree_matching_graphAut", "Challenge.audit_bounded_graph_qi_point", "Challenge.audit_extinction_of_not_supercritical", "Challenge.audit_full_classification_ae_iff", "Challenge.audit_mutual_embeddability", "Challenge.audit_twovalue_ae_tree_family", "Challenge.audit_twovalue_rate_tree", "Challenge.audit_markov_matching_finite", "Challenge.audit_markov_matching_zero", "Challenge.audit_markov_matching_finite_infinite", "Challenge.audit_markov_matching_zero_infinite" ] }, "id": "PALOMAR-2026-09-25-000002", "path": "entries/PALOMAR-2026-09-25-000002-v2.json", "preservation": { "repositories": [ { "commit": "35e78969d036b4f663fc9a96d8c22750834a6ac4", "fork_repository": "PalomarArchive/sascha2718--qi-trees-and-matching-lean--3baa1f2ce694", "source_repository": "sascha2718/qi-trees-and-matching-lean" } ] }, "published_at": "2026-09-30T11:52:38Z", "source": { "commit": "35e78969d036b4f663fc9a96d8c22750834a6ac4", "project_path": null, "repository": "sascha2718/qi-trees-and-matching-lean" }, "status": "registered", "title": "The quasi-isometry classes of Galton\u2013Watson trees: Lean formalisation", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "Kosc, Kuperberg, Rajon and Charlat (PNAS 122, e2421274122, 2025, Section 2.1) left open the complexity of detecting an autocatalytic core without a prescribed target species or allowed-food set. For finite reversible reaction networks with literal nonnegative integer reactant and product matrices, this unconstrained detection problem is proved NP-complete. A core is a nonempty entity/reaction pair, minimal under componentwise inclusion, with a selected reactant and product in every selected reaction and a signed real flow that strictly produces every selected entity. The input uses a canonical binary encoding. NP membership and a total polynomial-time many-one reduction from SAT are formalized using Complexitylib\u2019s multi-tape Turing machines, including the Cook\u2013Levin theorem.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "cs.CC", "q-bio.MN" ], "msc2020": [ "68Q17", "92C42" ] }, "formalization": { "theorem_names": [ "UnconstrainedPACDetection.complexity_root" ] }, "id": "PALOMAR-2026-09-30-000027", "path": "entries/PALOMAR-2026-09-30-000027-v1.json", "preservation": { "repositories": [ { "commit": "9a4508c8643310033182b18e1f348f5d786f7c28", "fork_repository": "PalomarArchive/mmislan--palomar-p081-productive-core-np-completeness--55af0db69bd1", "source_repository": "mmislan/palomar-p081-productive-core-np-completeness" } ] }, "published_at": "2026-09-30T11:33:17Z", "source": { "commit": "9a4508c8643310033182b18e1f348f5d786f7c28", "project_path": null, "repository": "mmislan/palomar-p081-productive-core-np-completeness" }, "status": "registered", "title": "Unconstrained detection of productive autocatalytic cores is NP-complete", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean-checked proof of the d = 3 case of FKS Problem 2: for every simple graph G on seven vertices, G or its complement has an injective placement in Euclidean three-space under which every edge has length 1. Non-edges are unconstrained and may also have length 1. The full question, for graphs on 2d + 1 vertices in dimension d, is stated and marked open. In the notation of Alon and Kupavskii, the proved case gives f_D(7) \u2264 3; together with the published lower bound it implies f_D(7) = 3. The formal target is the seven-vertex disjunction, not a definition or equality for f_D. The research audience is discrete geometers and extremal graph theorists studying unit-distance representations, graph dimension and distance Ramsey questions.", "authors": [ { "name": "Dishant Shah" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C62", "52C10" ] }, "formalization": { "theorem_names": [ "FKSProblem2.Palomar.target" ] }, "id": "PALOMAR-2026-09-30-000026", "path": "entries/PALOMAR-2026-09-30-000026-v1.json", "preservation": { "repositories": [ { "commit": "11413860db107522485d4e8e580ae406363c1164", "fork_repository": "PalomarArchive/Dishah3241--FKSProblem2--86c58af695b2", "source_repository": "Dishah3241/FKSProblem2" } ] }, "published_at": "2026-09-30T11:22:16Z", "source": { "commit": "11413860db107522485d4e8e580ae406363c1164", "project_path": null, "repository": "Dishah3241/FKSProblem2" }, "status": "registered", "title": "FKS Problem 2 in dimension three", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Nandan, Nghe and Unterberger (arXiv:2507.15546, Theorem 5.2 and the following remark) left uniqueness with degradation unresolved for Type II_l cores with l > 2 and, when some degradation rates vanish, for Type V cores. For the source-minimal Type II_l normal forms with l \u2265 3 and the Type V normal forms, unistationarity is established for reversible mass-action kinetics with positive forward and reverse rate constants and arbitrary nonnegative linear degradation. Any two strictly positive stationary concentration vectors coincide, including all internal species of the reversible unit chains. The Type II_l class includes separated stems and the coincident three-fork case. The proof combines passive-chain elimination and a ratio argument for Type V with boundary Jacobian nonsingularity and an implicit-function argument for Type II_l.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "92C42", "80A30", "15B48" ] }, "formalization": { "theorem_names": [ "MixedDegradation.paper_mixed_degradation_unistationarity" ] }, "id": "PALOMAR-2026-09-30-000025", "path": "entries/PALOMAR-2026-09-30-000025-v1.json", "preservation": { "repositories": [ { "commit": "f8d9da22ab2743054412b71177e4dbc5fb6ac993", "fork_repository": "PalomarArchive/mmislan--palomar-p026-nonnegative-degradation-unistationarity--f74794b0a996", "source_repository": "mmislan/palomar-p026-nonnegative-degradation-unistationarity" } ] }, "published_at": "2026-09-30T11:09:15Z", "source": { "commit": "f8d9da22ab2743054412b71177e4dbc5fb6ac993", "project_path": null, "repository": "mmislan/palomar-p026-nonnegative-degradation-unistationarity" }, "status": "registered", "title": "Unistationarity of Type II_l and Type V normal forms with nonnegative degradation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "An explicit local Hodge operator on the oriented coordinate four-plane 0123, extended by zero, together with the fifteen adjoint basis actions of the orthogonal algebra for diag(1,1,1,-1,1,-1), generates sl(15,K) over every field of characteristic different from two. A finite integral commutator certificate produces signed skew matrix units and one symmetric seed; a general propagation lemma then produces all trace-free matrices. Dimension 224 follows from generation. As a consequence, an endomorphism of this algebra commuting with the commutator actions of the same sixteen generators is scalar, with determinant c^224. The proof includes characteristics three and five without a simplicity assumption. The contribution is this constructive generation-and-rigidity theorem. PDT motivates the chosen operators; assigning a physical response magnitude or gravitational coupling is outside the selected claim.", "authors": [ { "name": "Stephanie Alexander" } ], "classification": { "arxiv": [ "math.RT" ], "msc2020": [ "17B05", "17B10" ] }, "formalization": { "theorem_names": [ "HorizonEinsteinClosure.geometricHodgeGeneration" ] }, "id": "PALOMAR-2026-09-30-000024", "path": "entries/PALOMAR-2026-09-30-000024-v1.json", "preservation": { "repositories": [ { "commit": "e2d8dde6f076b730b9a718c95ff1664c6aa88b51", "fork_repository": "PalomarArchive/stalex444--pdt-hodge-lie-generation--f565612039cb", "source_repository": "stalex444/pdt-hodge-lie-generation" } ] }, "published_at": "2026-09-30T10:36:49Z", "source": { "commit": "e2d8dde6f076b730b9a718c95ff1664c6aa88b51", "project_path": null, "repository": "stalex444/pdt-hodge-lie-generation" }, "status": "registered", "title": "Local Hodge generation of sl(15) and scalar response rigidity", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 formalization of a deterministic finite-path necessity theorem for graph-pebbling stackability: a positive-mass nonstackable path configuration above an explicit mass threshold must contain an actual directed integer branch message at most -h. The package also verifies the exact fixed-total corollary at total n*mu, forcing a message at most -(2*mu-1). The project's two-sided random path threshold theorem is not part of this Palomar result.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C99" ] }, "formalization": { "theorem_names": [ "ProbStack.Palomar.nonstackable_exists_directedMessage_le", "ProbStack.Palomar.nonstackable_total_mul_exists_directedMessage_le" ] }, "id": "PALOMAR-2026-09-30-000023", "path": "entries/PALOMAR-2026-09-30-000023-v1.json", "preservation": { "repositories": [ { "commit": "1897a76956168fba047f5943fa7998601f4fe459", "fork_repository": "PalomarArchive/jfairfaxball-348--ProbStack-Random-Stacking-on-Trees--ff7456044a49", "source_repository": "jfairfaxball-348/ProbStack-Random-Stacking-on-Trees" } ] }, "published_at": "2026-09-30T10:05:13Z", "source": { "commit": "1897a76956168fba047f5943fa7998601f4fe459", "project_path": null, "repository": "jfairfaxball-348/ProbStack-Random-Stacking-on-Trees" }, "status": "registered", "title": "ProbStack \u2014 Random Stacking on Trees", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The tower theory H is the many-sorted first-order theory of a tower of ZFC universes Z_0, Z_1, ... cut at consecutive inaccessible cardinals, with transition maps j_n identifying Z_n with the rank segment V(kappa_n) of Z_{n+1}. H is solid in Enayat's sense. For models M, N, P, with N interpreted in M and P interpreted in N, and an M-definable isomorphism between M and P, there is an M-definable isomorphism between M and N. The registered statement (Palomar/Challenge.lean) writes out the many-sorted first-order logic of H from scratch, with structures, formulas, satisfaction and definability with parameters. It states the axioms of H with Separation and Replacement as schemes over all definable classes, finitary interpretations in one-coordinate normal form (each interpreted sort is a definable subset of a single sort modulo a definable equivalence; the reduction from tuple domains is not formalized), presentations, and definable isomorphisms. The theorem is Theorem 2.1 of the accompanying paper, \"Solid idealized Lean\", where it is the tower half of the bi-interpretability of an idealized Lean with H. The proof (Palomar/Solution.lean) reduces the statement to the development's semantic solidity theorem tower_solid.", "authors": [ { "name": "Elliot Glazer" } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03C62", "03C40", "03F25", "03E30", "03E55", "03B38", "03B35" ] }, "formalization": { "theorem_names": [ "SolidLean.Palomar.tower_theory_solid" ] }, "id": "PALOMAR-2026-09-30-000022", "path": "entries/PALOMAR-2026-09-30-000022-v1.json", "preservation": { "repositories": [ { "commit": "337e7a5afd81ee7236c0e0dceadbc8cfc313f83f", "fork_repository": "PalomarArchive/elliotglazer--solid-lean--36b635673cfc", "source_repository": "elliotglazer/solid-lean" } ] }, "published_at": "2026-09-30T09:02:18Z", "source": { "commit": "337e7a5afd81ee7236c0e0dceadbc8cfc313f83f", "project_path": null, "repository": "elliotglazer/solid-lean" }, "status": "registered", "title": "solid-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For all finite subsets X,Y of an abelian group, |X-Y| <= |X+Y|^\u03bb with \u03bb = (9451e-3286)/(5378e+787) = 1.454277448906\u2026, less than the classical exponent 3/2. A multiscale transfer gives the upper bound (13524e-7359)/(9451e-3286) = 1.312373302115\u2026 for the Gyarmati\u2013Hennecart\u2013Ruzsa small-sumset exponent, less than their exponent 4/3. The proof of the first result combines a coupling with distinct differences, entropy inequalities for independent sums, a finite certificate using non-Shannon inequalities, and an explicit limiting certificate. The transfer uses localization and the Pl\u00fcnnecke-Ruzsa inequality over a sequence of scales.", "authors": [ { "name": "Logan J. Kleinwaks" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B30", "11B75", "94A17" ] }, "formalization": { "theorem_names": [ "SumDifferenceResults.pair_bound", "SumDifferenceResults.transfer_bound", "SumDifferenceResults.transfer_pair_exponent", "SumDifferenceResults.pair_exponent_bound", "SumDifferenceResults.theta_bound", "SumDifferenceResults.multiscale_bound" ] }, "id": "PALOMAR-2026-09-30-000021", "path": "entries/PALOMAR-2026-09-30-000021-v1.json", "preservation": { "repositories": [ { "commit": "9bd88c1a14ecedc6ecc6d96eb6403a62d9fc0989", "fork_repository": "PalomarArchive/kleinwaks--sum-difference--60abd0c4239b", "source_repository": "kleinwaks/sum-difference" } ] }, "published_at": "2026-09-30T08:28:23Z", "source": { "commit": "9bd88c1a14ecedc6ecc6d96eb6403a62d9fc0989", "project_path": null, "repository": "kleinwaks/sum-difference" }, "status": "registered", "title": "Improved sum\u2013difference inequalities in abelian groups", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Johnson's contact criterion for D-stability is expressed as an explicit principal-minor critical-point system. For every real square matrix and every choice of positive barrier weights, D-stability is equivalent to Hurwitz stability and the absence of positive solutions of the associated Fritz-John equations. The proof applies a proper exhaustion of the positive orthant to Johnson's contact variety, so that every nonempty contact set contains a critical point. In dimension five, the criterion is written explicitly as a Lagrange system and a tangency system, with coefficients given by principal minors. Both equivalences are formalized in Lean. The result applies the classical critical-point method to the characterization problem discussed by Kanovei and Logofet (1998) and Kushel (arXiv:2604.16526).", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS" ], "msc2020": [ "15A18", "34D20", "14P10" ] }, "formalization": { "theorem_names": [ "DStability5x5.dStable_iff_explicit", "DStability5x5.dStable_iff_five" ] }, "id": "PALOMAR-2026-09-30-000020", "path": "entries/PALOMAR-2026-09-30-000020-v1.json", "preservation": { "repositories": [ { "commit": "e0e376f5df0a639c619259a1e4a0a09648d25f93", "fork_repository": "PalomarArchive/mmislan--palomar-p087-d-stability-critical-points--c8b03e795d22", "source_repository": "mmislan/palomar-p087-d-stability-critical-points" } ] }, "published_at": "2026-09-30T08:16:26Z", "source": { "commit": "e0e376f5df0a639c619259a1e4a0a09648d25f93", "project_path": null, "repository": "mmislan/palomar-p087-d-stability-critical-points" }, "status": "registered", "title": "A critical-point criterion for D-stability in every dimension", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The computational difficulty of D-stability, raised by Chen, Fan and Yu (1995, doi:10.1016/0167-6911(94)00036-U), is addressed by an explicit integer-matrix reduction from PARTITION. For positive weights of total 2T, the construction produces a matrix of order m + 4 that is D-stable exactly when no subset has weight T. When an equal split exists, some positive diagonal scaling has an eigenvalue with strictly positive real part. A Pell recursion supplies the parameters, with a logarithmic search bound and an explicit bound on every matrix entry. These bounds give polynomial binary output length. Together with the classical NP-completeness of PARTITION, the reduction yields coNP-hardness of D-stability and D-semistability, and NP-hardness of strict D-instability, even for a four-dimensional core with attachments at one coordinate.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS" ], "msc2020": [ "15A18", "68Q17", "93D09" ] }, "formalization": { "theorem_names": [ "DStabilityHardness.dStability_coNP_reduction", "DStabilityHardness.partition_dichotomy" ] }, "id": "PALOMAR-2026-09-30-000019", "path": "entries/PALOMAR-2026-09-30-000019-v1.json", "preservation": { "repositories": [ { "commit": "4bca0dc5b1d87d12e789cfc817b91db5a84dec37", "fork_repository": "PalomarArchive/mmislan--palomar-p086-d-stability-conp-hard--e33c3a455fa7", "source_repository": "mmislan/palomar-p086-d-stability-conp-hard" } ] }, "published_at": "2026-09-30T06:03:55Z", "source": { "commit": "4bca0dc5b1d87d12e789cfc817b91db5a84dec37", "project_path": null, "repository": "mmislan/palomar-p086-d-stability-conp-hard" }, "status": "registered", "title": "Deciding D-stability is coNP-hard", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib proof that the 2-adic zeta value \u03b6\u2082(7) is irrational. The theorem has no hypotheses, and its proof uses only the axioms propext, Classical.choice and Quot.sound. The value is defined through the Volkenborn integral: zeta2 s = J(s\u22121)/((s\u22121)\u00b72^s), where J(s) is the 2-adic limit of the Riemann sums 2^(\u2212N) \u03a3_{x<2^N} (x+1/2)^(\u2212s), so zeta2 7 = J(6)/768; the theorem also proves that these limits exist. By Lemma 2.8 of Lai, Sprang and Zudilin (IMRN 2026), zeta2 s is the Kubota\u2013Leopoldt value \u03b6\u2082(s) = L\u2082(s, \u03c9^(1\u2212s)) for s \u2265 2. That identification is cited, not formalised; it only involves a non-zero rational factor, which does not affect irrationality. \u03b6\u2082(3) and \u03b6\u2082(5) are known to be irrational, and L. Lai (Int. J. Number Theory 2025) proved that at least one of \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11), \u03b6\u2082(13) is irrational; the theorem here implies that result and the two registered formalisations of the same maintainer for {\u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11)} and {\u03b6\u2082(7), \u03b6\u2082(9)}. An unrefereed draft by C. D. Long claims, by a different method, that \u03b6\u2082(s) is irrational for every odd s with 3 \u2264 s \u2264 29; it is not used here, and an internal review of it (docs/long_review.md) found a gap in its large-prime step, so we do not regard it as settling the question. No refereed proof that \u03b6\u2082(7) is irrational is known to us. The proof uses the linear forms S_n = \u222b R_n'''(t+1/2\u2212n/2) dt = \u03c1_n + Z_n \u03b6\u2082(7) of the rational functions R_n(t) = (2t+n) \u03a0_{m=1}^{6} \u03a0_{c\u2208C_m} (t+1/2+c) / \u03a0_{k=0}^{n} (t+k)^(a_k), with six nested well-poised numerator blocks C_m = [\u03b7_m n, (1\u2212\u03b7_m) n \u2212 1], \u03b7 = (5, 7, 7, 9, 11, 13)/40, and pole orders a_k \u2208 {1, 2, 3, 4} that ramp up from both ends; the evaluation at the centred point t + 1/2 \u2212 n/2 leaves \u03b6\u2082(7) as the only zeta value. The 2-adic size of S_n comes from Lai's and Sprang's \u0394-calculus, and the size of the coefficients from residue-level Cauchy bounds and a certified maximum of an explicit potential. The odd part of the denominators is bounded by a residue-class analysis of the partial fractions (a class lemma, a kernel identity and per-pole bounds, evaluated on local models of the residue classes whose values are certified in Lean on finitely many intervals) and the prime number theorem, proved here from a vendored Lean proof of the Wiener\u2013Ikehara theorem. If \u03b6\u2082(7) were rational, the valuations at the auxiliary prime \u2113 = 19n/20 \u2212 1 (an idea of Lai and Sprang) would make S_n non-zero for infinitely many n, and Lai's irrationality criterion gives the contradiction. The mathematics and the Lean code were produced by AI agents under human direction and have not been refereed.", "authors": [ { "name": "Gennaro Marco Devincenzis" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11S40", "11S80", "11M06" ] }, "formalization": { "theorem_names": [ "Zeta2.Z7.zeta2_7_irrational_palomar" ] }, "id": "PALOMAR-2026-09-30-000018", "path": "entries/PALOMAR-2026-09-30-000018-v1.json", "preservation": { "repositories": [ { "commit": "e26813f04b3e84e136c88c00753567e1b442d5db", "fork_repository": "PalomarArchive/gmDevi--zeta2-7-lean--d4961e7e615a", "source_repository": "gmDevi/zeta2-7-lean" } ] }, "published_at": "2026-09-30T05:52:42Z", "source": { "commit": "e26813f04b3e84e136c88c00753567e1b442d5db", "project_path": null, "repository": "gmDevi/zeta2-7-lean" }, "status": "registered", "title": "The 2-adic zeta value \u03b6\u2082(7) is irrational", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib formalization of the Krohn\u2013Rhodes prime decomposition theorem for finite monoids, in a form with aperiodic factors: every finite monoid M divides an iterated wreath product of finite monoids, each of which is aperiodic or a simple group, and every simple group among the factors divides M. The proof follows the local-divisor proof of Diekert, Kufleitner and Steinberg. Unlike the classical statement, the aperiodic factors are not reduced to copies of the three-element flip-flop monoid U\u2082; for an aperiodic M the statement holds with M as its only factor, and in general it separates the group structure of M, as simple groups dividing M, from aperiodic factors.", "authors": [ { "name": "Aditya Rao" } ], "classification": { "arxiv": [ "math.GR", "cs.FL" ], "msc2020": [ "20M10", "20M35", "20M20", "68Q70" ] }, "formalization": { "theorem_names": [ "KrohnRhodes.krohn_rhodes_prime_decomposition" ] }, "id": "PALOMAR-2026-09-30-000017", "path": "entries/PALOMAR-2026-09-30-000017-v1.json", "preservation": { "repositories": [ { "commit": "1f4a7e39ba2baede5434ac48c7571a442f8d521d", "fork_repository": "PalomarArchive/adii800--krohn-rhodes-lean--0944fc6761e8", "source_repository": "adii800/krohn-rhodes-lean" } ] }, "published_at": "2026-09-30T05:33:48Z", "source": { "commit": "1f4a7e39ba2baede5434ac48c7571a442f8d521d", "project_path": null, "repository": "adii800/krohn-rhodes-lean" }, "status": "registered", "title": "Krohn\u2013Rhodes decomposition of finite monoids into aperiodic and simple-group factors", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every integer d \u2265 4, every finite simple graph with fewer than C(d + 2, 2) edges has an injective placement in Euclidean d-space under which every edge has length 1, and the complete graph K_{d+2} has exactly C(d + 2, 2) edges and has no such placement. Equivalently, f(d) = C(d + 2, 2), where f(d) is the least number of edges in a finite simple graph with no unit-distance representation in R^d. Non-edges may also lie at distance 1. Finite simple graphs are represented as SimpleGraph (Fin n), including the empty graph. The proof is Frankl\u2013Kupavskii\u2013Swanepoel Theorem 3, already proved in the GraphDimension library and imported here. The spherical half of the same theorem is not stated. The question belongs to discrete geometry and extremal graph theory, specifically the study of unit-distance representations and the dimension of graphs.", "authors": [ { "name": "Dishant Shah" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C62", "52C10" ] }, "formalization": { "theorem_names": [ "FKSEdgeThreshold.Palomar.target" ] }, "id": "PALOMAR-2026-09-30-000016", "path": "entries/PALOMAR-2026-09-30-000016-v1.json", "preservation": { "repositories": [ { "commit": "b0577b932a2780463139c25aedde6ea222bb6bd1", "fork_repository": "PalomarArchive/Dishah3241--FKSEdgeThreshold--c82c6fe6c960", "source_repository": "Dishah3241/FKSEdgeThreshold" } ] }, "published_at": "2026-09-30T05:23:34Z", "source": { "commit": "b0577b932a2780463139c25aedde6ea222bb6bd1", "project_path": null, "repository": "Dishah3241/FKSEdgeThreshold" }, "status": "registered", "title": "Frankl-Kupavskii-Swanepoel Theorem 3: fewer than C(d+2,2) edges embeds in R^d", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the near-linear cycle theorem in \"Lov\u00e1sz conjecture holds asymptotically\". For every positive \u03b5, sufficiently large connected finite vertex-transitive graphs contain a simple cycle with at least n^(1\u2212\u03b5) edges. The development includes a weaker-constant Bondy\u2013Locke path-to-cycle comparison and the finite approximate-group and expansion arguments used in the path proofs.", "authors": [ { "name": "Matija Bucic" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C45", "05C38" ] }, "formalization": { "theorem_names": [ "PalomarLovasz.vertex_transitive_long_cycle", "PalomarLovasz.cayley_long_cycle", "PalomarLovasz.cayley_long_path", "PalomarLovasz.vertex_transitive_long_path" ] }, "id": "PALOMAR-2026-09-30-000015", "path": "entries/PALOMAR-2026-09-30-000015-v1.json", "preservation": { "repositories": [ { "commit": "7f2728d07ce648ed74cae08bbfa7084b9ec13f4b", "fork_repository": "PalomarArchive/mbucic4--Nearly-spanning-Lovasz--dc3fe5736fab", "source_repository": "mbucic4/Nearly-spanning-Lovasz" } ] }, "published_at": "2026-09-30T03:42:26Z", "source": { "commit": "7f2728d07ce648ed74cae08bbfa7084b9ec13f4b", "project_path": null, "repository": "mbucic4/Nearly-spanning-Lovasz" }, "status": "registered", "title": "Near-linear cycles in connected vertex-transitive graphs", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The red-blue pebble game of Hong Jia-Wei and H. T. Kung (STOC 1981) models a computation on a directed acyclic graph run with S words of fast memory and unlimited slow memory: a red pebble on a vertex is its value held in fast memory, a blue pebble its value held in slow memory, and each load or store costs one I/O operation. This formalisation defines the game, complete calculations and the minimum I/O time Q, and proves three families of results from the paper, with explicit constants and, where the paper states one, in its asymptotic form. Q is 0 by convention where no complete calculation exists, so each statement about Q is made only where one does. First, the partition lemma (Theorem 3.1 and Lemma 3.1): every complete calculation with at most S red pebbles and q I/O operations yields a 2S-partition of the graph into h sets with q <= S h <= q + S, so that Q >= S (P(2S) - 1) wherever a complete calculation exists. Read literally, the paper's Theorem 3.1 is false in two ways, since its rules let inputs be computed free of charge and let inputs be vertices other than the sources; here, as in Elango et al. (2014), the inputs are exactly the sources and cannot be computed. Second, the n-point FFT graph, for n = 2^k with k >= 1: every S-dominator partition has h sets with n (log2 n + 1) <= h S log2(2S) (Theorem 4.1, with a per-set bound sharper than the paper's, which also closes a gap in its proof), a complete calculation exists exactly when S >= 3, and every complete calculation has q >= 2n and n log2 n <= 5 q log2 S (Corollary 4.1 with an explicit constant); Q log S = Omega(n log n) holds with one constant for all such n and all S >= 3, and P_D(S) = Omega(n log n / (S log S)) with one constant for all such n and all S >= 2. Third, matrix multiplication by the ordinary algorithm, for every graph in which the m k n products A_il B_lj are vertices of their own, each with an edge from its two factors, and no vertex can be reached from the products of two different entries of the result; for m, k, n >= 1 this class contains every independent evaluation in the paper's sense. Every complete calculation has m k n <= 2 q sqrt(S) (Corollary 6.2 with an explicit constant), and for k >= 1 also q >= mk + kn + mn; Q sqrt(S) = Omega(m k n) holds with one constant over any collection of such graphs, for every graph of the collection and every S at which that graph has a complete calculation; and for all m, k, n >= 1 some such graph has a complete calculation exactly when S >= 3. Mathlib has no pebble games, so the statements rest on twelve definitions of the formalisation's own, which the Challenge restates for the reader. Not formalised: the paper's upper bounds (its Theorem 2.1 and section 8, which it states without proof), its sections 5 and 7, and its Theorem 6.1 for general expressions.", "authors": [ { "name": "George A. Constantinides" } ], "classification": { "arxiv": [ "cs.CC", "cs.DS" ], "msc2020": [ "68Q17", "68Q25", "05C57", "65T50", "65Y20" ] }, "formalization": { "theorem_names": [ "MiscMath.Computability.exists_partition_of_hasCompleteCalculation", "MiscMath.Computability.io_lower_bound_of_parts", "MiscMath.Computability.minIOTime_lower_bound", "MiscMath.Computability.fft_parts_lower_bound", "MiscMath.Computability.fft_parts_lower_bound_isBigO", "MiscMath.Computability.fft_complete_iff", "MiscMath.Computability.fft_io_bounds", "MiscMath.Computability.fft_io_lower_bound", "MiscMath.Computability.fft_io_lower_bound_isBigO", "MiscMath.Computability.exists_isMatMulEvaluation", "MiscMath.Computability.matMul_io_bounds", "MiscMath.Computability.matMul_io_lower_bound", "MiscMath.Computability.matMul_io_lower_bound_isBigO" ] }, "id": "PALOMAR-2026-09-30-000014", "path": "entries/PALOMAR-2026-09-30-000014-v1.json", "preservation": { "repositories": [ { "commit": "95624a5e96f93b05b744843f9fcd762503be1d85", "fork_repository": "PalomarArchive/geoconuk--lean-misc-math--79e5c7089a59", "source_repository": "geoconuk/lean-misc-math" } ] }, "published_at": "2026-09-30T03:30:54Z", "source": { "commit": "95624a5e96f93b05b744843f9fcd762503be1d85", "project_path": null, "repository": "geoconuk/lean-misc-math" }, "status": "registered", "title": "Hong and Kung's red-blue pebble game: the partition lemma and I/O lower bounds for the FFT and matrix multiplication", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the irrationality of Li\u2082(1/2) = \u03a3_{n\u22651} 2^(-n)/n\u00b2. The proof is inspired by A. Fauzan's work on the irrationality of \u03b6(5) and builds on Moritz Firsching's Lean formalization in mo271/Zeta5. The central idea is to establish the irrationality of a real number r by constructing a sequence of integer polynomials P_n whose values P_n(r) decay rapidly as n tends to infinity, while for every rational q, P_n(q) \u2260 0 for infinitely many n. Analytic umbral calculus was used to find suitable polynomials Q_n with rational coefficients. Random matrix theory provides rapid decay of Q_n(r) as n tends to infinity. Estimates of p-adic valuations, together with the prime number theorem, ensure that the values P_n(r) of the corresponding primitive integer polynomials also decay rapidly as n tends to infinity, completing the proof. Similar methods may apply to other values Li\u2082(1/n), but those cases are not included here. The compared theorem depends on propext, Classical.choice and Quot.sound and on nothing else, and there is no sorry outside the one deliberate hole in Challenge.lean.", "authors": [ { "name": "Qian Tang" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "33B30" ] }, "formalization": { "theorem_names": [ "li2_one_half_irrational" ] }, "id": "PALOMAR-2026-09-30-000013", "path": "entries/PALOMAR-2026-09-30-000013-v1.json", "preservation": { "repositories": [ { "commit": "950ee9d53914e1afb89c8f835a8bca6ccfaa0108", "fork_repository": "PalomarArchive/dtq1997--li2-half-irrationality--ba6c4988f6ba", "source_repository": "dtq1997/li2-half-irrationality" } ] }, "published_at": "2026-09-30T03:21:07Z", "source": { "commit": "950ee9d53914e1afb89c8f835a8bca6ccfaa0108", "project_path": null, "repository": "dtq1997/li2-half-irrationality" }, "status": "registered", "title": "Irrationality of Li\u2082(1/2)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "The kissing number of Euclidean 3-space is 12: at most twelve unit balls with disjoint interiors touch a given unit ball, and twelve do. The upper bound follows Musin's extension of the Delsarte linear-programming method. The Delsarte machinery on the sphere (positive-definite zonal kernels up to degree 9 and the LP bound, which by itself gives only 13) is built from scratch, since Mathlib has no spherical harmonics. The lower bound is the cuboctahedron.", "authors": [ { "name": "Marek Wrzos" } ], "classification": { "arxiv": [ "math.MG", "cs.LO" ], "msc2020": [ "52C17", "05B40", "94B65", "68V20" ] }, "formalization": { "theorem_names": [ "KissingNumber.kissingNumber_three", "KissingNumber.neighbors_le_twelve" ] }, "id": "PALOMAR-2026-09-30-000012", "path": "entries/PALOMAR-2026-09-30-000012-v1.json", "preservation": { "repositories": [ { "commit": "2e403b0d1afe84f41a90cb77d926bc31666a8972", "fork_repository": "PalomarArchive/Pendzoncymisio--kissing-number-lean--012113b2f3eb", "source_repository": "Pendzoncymisio/kissing-number-lean" } ] }, "published_at": "2026-09-30T03:10:53Z", "source": { "commit": "2e403b0d1afe84f41a90cb77d926bc31666a8972", "project_path": null, "repository": "Pendzoncymisio/kissing-number-lean" }, "status": "registered", "title": "kissing-number-lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Garc\u00eda Puente et al. (arXiv:2401.00078v3, Question 3.13) asked whether a unique positive zero-divisor value is necessary or sufficient for absolute concentration robustness (ACR) in at-most-bimolecular mass-action networks. Explicit networks with positive rational rates answer both questions negatively. A further three-species example shows that their candidate algorithm can miss a zero-divisor ACR value under an admissible elimination order, refuting the unrestricted elimination-order form of Conjecture 4.7. For block orders that compare the other species before the distinguished species, completeness is established: every real zero-divisor ACR value is returned when the finite basis consists of nonzero steady-ideal elements whose leading monomials cover the ideal. ACR includes existence of a positive steady state; zero-divisor values include coordinate differences lying in the steady-state ideal.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.AC", "q-bio.MN" ], "msc2020": [ "13P10", "13P25", "92C42" ] }, "formalization": { "theorem_names": [ "ACRZeroDivisors.question313_necessity_counterexample", "ACRZeroDivisors.question313_sufficiency_counterexample", "ACRZeroDivisors.OrderWitness.unrestricted_order_counterexample", "ACRZeroDivisors.mass_action_block_algorithm_complete" ] }, "id": "PALOMAR-2026-09-30-000011", "path": "entries/PALOMAR-2026-09-30-000011-v1.json", "preservation": { "repositories": [ { "commit": "a2bcfab8f3015df957bc20da1db66cbf1ae98bd0", "fork_repository": "PalomarArchive/mmislan--palomar-p082-acr-zero-divisors--53da94089871", "source_repository": "mmislan/palomar-p082-acr-zero-divisors" } ] }, "published_at": "2026-09-30T03:01:08Z", "source": { "commit": "a2bcfab8f3015df957bc20da1db66cbf1ae98bd0", "project_path": null, "repository": "mmislan/palomar-p082-acr-zero-divisors" }, "status": "registered", "title": "Zero-divisor criteria for absolute concentration robustness", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases (Ann. of Math. 165 (2007)), built on the repository's libraries ArithmeticHeights, DiophantineApproximation and ForMathlib, whose Subspace Theorem with algebraic coefficients it uses. The library AdamczewskiBugeaud2007 is sorry-free and within Lean's three standard axioms. It proves Theorem 1 (the complexity of an irrational algebraic number in an integer base satisfies p(n)/n \u2192 \u221e, with (1), p(n) \u2212 n \u2192 \u221e), Theorems 2\u20134 (automatic and morphic expansions), Theorem 5 in full (\u03b2 Pisot or Salem, bounded integer digits satisfying Condition (\u2217)_w: the number lies in \u211a(\u03b2) or is transcendental), Lemma 1, Theorems 1A\u20135A on \u03b2-expansions with K. Schmidt's periodicity theorem, and the p-adic Theorems 6 and 1B\u20134B, together with the finite automata and morphisms of \u00a73, and Theorem 7 in corrected form (u_k < min(b, p)) with Christol's theorem (algebraic \u21d2 automatic) and a counterexample to Theorem 7 as printed. The counterexample and the corrected hypothesis were worked out during this formalization, not taken from the cited works (see limitations). Thirty-two statements are compared.", "authors": [ { "name": "Ralf Stephan" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J81", "11A63", "11K16", "11R06", "68R15" ] }, "formalization": { "theorem_names": [ "Real.tendsto_complexity_div_atTop", "Real.tendsto_complexity_sub_atTop", "Real.transcendental_ofDigits_of_isGeneratedByAutomaton", "Real.transcendental_ofDigits_of_isAutomatic", "Real.transcendental_ofDigits_of_isGeneratedByMorphism", "Real.transcendental_ofDigits_of_isGeneratedByMorphism_of_isRecurrent", "Function.finite_kKernel_iff_isGeneratedByAutomaton", "Function.IsAutomatic.exists_complexity_le", "Function.IsAutomatic.isStammering", "Real.abs_sub_le_of_periodic", "Real.mem_adjoin_of_isStammering", "Real.mem_adjoin_or_transcendental_of_isPisot", "Real.mem_adjoin_or_transcendental_of_isSalem", "Real.tsum_mem_adjoin_of_isEventuallyPeriodic", "Real.tendsto_complexity_betaDigitsFin_div_atTop", "Real.not_isGeneratedByAutomaton_betaDigitsFin", "Real.not_isAutomatic_betaDigitsFin", "Real.not_isGeneratedByMorphism_of_betaDigits", "Real.not_isGeneratedByMorphism_betaDigitsFin_of_isRecurrent", "Real.isEventuallyPeriodic_betaDigits_of_mem_adjoin", "Real.transcendental_of_isStammering_betaDigits", "Padic.transcendental_of_isStammering_henselDigits", "Padic.transcendental_ofDigits_of_isStammering", "Padic.tendsto_complexity_henselDigits_div_atTop", "Padic.tendsto_complexity_div_atTop", "Padic.transcendental_ofDigits_of_isAutomatic", "Padic.transcendental_ofDigits_of_isGeneratedByMorphism", "Padic.transcendental_ofDigits_of_isGeneratedByMorphism_of_isRecurrent", "PowerSeries.finite_kKernel_coeff_of_isAlgebraic", "Real.isAlgebraic_and_isAlgebraic_iff", "exists_counterexample_theorem7", "not_forall_theorem7_as_printed" ] }, "id": "PALOMAR-2026-09-30-000010", "path": "entries/PALOMAR-2026-09-30-000010-v1.json", "preservation": { "repositories": [ { "commit": "ce289c64054c8ffcfb6fd37d5881312c6a5d577d", "fork_repository": "PalomarArchive/rwst--Subspace-Theorems--9bd9bba330e8", "source_repository": "rwst/Subspace-Theorems" } ] }, "published_at": "2026-09-30T02:38:23Z", "source": { "commit": "ce289c64054c8ffcfb6fd37d5881312c6a5d577d", "project_path": null, "repository": "rwst/Subspace-Theorems" }, "status": "registered", "title": "Adamczewski\u2013Bugeaud 2007: the complexity of algebraic numbers in integer bases", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Vassena (arXiv:2608.27081v2, Section 8) left open whether sequential distributive phosphorylation cycles with three or more sites admit Hopf bifurcation under mass-action kinetics. For every n >= 3, a componentwise-affine family of positive rate constants is constructed for the n-site cycle with one kinase and one phosphatase and a fixed positive equilibrium. A family of strictly positive, nonconstant periodic orbits bifurcates from this equilibrium, with amplitudes tending to zero and periods tending to a finite positive limit. Each orbit is orbitally asymptotically stable within its compatibility class: sufficiently nearby states with the same kinase, phosphatase and substrate totals have unique positive forward solutions that remain close to the orbit and converge to it.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "34C23", "37G15", "34D20", "92C45" ] }, "formalization": { "theorem_names": [ "ThreeSitePhosphorylation.AllSiteAttractingFamily.all_site_attracting_hopf" ] }, "id": "PALOMAR-2026-09-30-000009", "path": "entries/PALOMAR-2026-09-30-000009-v1.json", "preservation": { "repositories": [ { "commit": "99597afe8632cbd41f0ffec6c04f73a554ba9881", "fork_repository": "PalomarArchive/mmislan--palomar-p064-multisite-phosphorylation-oscillations--3703a9c14353", "source_repository": "mmislan/palomar-p064-multisite-phosphorylation-oscillations" } ] }, "published_at": "2026-09-30T02:26:38Z", "source": { "commit": "99597afe8632cbd41f0ffec6c04f73a554ba9881", "project_path": null, "repository": "mmislan/palomar-p064-multisite-phosphorylation-oscillations" }, "status": "registered", "title": "Attracting oscillations in sequential distributive phosphorylation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Vassena and Stadler (arXiv:2308.11486v3, Supplementary Example D) conjectured that sequential distributive phosphorylation cycles have only unstable-positive cores and that every invertible child-selection matrix has determinant of absolute value one. The determinant assertion is proved, and the restriction to unstable-positive cores is disproved. In the n-site cycle, every child-selection matrix has determinant 0, 1 or -1. For every n >= 3, a six-species child-selection matrix is minimally Hurwitz-unstable and has determinant 1, giving an unstable-negative core. For every n >= 2, there is an unstable-positive core on 2n+1 species, so their sizes are unbounded; every proper principal submatrix remains nonunstable under every positive column scaling. For every n >= 3, there is also a five-species D-unstable core: a positive column scaling makes it unstable, while every proper principal submatrix remains nonunstable under all such scalings.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS", "q-bio.MN" ], "msc2020": [ "92C42", "15A15", "15A18", "34D20" ] }, "formalization": { "theorem_names": [ "FutileCycle.resolution", "FutileCycle.positive_core_all_n", "FutileCycle.negativeFive_core_all_n" ] }, "id": "PALOMAR-2026-09-30-000008", "path": "entries/PALOMAR-2026-09-30-000008-v1.json", "preservation": { "repositories": [ { "commit": "87366c2f6997480f4127b634c813e8daf8199abf", "fork_repository": "PalomarArchive/mmislan--palomar-p038-futile-cycle-child-selections--c80de064fed9", "source_repository": "mmislan/palomar-p038-futile-cycle-child-selections" } ] }, "published_at": "2026-09-30T02:15:36Z", "source": { "commit": "87366c2f6997480f4127b634c813e8daf8199abf", "project_path": null, "repository": "mmislan/palomar-p038-futile-cycle-child-selections" }, "status": "registered", "title": "Determinants and unstable cores in sequential distributive phosphorylation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib proof that at least one of the eight values \u03b6(7), \u03b6(9), \u03b6(11), \u03b6(13), \u03b6(15), \u03b6(17), \u03b6(19), \u03b6(21) of the Riemann zeta function is irrational. The statement uses only Mathlib's riemannZeta, the theorem has no hypotheses, and its proof uses only the axioms propext, Classical.choice and Quot.sound. The shortest list of odd zeta values starting at \u03b6(7) that our literature search found proved to contain an irrational number was \u03b6(7), \u03b6(9), \u2026, \u03b6(35) (W. Zudilin, Math. Notes 2001). The proof uses the very-well-poised linear forms of Zudilin (J. Th\u00e9or. Nombres Bordeaux 2004, \u00a78) with five derivatives, 23 factors and the direction vector \u03b7\u2080 = 160, \u03b7 = (47, 47, 47, 47, 48, 50, 50, 51, 52, \u2026, 66); they are linear forms in 1, \u03b6(7), \u03b6(9), \u2026, \u03b6(21) with rational coefficients. Zudilin anticipated such an improvement but could not prove the asymptotics of his forms (his Lemma 20) for more than three derivatives. The proof here does not need them. The denominators of the coefficients are bounded by Zudilin's arithmetic lemma, the prime number theorem (proved from a vendored Lean proof of the Wiener\u2013Ikehara theorem) and a kernel-checked table of 2602 pieces of Zudilin's function \u03c6. For large n the forms are bounded by e^(\u2212748.1n), through a contour integral on a fixed vertical line, pointwise estimates and certified numerical inequalities, without saddle-point analysis. They are non-zero if the eight values are rational, because at the auxiliary primes \u2113 = 63n \u2212 1, of which there are infinitely many by Dirichlet's theorem, the constant term has \u2113-adic valuation \u22125 while the coefficients of the zeta values are \u2113-integral. The theorem does not say which value is irrational. The mathematics and the Lean code were produced by AI agents under human direction and have not been refereed.", "authors": [ { "name": "Gennaro Marco Devincenzis" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11M06", "33C20" ] }, "formalization": { "theorem_names": [ "ZetaWindow.zeta_7_to_21_not_all_rational_palomar" ] }, "id": "PALOMAR-2026-09-30-000007", "path": "entries/PALOMAR-2026-09-30-000007-v1.json", "preservation": { "repositories": [ { "commit": "a3b69fc43aea3dd140540f6e8380c7d0fecce028", "fork_repository": "PalomarArchive/gmDevi--zeta-7-21-lean--801a5a28c754", "source_repository": "gmDevi/zeta-7-21-lean" } ] }, "published_at": "2026-09-30T01:55:49Z", "source": { "commit": "a3b69fc43aea3dd140540f6e8380c7d0fecce028", "project_path": null, "repository": "gmDevi/zeta-7-21-lean" }, "status": "registered", "title": "At least one of \u03b6(7), \u03b6(9), \u2026, \u03b6(21) is irrational", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 proof development of reduction idea toward NP-hardness of feature selection for relative Gaussian c-separation in the formulation of Borozan et al. The argument starts from NP-completeness of equal-cardinality partition of positive squares, giving integral Gaussian data via integer roots and an even paired total. Three Gaussians enforce partition balance, and a schedule of d+1 Gaussians enforces binary selectors. Explicit separation of the two blocks makes cross-block hinge penalties vanish. The fixed threshold is (dot s 1 / 2)*d*(d+1)/2 - 3*d/4, an integer multiple of 1/4 for natural input weights. Here d is the full dimension. The main proved result is the pointwise gadget iff. The full complexity reduction is not formalized. The threshold depends only on the input. General Karamata inequality and an unconditional ordinary-PARTITION to perfect-square equal-cardinality partition equivalence are auxiliary results stated in Solution.lean. The square encoding has proved injectivity and numerical size bounds; its NP-completeness argument is documented prose. The original Gaussian gadget predicate also has a proved binary certificate verifier with soundness, completeness, and quadratic work in a bit-cost model. The Gaussian gadget equivalence and square-partition reduction were developed and first presented by the author in this formalization. Their mathematical novelty and priority relative to earlier literature are unknown; no systematic literature search or comparison establishing them has been documented.", "authors": [ { "name": "Tomislav Prusina" } ], "classification": { "arxiv": [ "cs.LO", "cs.CC" ], "msc2020": [ "03B35", "68Q17" ] }, "formalization": { "theorem_names": [ "CSeparatedNPComplete.partition_gadget_schedule_partition_iff" ] }, "id": "PALOMAR-2026-09-30-000006", "path": "entries/PALOMAR-2026-09-30-000006-v1.json", "preservation": { "repositories": [ { "commit": "d70de3f969dd580506e0835f7d7773890b0a6eaa", "fork_repository": "PalomarArchive/tomo61098--c-separated-np-complete-lean--1698d4b0f764", "source_repository": "tomo61098/c-separated-np-complete-lean" } ] }, "published_at": "2026-09-30T01:45:33Z", "source": { "commit": "d70de3f969dd580506e0835f7d7773890b0a6eaa", "project_path": null, "repository": "tomo61098/c-separated-np-complete-lean" }, "status": "registered", "title": "NP-hardness of Gaussian c-separation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean 4 results (core Lean, no dependencies, no axioms beyond the standard three) about law 1518 of the Equational Theories Project, x = (y \u25c7 y) \u25c7 (x \u25c7 (y \u25c7 x)). Theorem A: in every magma satisfying 1518 and 3862 (x \u25c7 x = (x \u25c7 (x \u25c7 x)) \u25c7 x), writing S x = x \u25c7 x, the set {x, S x, S (S x)} is closed under \u25c7 with u \u25c7 v = S v and S (S (S x)) = x; hence every one-generated such magma is the trivial magma or the three-element cyclic shift x \u25c7 y = y + 1. This proves a classification conjectured by Terence Tao on the Lean Zulip on 2024-11-29 (message 485148288). That message assumed 1518 together with all four of 47, 614, 817 and 3862 and carried no finiteness hypothesis; assuming 3862 alone is no weaker, because Matthew Bolan reported the same day (message 485153888) that 1518 and 3862 already imply the other three. What is added here is a machine-checked proof, not a weaker hypothesis: no proof of the classification was found in the venues searched. Theorem A is what the compared declarations state. Outside them the repository holds an explicit family of finite 1518-magmas that do violate the four targets, with two members checked by kernel evaluation, and kernel-checked certificate identities behind a cohomological obstruction theorem (every 1518-cocycle over the shift with finite fibre is a coboundary), whose reduction is a written proof, and the computations classifying base-dependent extensions of the shift.", "authors": [ { "name": "Carlo Perassi" } ], "classification": { "arxiv": [ "math.RA", "math.LO" ], "msc2020": [ "20N02", "08B05", "03C05", "68V20" ] }, "formalization": { "theorem_names": [ "Magma1518.table", "Magma1518.cube", "Magma1518.words_in_T", "Magma1518.one_or_three" ] }, "id": "PALOMAR-2026-09-30-000005", "path": "entries/PALOMAR-2026-09-30-000005-v1.json", "preservation": { "repositories": [ { "commit": "eb2bbde8c137570d0e281a237187f5323a1b0a2d", "fork_repository": "PalomarArchive/carlok--magma-1518-obstruction-lean--763845730337", "source_repository": "carlok/magma-1518-obstruction-lean" } ] }, "published_at": "2026-09-30T01:30:28Z", "source": { "commit": "eb2bbde8c137570d0e281a237187f5323a1b0a2d", "project_path": null, "repository": "carlok/magma-1518-obstruction-lean" }, "status": "registered", "title": "One-generated (1518 + 3862)-magmas are trivial or the Z/3 shift; an explicit family of finite 1518-magmas violating laws 47, 614, 817, 3862", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean proof resolving the Burr\u2013Erd\u0151s\u2013Graham\u2013S\u00f3s conjecture (Erd\u0151s Problem 809): for every fixed odd cycle of length at least seven, the maximal anti-Ramsey threshold at floor(n\u00b2/4) + 1 edges is n\u00b2/8 + o(n\u00b2). The seven-cycle case is new to our knowledge. The development proves it and formalizes the stronger full-density result of Buci\u0107, Chen, and Ma for odd cycles of length at least nine.", "authors": [ { "name": "Jacob Parish" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15", "05C35" ] }, "formalization": { "theorem_names": [ "Erdos809.main_result" ] }, "id": "PALOMAR-2026-09-30-000004", "path": "entries/PALOMAR-2026-09-30-000004-v1.json", "preservation": { "repositories": [ { "commit": "0f27743fabd97b262d2fce7946de9f0b3325a625", "fork_repository": "PalomarArchive/plasma-ai--erdos-809--95e57efacaad", "source_repository": "plasma-ai/erdos-809" } ] }, "published_at": "2026-09-30T01:21:55Z", "source": { "commit": "0f27743fabd97b262d2fce7946de9f0b3325a625", "project_path": null, "repository": "plasma-ai/erdos-809" }, "status": "registered", "title": "Erd\u0151s Problem 809: rainbow odd cycles", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Erd\u0151s problem #1219 (erdosproblems.com/1219; Erd\u0151s\u2013Hajnal 1971, Problem 3, asked by Erd\u0151s, Hajnal and Rado): let (n_k) be an increasing sequence of integers such that 2^{\u2135_{n_k}} is strictly increasing and 2^{\u2135_{n_0}} > \u2135_\u03c9; is it true that \u03a3_k 2^{\u2135_{n_k}} \u2192 (\u2135_\u03c9)\u00b2? Shelah (Notes on partition calculus, 1975, \u00a71) proved that the answer is yes. This project formalizes that proof in Lean 4 with Mathlib: the comparator target erdos_1219 states that for every strictly increasing n : \u2115 \u2192 \u2115 with k \u21a6 2 ^ \u2135_(n k) strictly increasing and \u2135_\u03c9 < 2 ^ \u2135_(n 0), the binary partition relation PartitionArrow (\u03a3_k 2 ^ \u2135_(n k)) \u2135_\u03c9 holds, i.e. every 2-colouring of the 2-element subsets of a set of cardinality \u03a3_k 2 ^ \u2135_(n k) has a monochromatic subset of cardinality \u2135_\u03c9 (PartitionArrow is the binary case of formal-conjectures' Combinatorics.cardinalPartitionRel). The proof follows Shelah's Canonization Lemma 1.1 and Theorem 1.2: the unbalanced Erd\u0151s\u2013Rado theorem (2^\u03bc)\u207a \u2192 ((2^\u03bc)\u207a, \u03bc\u207a)\u00b2 (proved here with a critical-ordinal / closure-point argument), a canonization of the colouring across blocks of sizes (2^{\u2135_{n_k}})\u207a by means of guides generic over all small subsets of the earlier blocks, and Ramsey's theorem for pairs on \u2115. The target is proved from propext, Classical.choice and Quot.sound only and passes the Lean comparator (landrun sandbox, NanoDa kernel).", "authors": [ { "name": "Ji Ho Bae (JRTI)" } ], "classification": { "arxiv": [ "math.LO", "math.CO" ], "msc2020": [ "03E02", "03E05", "03E10", "05D10" ] }, "formalization": { "theorem_names": [ "erdos_1219" ] }, "id": "PALOMAR-2026-09-30-000003", "path": "entries/PALOMAR-2026-09-30-000003-v1.json", "preservation": { "repositories": [ { "commit": "00e696a9cd2bbd8cad33d2cdc90ad385058c1466", "fork_repository": "PalomarArchive/jbaelaw--erdos1219-lean--9297707161f0", "source_repository": "jbaelaw/erdos1219-lean" } ] }, "published_at": "2026-09-30T01:10:42Z", "source": { "commit": "00e696a9cd2bbd8cad33d2cdc90ad385058c1466", "project_path": null, "repository": "jbaelaw/erdos1219-lean" }, "status": "registered", "title": "Erd\u0151s Problem #1219 in Lean 4: \u03a3_k 2^{\u2135_{n_k}} \u2192 (\u2135_\u03c9)\u00b2 (Shelah 1975)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 characterization of two-fold Langford sequences. An m-fold Langford sequence of order l and defect d fills the positions 1, ..., 2ml so that every p in [d, d + l - 1] occupies the endpoints of m pairwise disjoint pairs of positions at distance p. For d, l >= 1 a two-fold sequence exists exactly when 3l >= 2d - 1. For every m, (2m - 1)l >= 2d - 1 is necessary, with l(2d + l + 1) = 0 (mod 4) for odd m, and the bound is attained whenever 2d + l = 2ml + 1; the row of order one exists exactly when d divides m; no three-fold sequence of order 3 and defect 6 exists; and for every m >= 3 the necessary conditions are not sufficient. For every T >= 1, with r = ml mod T, every m-fold sequence satisfies the residue bound r(T - r) <= m * sum |p - T| over p in [d, d + l - 1], which contains the counting bound and the necessity half of the order-one theorem. Every m-fold sequence satisfies the forced-endpoint bound 6mld + ml <= 4d^2 + 2(ml)^2 + ml^2. For every m >= 3 the cell (d, l) = (3m - 3, 3) meets the necessary conditions and admits no m-fold sequence. Equality 2d + l = 2ml + 1 holds exactly when every pair straddles the middle, that is ml < i + s_i for every position i <= ml. Alkasasbeh, Dyer and Howell define these sequences and ask for necessary and sufficient conditions when m >= 2; this repository answers the question for m = 2. Necessity is a distance sum. Sufficiency at m = 2 is strong induction on l: sixteen literal cells, concatenation, and otherwise a mirror-symmetric sequence built by Lemma S from a signed permutation, which is found on the whole cone by inversion, the reversal, a descent along lines of fixed mu, and in the band by 21 block-reversal families plus the two-block reversal tau_l, eight sporadic cells and three bases. The tight line is m interleaved copies of the source's Table 1.", "authors": [ { "name": "JD Jones" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A05", "11B83" ] }, "formalization": { "theorem_names": [ "Langford.twoFold_exists_iff", "Langford.necessary", "Langford.tight_exists", "Langford.order_one_iff", "Langford.not_threeFold_six_three", "Langford.not_sufficient", "Langford.residue_bound", "Langford.forced_endpoint", "Langford.not_order_three", "Langford.tight_iff_straddle" ] }, "id": "PALOMAR-2026-09-30-000002", "path": "entries/PALOMAR-2026-09-30-000002-v1.json", "preservation": { "repositories": [ { "commit": "c4cbeafac7e44ebcbbee9a8839f70278bb19b6ab", "fork_repository": "PalomarArchive/JD-Jones-ASES--l2-lean--fd9f0be0adc0", "source_repository": "JD-Jones-ASES/l2-lean" } ] }, "published_at": "2026-09-30T01:00:15Z", "source": { "commit": "c4cbeafac7e44ebcbbee9a8839f70278bb19b6ab", "project_path": null, "repository": "JD-Jones-ASES/l2-lean" }, "status": "registered", "title": "Two-fold Langford sequences exist exactly when 3l \u2265 2d \u2212 1; the m-fold counting bound and its attainment, and a residue bound", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We answer the set-theoretic characterization question for digraph support families posed by Mike Steel in Section 4 of Interior operators and their relationship to autocatalytic networks (arXiv:2305.16714v4, 2023). A digraph on a finite set E is given by a predecessor map P assigning to each vertex e a set P(e) of in-neighbours, loops allowed, and a set S is supported when every element e of S has some element of S in P(e). The characterization refers only to the family F. Start from the full power set of E with no roots used. A legal step chooses a remaining set S outside F that is inclusion-maximal among the remaining sets outside F, chooses a root r in S that has not been used before and lies in no member of F contained in S, and deletes every remaining set X with r in X and X contained in S. A family F of subsets of E is exactly the support family of some predecessor map on E if and only if some finite sequence of legal steps ends exactly at F.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C20", "06A15", "05D05" ] }, "formalization": { "theorem_names": [ "DigraphRealizability.publication_resolution" ] }, "id": "PALOMAR-2026-09-30-000001", "path": "entries/PALOMAR-2026-09-30-000001-v1.json", "preservation": { "repositories": [ { "commit": "579ba54f592330e8fd972d9e5af02699f7fead6c", "fork_repository": "PalomarArchive/mmislan--palomar-p029-digraph-support-families--cb671abcfe16", "source_repository": "mmislan/palomar-p029-digraph-support-families" } ] }, "published_at": "2026-09-30T00:48:43Z", "source": { "commit": "579ba54f592330e8fd972d9e5af02699f7fead6c", "project_path": null, "repository": "mmislan/palomar-p029-digraph-support-families" }, "status": "registered", "title": "Characterizing digraph support families by rooted-interval deletion", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A formalization in Lean 4 and Mathlib of the Erd\u0151s discrepancy theorem, proved by Terence Tao in 2015: for every sequence f(1), f(2), \u2026 of signs \u00b11 and every C there are d \u2265 1 and n with |f(d) + f(2d) + \u22ef + f(nd)| > C, so every \u00b11 sequence has unbounded discrepancy along homogeneous arithmetic progressions. This is the original \u00b11 formulation, Corollary 1.2 of Tao's paper; its Hilbert-space-valued Theorem 1.1 is not formalized. The proof follows the architecture of Tao's argument: a Fourier-analytic reduction from a sequence of bounded discrepancy to a stochastic completely multiplicative function with bounded mean-square partial sums; a van der Corput argument, driven by the logarithmically averaged two-point Elliott estimate, forcing such a function to pretend to be a twisted Dirichlet character; and a generalized Borwein\u2013Choi\u2013Coons argument excluding that case. The analytic number theory these steps use is proved in the same development rather than assumed: the entropy-decrement proof of the two-point Elliott estimate from Tao's companion paper, the Matom\u00e4ki\u2013Radziwi\u0142\u0142 short-interval estimate on the major arcs via the Dirichlet-polynomial route of Matom\u00e4ki, Radziwi\u0142\u0142 and Tao, the Hal\u00e1sz\u2013Montgomery large-values inequality, a large-values bound for Dirichlet polynomials over primes from a zero-free region of the Riemann zeta function, that zero-free region from Weyl-sum bounds on zeta, and a form of Vinogradov's mean value theorem. Where a published route did not fit, the development adapts it; the adaptations are documented. The theorem depends only on Lean's three standard axioms. This is a formalization of a published result, not a new proof; it was produced by AI agents working through pull requests checked by continuous integration in the ProofFleet repository (formerly MoltResearch).", "authors": [ { "name": "Sean Huver" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11K38", "11N37", "11M26", "11L15" ] }, "formalization": { "theorem_names": [ "EDP.erdos_discrepancy" ] }, "id": "PALOMAR-2026-09-29-000002", "path": "entries/PALOMAR-2026-09-29-000002-v1.json", "preservation": { "repositories": [ { "commit": "13853222d90d7cff0608c6380d0ee75b0e12473d", "fork_repository": "PalomarArchive/ProofFleet--prooffleet--f3efc1a69e0c", "source_repository": "ProofFleet/prooffleet" } ] }, "published_at": "2026-09-29T01:01:26Z", "source": { "commit": "13853222d90d7cff0608c6380d0ee75b0e12473d", "project_path": null, "repository": "ProofFleet/prooffleet" }, "status": "registered", "title": "A Lean formalization of the Erd\u0151s discrepancy theorem", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean proofs of strong duality and exact block composition for the negative-weight adversary bound on total Boolean functions, explicit constant-factor characterizations of bounded-error quantum query complexity in value and Boolean XOR oracle models, and uniform extraction of query algorithms from supplied promise dual certificates. The submission also proves that every output probability of a q-query algorithm on Boolean inputs has a real polynomial representation of degree at most 2q in each model. Seven statements are compared against a standalone Mathlib-only specification of the finite quantum model.", "authors": [ { "name": "Troy Lee" } ], "classification": { "arxiv": [ "quant-ph", "cs.CC" ], "msc2020": [ "68Q12", "81P68" ] }, "formalization": { "theorem_names": [ "PalomarQuantumQuery.strong_duality", "PalomarQuantumQuery.composition", "PalomarQuantumQuery.value_characterization", "PalomarQuantumQuery.xor_characterization", "PalomarQuantumQuery.uniform_dual_upper_bound", "PalomarQuantumQuery.value_probability_polynomial", "PalomarQuantumQuery.xor_probability_polynomial" ] }, "id": "PALOMAR-2026-09-29-000001", "path": "entries/PALOMAR-2026-09-29-000001-v1.json", "preservation": { "repositories": [ { "commit": "517ba85eb211232771429804d8583e3f52b9d541", "fork_repository": "PalomarArchive/troyjlee--quantum-query-complexity--2f2b6ce61352", "source_repository": "troyjlee/quantum-query-complexity" } ] }, "published_at": "2026-09-29T00:40:42Z", "source": { "commit": "517ba85eb211232771429804d8583e3f52b9d541", "project_path": null, "repository": "troyjlee/quantum-query-complexity" }, "status": "registered", "title": "Adversary bounds and the polynomial method for quantum queries", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean formalization of the implication from five explicitly stated literature inputs to a homeomorphism between the ordinary Gromov--Hausdorff space of nonempty compact metric spaces and real square-summable sequences. The proof constructs invariant full-support probability assignments, finite dimensional local models, global AR domination and discrete approximation. The five literature inputs are assumptions, not theorems proved in this snapshot.", "authors": [ { "name": "Yoshito Ishiki" } ], "classification": { "arxiv": [ "math.GN", "math.MG" ], "msc2020": [ "54E35", "54C55" ] }, "formalization": { "theorem_names": [ "PalomarPaperN.main" ] }, "id": "PALOMAR-2026-09-28-000004", "path": "entries/PALOMAR-2026-09-28-000004-v1.json", "preservation": { "repositories": [ { "commit": "8b25d3fc5b9f2a61dffa59e1daa8c9c483f70a46", "fork_repository": "PalomarArchive/yoshito-ishiki-math--lean-gh-space-hilbert--6ad5ce4534be", "source_repository": "yoshito-ishiki-math/lean-gh-space-hilbert" } ] }, "published_at": "2026-09-28T06:50:30Z", "source": { "commit": "8b25d3fc5b9f2a61dffa59e1daa8c9c483f70a46", "project_path": null, "repository": "yoshito-ishiki-math/lean-gh-space-hilbert" }, "status": "registered", "title": "Conditional Hilbert-space classification of Gromov--Hausdorff space", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalization in Lean 4 / mathlib of the existing paper by Furber, Mardare, Panangaden, and Scott, Interpreting Lambda Calculus in Domain-Valued Random Variables (LIPIcs, Vol. 363, CSL 2026, Article 48). This repository formalizes that published mathematics; it does not present the paper's results as new mathematics originating in this repository. Three capstones sit on top of the rest of the paper: Theorem 26 (internal \u03bb-interpretation in SetoidF_A), Corollary 34 (the Engeler model with numerals in V^A), and Theorem 43 (two fair-coin subsets of \u2115 that are incomparable under the \u03bb-definable many-one preorder). Domain-theory background is imported from a vendored copy of scott1972. The Palomar Challenge locks Theorem 43 as csl2026 and proposition_36_i on the Engeler graph model. Full Corollary 34 remains kernel-checked in the library and is not represented by a weaker Mathlib-only proxy. Lean is written by AI agents under the direction and review of the author. Dana Scott gave the author the paper. This formalization does not claim that the paper's authors participated in it or endorsed it. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake require.", "authors": [ { "name": "Lars Warren Ericson" } ], "classification": { "arxiv": [ "cs.LO", "cs.PL", "math.LO" ], "msc2020": [ "68Q55", "06D99", "03B70", "68V20" ] }, "formalization": { "theorem_names": [ "Scott2026.csl2026" ] }, "id": "PALOMAR-2026-09-28-000003", "path": "entries/PALOMAR-2026-09-28-000003-v1.json", "preservation": { "repositories": [ { "commit": "4b57681f2804e6390c73bae4149f54a92029c50d", "fork_repository": "PalomarArchive/catskillsresearch--scott2026--c9ae37f0429f", "source_repository": "catskillsresearch/scott2026" } ] }, "published_at": "2026-09-28T06:12:25Z", "source": { "commit": "4b57681f2804e6390c73bae4149f54a92029c50d", "project_path": null, "repository": "catskillsresearch/scott2026" }, "status": "registered", "title": "Lean Formalization of Interpreting Lambda Calculus in Domain-Valued Random Variables (CSL 2026)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every finite abelian extension L/K of a nonarchimedean local field, the multiplicative group K\u00d7 modulo the field-norm subgroup N_{L/K}(L\u00d7) is isomorphic as a topological group to Gal(L/K). The registered declaration is the finite norm-quotient form of local reciprocity proved in the included ClassFieldTheory development. Stronger compatibility, normalization and existence results in that development are documented separately and are not part of the compared declaration.", "authors": [ { "name": "Naganori Yamaguchi" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11S31" ] }, "formalization": { "theorem_names": [ "ClassFieldTheory.finiteAbelianLocalReciprocity_quotient" ] }, "id": "PALOMAR-2026-09-28-000002", "path": "entries/PALOMAR-2026-09-28-000002-v1.json", "preservation": { "repositories": [ { "commit": "3b402b69622b131ebdc7ffe746c8975c593dc330", "fork_repository": "PalomarArchive/n-yamaguchi-0729--FiniteAbelianLocalReciprocityv2--fea68c6813ec", "source_repository": "n-yamaguchi-0729/FiniteAbelianLocalReciprocityv2" } ] }, "published_at": "2026-09-28T05:35:39Z", "source": { "commit": "3b402b69622b131ebdc7ffe746c8975c593dc330", "project_path": null, "repository": "n-yamaguchi-0729/FiniteAbelianLocalReciprocityv2" }, "status": "registered", "title": "Local class field theory in Lean 4.35.0-rc2", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every finite abelian extension L/K of number fields, the id\u00e8le class group C_K modulo the image of the id\u00e8le-class norm from C_L is isomorphic as a topological multiplicative group to Gal(L/K). The Challenge defines the norm as the determinant on the relative ad\u00e8le algebra and descends it through principal id\u00e8les; the included ClassFieldTheory development supplies the proof of the resulting reciprocity theorem.", "authors": [ { "name": "Naganori Yamaguchi" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R37" ] }, "formalization": { "theorem_names": [ "ClassFieldTheory.GlobalClassFieldComparison.finiteAbelianGlobalReciprocity_relativeNormQuotient" ] }, "id": "PALOMAR-2026-09-28-000001", "path": "entries/PALOMAR-2026-09-28-000001-v1.json", "preservation": { "repositories": [ { "commit": "01b4bcb786c458c9af029860c9bbdbd67dd7c978", "fork_repository": "PalomarArchive/n-yamaguchi-0729--FiniteAbelianGlobalReciprocityv2--ccc6f4bb14f4", "source_repository": "n-yamaguchi-0729/FiniteAbelianGlobalReciprocityv2" } ] }, "published_at": "2026-09-28T04:50:22Z", "source": { "commit": "01b4bcb786c458c9af029860c9bbdbd67dd7c978", "project_path": null, "repository": "n-yamaguchi-0729/FiniteAbelianGlobalReciprocityv2" }, "status": "registered", "title": "Global class field theory in Lean 4.35.0-rc2", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Glauberman and Norton (Proc. Amer. Math. Soc. 119, 1993) end their paper on a combinatorial problem that arises in the proof of the Feit\u2013Thompson odd order theorem with a question of P\u00e9terfalvi: can the hypothesis of their Proposition 9 be satisfied for p = 3? The same question is Problem 1 in Appendix C of Bender and Glauberman, Local Analysis for the Odd Order Theorem (1994). Let F be the field with p^q elements, P its additive group, U its group of units of norm one over the prime field, H the semidirect product P \u22ca U, and P\u2080 the image of the additive group of the prime field in H. Hypothesis (B) asks for an injective homomorphism \u03c3 from H into a group G, a finite abelian subgroup Q of G whose order is prime to p, and an element y of Q, such that \u03c3(P\u2080) normalizes Q and \u03c3(P\u2080)^y normalizes \u03c3(U). Proposition 9 also assumes condition (A): q does not divide p \u2212 1. This project proves in Lean 4 that the answer is negative: for every prime q with q \u2224 3 \u2212 1, no group G, finite or infinite, satisfies hypothesis (B) with p = 3. It also proves P\u00e9terfalvi's example from the same paper, which shows that SL(2, 2^q) satisfies hypothesis (B) with p = 2, so the hypothesis is not empty. The proof is new. It turns a witness of hypothesis (B) into relations between three copies of the additive group of the field with 3^q elements, and it rules out these relations by arithmetic in that field.", "authors": [ { "name": "Yawara Ishida" } ], "classification": { "arxiv": [ "math.GR", "math.NT" ], "msc2020": [ "20D05", "20D10", "11T06" ] }, "formalization": { "theorem_names": [ "PeterfalviProblem.not_hypothesisB_three", "PeterfalviProblem.hypothesisB_two" ] }, "id": "PALOMAR-2026-09-27-000006", "path": "entries/PALOMAR-2026-09-27-000006-v1.json", "preservation": { "repositories": [ { "commit": "698b7ae0af649f7fad48c482aa8ff0d2d14336bb", "fork_repository": "PalomarArchive/yawara--peterfalvi-problem--a99f0c3a84ab", "source_repository": "yawara/peterfalvi-problem" } ] }, "published_at": "2026-09-27T02:21:31Z", "source": { "commit": "698b7ae0af649f7fad48c482aa8ff0d2d14336bb", "project_path": null, "repository": "yawara/peterfalvi-problem" }, "status": "registered", "title": "P\u00e9terfalvi's Problem for p = 3: No Group Satisfies Hypothesis (B)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib formalization of the Four Color Theorem, ported from Georges Gonthier's Coq proof by Claude Opus 5.5, Anthropic's AI model, working as an agent under the direction of Robert D. Barish. The main theorem, FourColor.RealPlane.four_color, states that every simple map of the real plane can be colored with at most four colors so that adjacent regions get different colors. The map may have infinitely many regions, bounded or not; each region must be open and connected, and two regions are adjacent when their closures share a point that lies in the closure of no third region. As in Coq, a compactness argument reduces the theorem to finite maps, which are turned into planar bridgeless hypermaps. A smallest hypermap that cannot be four-colored cannot exist: discharging shows that it would contain one of the 633 configurations of Robertson, Sanders, Seymour and Thomas, and each of these is reducible.\nThe port differs from the Coq proof in three main ways. Its statement uses Mathlib's real numbers rather than an arbitrary model of Coq's axiomatized reals; the two are equivalent, because all such models are isomorphic and the definitions use only the order. Coq checks that the 633 configurations are reducible with a procedure on trees of colorings, run by its virtual machine. The port replaces it with a new bit-parallel engine whose certificates, generated during the build, are checked by the Lean kernel itself. Finally, the case analysis of the unavoidability proof, written as tactic scripts in Coq, becomes case trees checked by a checker proved correct once. Every computation is carried out by the Lean kernel, without native_decide, and the proof uses only the axioms propext, Classical.choice and Quot.sound.", "authors": [ { "name": "Robert D. Barish" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15", "05C10", "68V20", "68V15" ] }, "formalization": { "theorem_names": [ "FourColor.RealPlane.four_color" ] }, "id": "PALOMAR-2026-09-27-000005", "path": "entries/PALOMAR-2026-09-27-000005-v1.json", "preservation": { "repositories": [ { "commit": "20fa34599f5189dd569d96dd22f306c94ab1c446", "fork_repository": "PalomarArchive/RBarish-UTokyo--FourColorTheorem-Lean4--2ebb41eff675", "source_repository": "RBarish-UTokyo/FourColorTheorem-Lean4" } ] }, "published_at": "2026-09-27T01:43:13Z", "source": { "commit": "20fa34599f5189dd569d96dd22f306c94ab1c446", "project_path": null, "repository": "RBarish-UTokyo/FourColorTheorem-Lean4" }, "status": "registered", "title": "The Four Color Theorem (Lean 4 port of Gonthier's Coq proof)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 proof that the zero-forcing number of the generalized Petersen graph P(n,3) is exactly 8 for every natural number n >= 13. The formalized result is the statement recorded as Conjecture 5 in Arnav Krishnan's July 2026 correction preprint; this repository supplies and kernel-checks a proof of that conjectured statement rather than formalizing a proof already given in the cited preprint.", "authors": [ { "name": "John Fairfax-Ball" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50" ] }, "formalization": { "theorem_names": [ "PetersenZeroForcing.zero_forcing_number_eq_eight" ] }, "id": "PALOMAR-2026-09-27-000004", "path": "entries/PALOMAR-2026-09-27-000004-v1.json", "preservation": { "repositories": [ { "commit": "5dd760e79ab33b3fcf2feb93eb37e23509d184ed", "fork_repository": "PalomarArchive/jfairfaxball-348--Petersen-Zero-Forcing--092c328236aa", "source_repository": "jfairfaxball-348/Petersen-Zero-Forcing" } ] }, "published_at": "2026-09-27T01:34:24Z", "source": { "commit": "5dd760e79ab33b3fcf2feb93eb37e23509d184ed", "project_path": null, "repository": "jfairfaxball-348/Petersen-Zero-Forcing" }, "status": "registered", "title": "Zero forcing number of the generalized Petersen graphs P(n,3)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "In a catalytic reaction system with finitely many reactions, a RAF is a nonempty set of reactions whose reactants are produced from the food set by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Steel (arXiv:2305.16714v4, Section 4) asked whether some reaction always lies in at least half of the RAFs. Call a reaction food-ready if all its reactants are food, and abundant if it lies in at least half of the sets in the family formed by the RAFs and the empty set. If some RAF U consists entirely of food-ready reactions, then some reaction of U is abundant, with no condition on the other reactions. If every reaction of the system, except at most two, is food-ready and the system has a RAF, then some reaction is abundant. Finally, the statement that every catalytic reaction system with finitely many reactions and a RAF has an abundant reaction is equivalent to Frankl's union-closed sets conjecture in the form: every finite union-closed family containing the empty set and a nonempty set has an element lying in at least half of its members.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.CO", "q-bio.MN" ], "msc2020": [ "05D05", "92C42", "68V20" ] }, "formalization": { "theorem_names": [ "RAF.Frankl.elementary_core_exists_abundant", "RAF.Frankl.two_nonfood_rafFixedFrankl", "RAF.Frankl.PairGadget.all_rafFixedFrankl_iff_all_unionClosedFrankl" ] }, "id": "PALOMAR-2026-09-27-000003", "path": "entries/PALOMAR-2026-09-27-000003-v1.json", "preservation": { "repositories": [ { "commit": "c1ebf53f80cc4a45af67142aa1e33f1a53494b8c", "fork_repository": "PalomarArchive/mmislan--palomar-p020-raf-abundant-reactions--8b8f0cbe7d09", "source_repository": "mmislan/palomar-p020-raf-abundant-reactions" } ] }, "published_at": "2026-09-27T01:24:40Z", "source": { "commit": "c1ebf53f80cc4a45af67142aa1e33f1a53494b8c", "project_path": null, "repository": "mmislan/palomar-p020-raf-abundant-reactions" }, "status": "registered", "title": "Equivalence of the RAF abundance conjecture and Frankl's union-closed sets conjecture", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib formalization of the whole of P. Corvaja and U. Zannier, On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mend\u00e8s France (Acta Math. 193 (2004)), built on the repository's libraries ArithmeticHeights and DiophantineApproximation, whose Subspace Theorem it uses. The library CorvajaZannier2004 is sorry-free and within Lean's three standard axioms. It proves the Main Theorem (finitely many non-pseudo-Pisot \u03b4qu that are very close to integers, u in a finitely generated group of algebraic numbers), Theorem 1 (Mahler's question: if \u2016\u03b1\u207f\u2016 < l\u207f infinitely often, some power of \u03b1 is a Pisot number), Lemmas 1\u20135, Theorem 2 (Mend\u00e8s France's question: the period of the continued fraction of \u03b1\u207f tends to infinity unless \u03b1 is a unit or a square root of a rational, and is bounded for units), and the Appendix (the word \"algebraic\" cannot be dropped from Theorem 1). Eleven statements are compared.", "authors": [ { "name": "Ralf Stephan" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J68", "11R06", "11J70", "11A55", "11J87" ] }, "formalization": { "theorem_names": [ "finite_setOf_not_isPseudoPisot", "exists_isPisot_pow_of_frequently", "isIntegral_or_exists_pow_eq_ratCast", "isIntegral_of_trace_mul_pow", "Real.isPisot_or_tendsto_log_cfQuot", "Real.isPisot_or_tendsto_log_cfQuot_odd", "Real.tendsto_cfPeriod_pow", "Real.tendsto_cfPeriod_pow_odd", "Real.bddAbove_cfPeriod_pow", "Real.exists_frequently_not_isPisot", "Real.exists_transcendental_frequently_not_isPisot" ] }, "id": "PALOMAR-2026-09-27-000002", "path": "entries/PALOMAR-2026-09-27-000002-v1.json", "preservation": { "repositories": [ { "commit": "d7a11cb1dcab3883fd1008db27a8382c95455b0a", "fork_repository": "PalomarArchive/rwst--Subspace-Theorems--9bd9bba330e8", "source_repository": "rwst/Subspace-Theorems" } ] }, "published_at": "2026-09-27T01:14:44Z", "source": { "commit": "d7a11cb1dcab3883fd1008db27a8382c95455b0a", "project_path": null, "repository": "rwst/Subspace-Theorems" }, "status": "registered", "title": "Corvaja\u2013Zannier 2004: rational approximations to the powers of an algebraic number", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We prove uniform cancellation in prime-weighted exponential sums at the 2/15 scale, improving the 8/33 local Fourier scale in the Matom\u00e4ki\u2013Radziwi\u0142\u0142\u2013Tao machinery for averaged prime-pair correlations. On the minor-arc part of every centered arc of radius 1/(2H), the squared von Mangoldt exponential sum over X < n <= 2X has integral at most C X (log X)^(-A), for every A > 0 and epsilon > 0, once H >= X^(2/15+epsilon) and X is large enough. The center may lie anywhere on the circle. The same cutoffs and constants work for every center and are chosen before X and H. This gives uniform minor-arc cancellation at the 2/15 threshold. Since 2/15 < 8/33, the admissible H is smaller, so the frequency windows of length 1/H are wider and the shift-averaging intervals are shorter. The constants C and X0 are ineffective because the proof uses Siegel's theorem. The development also contains an internal proof of the epsilon-form of Guth\u2013Maynard's large-value estimate for Dirichlet polynomials. The recorded 20 September Lean 4.30 kernel replays checked 100,596 declarations for MAP and 75,874 for Guth\u2013Maynard, using only standard axioms. The formal statement is AllCenterMAP.map_two_fifteenths, manuscript Theorem 1.1.", "authors": [ { "name": "Conor Grogan" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37" ] }, "formalization": { "theorem_names": [ "AllCenterMAP.map_two_fifteenths" ] }, "id": "PALOMAR-2026-09-27-000001", "path": "entries/PALOMAR-2026-09-27-000001-v1.json", "preservation": { "repositories": [ { "commit": "f369f267b4dcfebf010c8e7baa2c9602e2960eba", "fork_repository": "PalomarArchive/jconorgrogan--prime-minor-arcs-2-15--b3b69567282b", "source_repository": "jconorgrogan/prime-minor-arcs-2-15" } ] }, "published_at": "2026-09-27T01:05:19Z", "source": { "commit": "f369f267b4dcfebf010c8e7baa2c9602e2960eba", "project_path": null, "repository": "jconorgrogan/prime-minor-arcs-2-15" }, "status": "registered", "title": "Uniform Minor-Arc Cancellation for Primes at the X^{2/15+\u03b5} Scale", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Lean proofs that every metric TSP instance on at least three vertices has a Hamiltonian tour costing at most (3/2 - 2.05522e-30) times any feasible subtour-LP objective, with one strictly larger saving uniform over all instances. The submission also exposes the exact finite-layer saving and a KKO-framework bound using GKL capacity estimates.", "authors": [ { "name": "Troy Lee" } ], "classification": { "arxiv": [ "cs.DS", "math.CO" ], "msc2020": [ "90C27", "05C45" ] }, "formalization": { "theorem_names": [ "PalomarTSPGap.kko_gap", "PalomarTSPGap.song_gap", "PalomarTSPGap.song_gap_strict", "PalomarTSPGap.song_gap_exact", "PalomarTSPGap.exactGain_gt_target" ] }, "id": "PALOMAR-2026-09-26-000005", "path": "entries/PALOMAR-2026-09-26-000005-v1.json", "preservation": { "repositories": [ { "commit": "55d7c9c11ee4240328a71b1ab43d66f27a976da6", "fork_repository": "PalomarArchive/troyjlee--tcs-formalizations--200ed78cfc67", "source_repository": "troyjlee/tcs-formalizations" } ] }, "published_at": "2026-09-26T13:32:24Z", "source": { "commit": "55d7c9c11ee4240328a71b1ab43d66f27a976da6", "project_path": null, "repository": "troyjlee/tcs-formalizations" }, "status": "registered", "title": "Metric TSP subtour-LP integrality gap", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "This project formalizes the existence of a model companion for the first-order theory of groups satisfying x^3 = 1. Its principal group-theoretic result gives a uniform, explicit bound on the number of generators needed to witness a failure of amalgamation with an existentially closed exponent-three group. The development follows the accompanying manuscript through free groups, associated graded Lie algebras, strict extensions, and the bounded-amalgamation criterion.", "authors": [ { "name": "Yawara Ishida" } ], "classification": { "arxiv": [ "math.LO", "math.GR" ], "msc2020": [ "03C60", "20A15" ] }, "formalization": { "theorem_names": [ "T3.exists_bounded_nonamalgamation_witness", "T3.has_model_companion" ] }, "id": "PALOMAR-2026-09-26-000004", "path": "entries/PALOMAR-2026-09-26-000004-v2.json", "preservation": { "repositories": [ { "commit": "7d38e7d9fe9d131e713ce160853983321970e598", "fork_repository": "PalomarArchive/yawara--t3-model-companion--87b9fb97bd12", "source_repository": "yawara/t3-model-companion" } ] }, "published_at": "2026-09-26T12:58:07Z", "source": { "commit": "7d38e7d9fe9d131e713ce160853983321970e598", "project_path": null, "repository": "yawara/t3-model-companion" }, "status": "registered", "title": "Existence of a Model Companion for Groups of Exponent 3: Lean Formalization", "trust": { "level": "high" }, "version": 2, "versions": 2 }, { "abstract": "A complete Lean 4 / Mathlib formalization of the following theorem: there are absolute constants C, n0 > 0 such that every connected Cayley graph Cay(G, S) on n >= n0 vertices whose degree d = |S| is at least C (log n)^13 / log log n contains a Hamilton cycle (Lovasz.hamiltonian_of_polylog_degree). Here G is a finite group, S is a connection set (S = S^-1 and 1 not in S, so the graph is |S|-regular), log is the natural logarithm, and connectivity is equivalent to S generating G. The theorem is a case of the Lov\u00e1sz conjecture (every connected vertex-transitive graph has a Hamilton path) in its Cayley-graph form (every connected Cayley graph on at least three vertices is Hamiltonian). For arbitrary connected Cayley graphs the previously published degree thresholds were linear degree (Christofides, Hladk\u00fd and M\u00e1th\u00e9, 2014) and degree at least n^(1-c) (Bedert, Dragani\u0107, M\u00fcyesser and Pavez-Sign\u00e9, 2026). The proof follows the source manuscript: it extracts a connected bipartite template with spectral gap of order (log n)^-2, allocates the vertices to weighted translates of the template by signed rounding (a robust signed integrality theorem derived from the capacitated b-matching polytope, swap rounding, and a trace-exponential matrix concentration bound), connects the pieces by a sparse matching chosen by mutual nominations and a signed circulation, absorbs the outside matching inside every piece with Hamilton routers built from even-cycle comparators, and merges cycles. Every lemma of the manuscript is proved in the repository, as are the classical results it uses without proof (normalized Cheeger inequality, Haxell's hypergraph matching theorem, the b-matching polytope, Watkins's connectivity bound, the Nash-Williams\u2013Tutte tree-packing theorem, Hoffman's circulation criterion, scalar Chernoff bounds); Hall's theorem is taken from Mathlib, and matrix Bernstein is avoided by deriving the one lemma that used it from the manuscript's own concentration bound. The source manuscript is an unpublished research draft that describes its own proof as not independently verified; this formalization is, to the maintainer's knowledge, the first verification of the theorem. The proved theorem has no sorry and depends only on propext, Classical.choice and Quot.sound.", "authors": [ { "name": "Shengtong Zhang" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C45", "05C25", "05C48", "05C50", "05C70", "05C80" ] }, "formalization": { "theorem_names": [ "Lovasz.hamiltonian_of_polylog_degree" ] }, "id": "PALOMAR-2026-09-26-000003", "path": "entries/PALOMAR-2026-09-26-000003-v1.json", "preservation": { "repositories": [ { "commit": "ee0085ddec79996279c7b52044659032e5cf8431", "fork_repository": "PalomarArchive/ShengtongZhang-alt--lovasz-opus--32b678aa320e", "source_repository": "ShengtongZhang-alt/lovasz-opus" } ] }, "published_at": "2026-09-26T03:17:12Z", "source": { "commit": "ee0085ddec79996279c7b52044659032e5cf8431", "project_path": null, "repository": "ShengtongZhang-alt/lovasz-opus" }, "status": "registered", "title": "Hamilton cycles in Cayley graphs of polylogarithmic degree", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalizes the continuous-type bilateral-trade impossibility result: no mechanism can satisfy ex post efficiency (almost surely), Bayesian incentive compatibility, interim individual rationality, and weak ex ante budget balance when buyer and seller supports overlap. The proof uses payoff equivalence and envelope identities, together with the strict rent-gap identity B + C > S.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "econ.TH", "cs.GT" ], "msc2020": [ "91B26" ] }, "formalization": { "theorem_names": [ "MS.Palomar.myersonSatterthwaite" ] }, "id": "PALOMAR-2026-09-26-000002", "path": "entries/PALOMAR-2026-09-26-000002-v1.json", "preservation": { "repositories": [ { "commit": "f4dae9744e26a499679e4b8a7aeb9d82dd3f11ee", "fork_repository": "PalomarArchive/Arthur742Ramos--myerson-satterthwaite-lean--29d4c4ff61d0", "source_repository": "Arthur742Ramos/myerson-satterthwaite-lean" } ] }, "published_at": "2026-09-26T02:53:18Z", "source": { "commit": "f4dae9744e26a499679e4b8a7aeb9d82dd3f11ee", "project_path": null, "repository": "Arthur742Ramos/myerson-satterthwaite-lean" }, "status": "registered", "title": "Myerson-Satterthwaite impossibility theorem (continuous bilateral-trade version)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Erd\u0151s problem #1220 (erdosproblems.com/1220; Erd\u0151s\u2013Hajnal 1971): let \u03bb be a singular cardinal such that both \u03bb and cf(\u03bb) are \u2135\u2080-inaccessible (\u03bc^\u2135\u2080 < \u03ba for all \u03bc < \u03ba); does \u03bb \u2192 (\u03bb, \u2135\u2081)\u00b2 hold? This formalization settles the question as asked on erdosproblems.com in the negative sense \"not provable\": ZFC does not prove that every such \u03bb satisfies \u03bb \u2192 (\u03bb, \u2135\u2081)\u00b2 (erdos1220_not_provable : \u00ac (ZFC \u22a8\u1d47 Erdos1220)), i.e. there is a model of ZFC with a singular \u03bb, \u03bb and cf(\u03bb) \u2135\u2080-inaccessible, and \u03bb \u219b (\u03bb, \u2135\u2081)\u00b2. Shelah and Stanley (Sh:258, Ann. Pure Appl. Logic 36 (1987)) showed that a negative answer is consistent with ZFC. This project formalizes that consistency result: the comparator target erdos1220_not_provable : \u00ac (ZFC \u22a8\u1d47 Erdos1220) is stated for literally the same first-order theory ZFC, language and ZFSet interpretation as elliotglazer/erdos501's comparator challenge (imported from its statement module Erdos501.FOL.Statement), together with a faithfulness theorem erdos1220_sentence_faithful: in Mathlib's ZFSet the sentence Erdos1220 holds iff the Mathlib-level statement Erdos1220.Problem1220 holds. The witness is the ground-model singular strong-limit cardinal \u03bb = \u2136_{\ud835\udd20\u207a} (cf \u03bb = \ud835\udd20\u207a; \u03bb and cf \u03bb are \u2135\u2080-inaccessible without any deep cardinal-arithmetic input), not \u2135_{\ud835\udd20\u207a}. The forcing is the Shelah\u2013Stanley \u00a73 forcing, formalized as block-preserving, block-covering histories of basic conditions (weak extension order), with the \u03bc\u207a-strategic closure (hence \u03bc-distributivity) and the (2^\u03bc)\u207a-chain condition proved in ZFC without GCH (\u0394-system lemma for (2^\u03bc)\u207a sets of size \u2264 \u03bc, blueprint counting (2^\u03bc)^\u03bc = 2^\u03bc). In the Boolean-valued model of its regular-open algebra (built with the Lean 4 port of Han\u2013van Doorn's Flypitch vendored by erdos501) the hypotheses on \u03bb are preserved, the generic colouring of [\u03bb]\u00b2 has neither a red homogeneous set of size \u03bb nor a blue homogeneous set of size \u2135\u2081, and Flypitch's completeness theorem turns this into a two-valued model of ZFC satisfying \u00acErdos1220. Both targets are proved from propext, Classical.choice and Quot.sound only and pass the Lean comparator. Only the negative direction is claimed: independence (consistency of a positive answer) is NOT claimed.", "authors": [ { "name": "Ji Ho Bae (JRTI)" } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03E35", "03E02", "03E05", "03E40", "03E55" ] }, "formalization": { "theorem_names": [ "erdos1220_not_provable", "erdos1220_sentence_faithful" ] }, "id": "PALOMAR-2026-09-26-000001", "path": "entries/PALOMAR-2026-09-26-000001-v1.json", "preservation": { "repositories": [ { "commit": "9cb81ffa48ffb766127c7e485ccef77bd4160868", "fork_repository": "PalomarArchive/jbaelaw--erdos1220-lean--9ed9443659d9", "source_repository": "jbaelaw/erdos1220-lean" } ] }, "published_at": "2026-09-26T02:04:54Z", "source": { "commit": "9cb81ffa48ffb766127c7e485ccef77bd4160868", "project_path": null, "repository": "jbaelaw/erdos1220-lean" }, "status": "registered", "title": "Erd\u0151s Problem #1220 in Lean 4: the affirmative statement is not provable in ZFC", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalizes the Gibbard-Satterthwaite theorem for finite electorates and finite alternative sets with at least three alternatives: every onto, strategy-proof social choice function is dictatorial. The proof constructs a social welfare function and reduces to Arrow's impossibility theorem.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "math.LO", "cs.GT" ], "msc2020": [ "91B14" ] }, "formalization": { "theorem_names": [ "GibbardSatterthwaite.gibbardSatterthwaite" ] }, "id": "PALOMAR-2026-09-25-000028", "path": "entries/PALOMAR-2026-09-25-000028-v1.json", "preservation": { "repositories": [ { "commit": "8398e65a99cd3d5e973c64963c4bab8c0491947c", "fork_repository": "PalomarArchive/Arthur742Ramos--gibbard-satterthwaite-lean--fe2cb7e0c225", "source_repository": "Arthur742Ramos/gibbard-satterthwaite-lean" } ] }, "published_at": "2026-09-25T21:11:21Z", "source": { "commit": "8398e65a99cd3d5e973c64963c4bab8c0491947c", "project_path": null, "repository": "Arthur742Ramos/gibbard-satterthwaite-lean" }, "status": "registered", "title": "Gibbard-Satterthwaite theorem (strategy-proof social choice)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We establish three sets of results addressing published questions about the composition, enumeration and degradation of autocatalytic reaction networks. In response to the composition questions of Gagrani, Wang, De Decker and Lacoste (arXiv:2603.02627, Discussion and outlook), we give exact criteria for strict increases in the maximum amplification factor (MAF) under reaction addition, nonnegative species aggregation and shared-species coupling, assuming nonnegative inputs and attained MAFs. A block direct sum attains the larger component MAF, while an explicit example shows that component MAFs alone do not determine the MAF of a coupled network. Addressing the CS-core enumeration direction of Golnik, Gatter, Stadler and Vassena (arXiv:2603.02770, Section 6), we prove exact reconstruction from a complete list of ordinary-core anchor matchings, each unique on its vertex set, by alternating-path generation and certified rational semipositivity filtering. We establish existence of the certified test, its rejection of the empty matching, and an explicit nonempty instance for A \u2192 2A. For the arbitrary-degradation question discussed by Nandan, Nghe and Unterberger (arXiv:2507.15546, Section 6), we characterize growth, criticality and extinction for irreducible linear diluted source networks with nonnegative unary reaction data and species degradation d \u2265 0: the real spectral bound of A \u2212 diag(d) is positive, zero or negative exactly when d is componentwise strictly below, equal to or strictly above (Av)_i/v_i for some positive v. We also characterize partial-actuation stabilizability of Metzler matrices through a positive-vector certificate on the unactuated block. All eleven selected statements are formalized in Lean 4 with Mathlib.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "q-bio.MN", "math.CO" ], "msc2020": [ "92C42", "15A39", "05C70", "15B48" ] }, "formalization": { "theorem_names": [ "MAFComposition.reactionEdit", "MAFComposition.aggregate", "MAFComposition.directSum", "MAFComposition.parallel", "MAFComposition.scalarNoncompositionality", "DegradationControl.sourceNetwork_arbitraryDegradationRegions", "DegradationControl.partialActuation_stabilizable_iff_unactuatedCertificate", "AutocatalyticCS.sourceDirectCSCoreEnum_exact", "AutocatalyticCS.certifiedAutocatalyticTest_nonempty", "AutocatalyticCS.certifiedAutocatalyticTest_rejects_empty", "AutocatalyticCS.OneSpeciesExample.sourceDirectCSCoreEnum_example" ] }, "id": "PALOMAR-2026-09-25-000027", "path": "entries/PALOMAR-2026-09-25-000027-v1.json", "preservation": { "repositories": [ { "commit": "1a6347f281c134c9561eb883ca726876b4a88c07", "fork_repository": "PalomarArchive/mmislan--palomar-p011-modular-autocatalytic-interfaces--5eb3a0b13fdd", "source_repository": "mmislan/palomar-p011-modular-autocatalytic-interfaces" } ] }, "published_at": "2026-09-25T20:44:40Z", "source": { "commit": "1a6347f281c134c9561eb883ca726876b4a88c07", "project_path": null, "repository": "mmislan/palomar-p011-modular-autocatalytic-interfaces" }, "status": "registered", "title": "MAF composition criteria, exact enumeration of child-selection autocatalytic cores, and degradation regions of linear diluted source networks", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib proof that at least one of the 2-adic zeta values \u03b6\u2082(7), \u03b6\u2082(9) is irrational. The theorem has no hypotheses, and its proof uses only the axioms propext, Classical.choice and Quot.sound. The values are defined through the Volkenborn integral: zeta2 s = J(s\u22121)/((s\u22121)\u00b72^s), where J(s) is the 2-adic limit of the Riemann sums 2^(\u2212N) \u03a3_{x<2^N} (x+1/2)^(\u2212s), and the theorem also proves that these limits exist. By Lemma 2.8 of Lai, Sprang and Zudilin (IMRN 2026), zeta2 s is the Kubota\u2013Leopoldt value \u03b6\u2082(s) = L\u2082(s, \u03c9^(1\u2212s)) for s \u2265 2. That identification is cited, not formalised; it only involves a non-zero rational factor, which does not affect irrationality. The result sharpens Theorem 1.1 (case s = 3) of L. Lai (Int. J. Number Theory 2025), which shows that at least one of \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11), \u03b6\u2082(13) is irrational, and it implies the companion result, formalised in a separate repository, that at least one of \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11) is irrational. The proof uses linear forms S_n = \u222b R_n'''(t+1/2) dt in 1, \u03b6\u2082(7), \u03b6\u2082(9) built from the shifted well-poised rational functions R_n(t) = (2t+n) \u03a0_{m=1}^{6} (t+1/2\u2212h_m)_(n+2h_m) / (t)_(n+1)^6 with h = (n/40)(\u221217, 1, 2, 3, 5, 6). The 2-adic size of S_n comes from Lai's and Sprang's \u0394-calculus. The size of the coefficients comes from a contour-integral representation and a saddle-point bound, whose numerical inequalities are certified in Lean. The odd part of the denominators is bounded by Legendre-type counts at the poles, the vanishing of R_n at half-integers, kernel-checked certificates and the prime number theorem (proved here from a vendored Lean proof of the Wiener\u2013Ikehara theorem). If both values were rational, the valuations of the coefficients at the prime n \u2212 1 (an idea of Lai and Sprang) would make S_n non-zero for infinitely many n, and Lai's irrationality criterion gives the contradiction. The mathematics and the Lean code were produced by AI agents under human direction and have not been refereed.", "authors": [ { "name": "Gennaro Marco Devincenzis" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11S40", "11S80", "11M06" ] }, "formalization": { "theorem_names": [ "Zeta2.Pair.zeta2_7_9_not_both_rational_palomar" ] }, "id": "PALOMAR-2026-09-25-000026", "path": "entries/PALOMAR-2026-09-25-000026-v1.json", "preservation": { "repositories": [ { "commit": "3797cb6e4f8653744c0aa8e66ec8eb3645e75dcd", "fork_repository": "PalomarArchive/gmDevi--zeta2-7-9-lean--782bf3edf3d4", "source_repository": "gmDevi/zeta2-7-9-lean" } ] }, "published_at": "2026-09-25T20:15:02Z", "source": { "commit": "3797cb6e4f8653744c0aa8e66ec8eb3645e75dcd", "project_path": null, "repository": "gmDevi/zeta2-7-9-lean" }, "status": "registered", "title": "At least one of the 2-adic zeta values \u03b6\u2082(7), \u03b6\u2082(9) is irrational", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Addressing the RAF-realizability question in Section 4 of Mike Steel's Interior Operators and Their Relationship to Autocatalytic Networks (arXiv:2305.16714v4, 2023), we characterize which interior operators on the subsets of a finite set E are maxRAF operators of catalytic reaction systems whose reaction set is exactly E. An interior operator is represented by its family of fixed sets, a union-closed family containing the empty set. A same-ground realization is a system with one reaction per element of E, finitely many molecule types, a food set and a catalysis relation; it realizes the empty set together with its RAFs, the nonempty reaction sets whose reactants arise from food by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Such a family has a same-ground realization if and only if it is the intersection of an accessible union-closed family A and a predecessor-support family. Here A contains the empty set, is closed under unions, and every nonempty member of A has an element whose removal leaves a member of A; the members of A need not cover E. A set S is supported by a predecessor map P on E when every element e of S has some element of S in P(e), loops allowed. The family consisting of the empty set and all subsets of size at least three has a same-ground realization if and only if E has at most four elements. This gives the sharp threshold for the family considered in Proposition 5 and its subsequent remarks in Steel (2023). Both results are formalized in Lean 4 with Mathlib.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B35", "05C20" ] }, "formalization": { "theorem_names": [ "RAFInteriorRealizability.palomar_same_ground_family_characterization", "RAFInteriorRealizability.palomar_cplusThree_threshold" ] }, "id": "PALOMAR-2026-09-25-000025", "path": "entries/PALOMAR-2026-09-25-000025-v1.json", "preservation": { "repositories": [ { "commit": "ebc29d42debba48f8ffc90624d7bd14cdb4a281e", "fork_repository": "PalomarArchive/mmislan--palomar-p005-raf-interior-realizability--94f206c5ba90", "source_repository": "mmislan/palomar-p005-raf-interior-realizability" } ] }, "published_at": "2026-09-25T20:03:47Z", "source": { "commit": "ebc29d42debba48f8ffc90624d7bd14cdb4a281e", "project_path": null, "repository": "mmislan/palomar-p005-raf-interior-realizability" }, "status": "registered", "title": "Same-ground RAF realizability of finite interior operators and the four-reaction threshold for the empty-or-at-least-three family", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "Formalizes Arrow's impossibility theorem for finitely many voters and finitely many alternatives (at least three): every social welfare function satisfying unanimity and independence of irrelevant alternatives is dictatorial. The proof follows the classical decisive-coalition argument: field expansion (a coalition decisive for one pair is decisive for every pair) and group contraction (a decisive coalition with at least two voters contains a smaller decisive coalition), iterated down to a singleton dictator. This is a reusable formalization artifact, not a claim of new mathematics.", "authors": [ { "name": "Arthur Freitas Ramos" } ], "classification": { "arxiv": [ "math.LO", "cs.GT" ], "msc2020": [ "91B14" ] }, "formalization": { "theorem_names": [ "Arrow.Palomar.decisiveUniv", "Arrow.Palomar.fieldExpansion", "Arrow.Palomar.groupContraction", "Arrow.Palomar.arrowImpossibility" ] }, "id": "PALOMAR-2026-09-25-000024", "path": "entries/PALOMAR-2026-09-25-000024-v1.json", "preservation": { "repositories": [ { "commit": "4f405d27d0b244574bbee5d3d79bae0c66bda16e", "fork_repository": "PalomarArchive/Arthur742Ramos--arrow-impossibility-lean--5d0750e4e794", "source_repository": "Arthur742Ramos/arrow-impossibility-lean" } ] }, "published_at": "2026-09-25T19:02:37Z", "source": { "commit": "4f405d27d0b244574bbee5d3d79bae0c66bda16e", "project_path": null, "repository": "Arthur742Ramos/arrow-impossibility-lean" }, "status": "registered", "title": "Arrow's impossibility theorem (general finite version)", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 and Mathlib proof that at least one of the 2-adic zeta values \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11) is irrational. The theorem has no hypotheses, and its proof uses only the axioms propext, Classical.choice and Quot.sound. The values are defined through the Volkenborn integral: zeta2 s = J(s\u22121)/((s\u22121)\u00b72^s), where J(s) is the 2-adic limit of the Riemann sums 2^(\u2212N) \u03a3_{x<2^N} (x+1/2)^(\u2212s), and the theorem also proves that these limits exist. By Lemma 2.8 of Lai, Sprang and Zudilin (IMRN 2026), zeta2 s is the Kubota\u2013Leopoldt value \u03b6\u2082(s) = L\u2082(s, \u03c9^(1\u2212s)) for s \u2265 2. That identification is cited, not formalised; it only involves a non-zero rational factor, which does not affect irrationality. The result sharpens Theorem 1.1 (case s = 3) of L. Lai (Int. J. Number Theory 2025), which shows that at least one of \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11), \u03b6\u2082(13) is irrational. The proof uses linear forms S_n = \u2212\u222b R_n'''(t+1/2) dt in 1, \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11), with R_n(t) = 2^(16n)(2t+n)(t+1/2)_n^8/(t)_(n+1)^8. Their denominators are controlled by a one-power saving that comes from Andrews' multiple-series transformation (proved here in Lean) and by the prime number theorem (proved here from a vendored Lean proof of the Wiener\u2013Ikehara theorem). The exact 2-adic valuation of S_n along n = 2^m \u2212 1 shows that S_n \u2260 0, and Lai's irrationality criterion concludes. The mathematics and the Lean code were produced by AI agents under human direction and have not been refereed.", "authors": [ { "name": "Gennaro Marco Devincenzis" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11S40", "11S80", "11M06", "33C20" ] }, "formalization": { "theorem_names": [ "Zeta2.zeta2_7_9_11_not_all_rational_palomar" ] }, "id": "PALOMAR-2026-09-25-000023", "path": "entries/PALOMAR-2026-09-25-000023-v1.json", "preservation": { "repositories": [ { "commit": "2ae1bccb82f1f72ed7a7aeab3196aa1cdbee165a", "fork_repository": "PalomarArchive/gmDevi--zeta2-7-9-11-lean--c7bc8f42d840", "source_repository": "gmDevi/zeta2-7-9-11-lean" } ] }, "published_at": "2026-09-25T17:53:11Z", "source": { "commit": "2ae1bccb82f1f72ed7a7aeab3196aa1cdbee165a", "project_path": null, "repository": "gmDevi/zeta2-7-9-11-lean" }, "status": "registered", "title": "At least one of the 2-adic zeta values \u03b6\u2082(7), \u03b6\u2082(9), \u03b6\u2082(11) is irrational", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "A Lean 4 / Mathlib development of Schmidt's Subspace Theorem and the theory beneath it, following Bombieri\u2013Gubler, Heights in Diophantine Geometry (Chapters 1\u20137), and Schmidt's route in general. Two libraries, sorry-free and within Lean's three standard axioms. ArithmeticHeights builds heights of points, polynomials, matrices and linear subspaces, Northcott and Kronecker, Minkowski's second theorem, Siegel's lemma and the Bombieri\u2013Vaaler theorems. DiophantineApproximation builds on it Roth's theorem over a number field with several places (and Ridout, the p-adic form, targets at infinity, moving targets), the geometry of numbers of approximation domains, and the Subspace Theorem: parametric, with coefficients in K, with algebraic coefficients, for S-integral points, and in Vojta's general-position form. It then proves the applications \u2014 Schmidt's simultaneous approximation, a transcendence criterion, unit equations, decomposable-form, Thue\u2013Mahler, hyperelliptic and norm-form equations, the Corvaja\u2013Zannier gcd bound \u2014 and Evertse's quantitative steps: systems of inequalities, the gap principles, the count of small solutions, and the passage from height intervals to subspaces. Twenty-six headline statements are compared: seven forms of Siegel's lemma and Bombieri\u2013Vaaler, seven of Roth's theorem, seven of the Subspace Theorem, and five of Evertse's counting of the subspaces.", "authors": [ { "name": "Ralf Stephan" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J87", "11J68", "11G50", "11J25", "11H06", "11D61", "11D57", "11D59" ] }, "formalization": { "theorem_names": [ "Int.Matrix.exists_ne_zero_mulVec_eq_zero_iSup_abs_le", "NumberField.exists_forall_exists_ne_zero_mulVec_eq_zero_absMulHeight_le", "Int.Matrix.exists_linearIndependent_mulVec_eq_zero_prod_iSup_abs_le", "Int.Matrix.exists_ne_zero_mulVec_eq_zero_iSup_abs_le_det_rpow", "NumberField.exists_ne_zero_mem_ker_absMulHeight_le", "NumberField.exists_linearIndependent_mem_ker_prod_absMulHeight_le", "MvPolynomial.exists_ne_zero_mem_ker_mulHeight_rpow_le", "NumberField.finite_setOf_prod_min_one_le", "Real.irrationalityExponent_eq_two", "Rat.finite_setOf_ridout", "Rat.finite_setOf_apply_intCast_sub_le", "NumberField.finite_setOf_prod_onePointApprox_le", "NumberField.finite_setOf_prod_min_one_le_card_two", "NumberField.finite_setOf_prod_min_one_le_of_isLittleO", "NumberField.exists_finset_submodule_of_approxProd_le_card_two", "NumberField.exists_finset_submodule_forall_approxDomain_subset", "NumberField.exists_finset_submodule_of_approxProd_le", "NumberField.exists_finset_submodule_of_approxProd_le_extension", "NumberField.exists_finset_submodule_of_integer_of_affineProd_le", "NumberField.exists_finset_submodule_of_generalProd_le", "NumberField.exists_finset_submodule_of_approxProd_le_card_two_of_extension", "NumberField.exists_finset_submodule_of_systemWeight_neg", "NumberField.exists_finset_submodule_of_not_isLargeSolution", "Rat.exists_finset_submodule_of_not_isLargeSolution", "NumberField.exists_finset_submodule_of_forall_mem_interval", "NumberField.exists_finset_submodule_of_card_eq_two" ] }, "id": "PALOMAR-2026-09-25-000022", "path": "entries/PALOMAR-2026-09-25-000022-v1.json", "preservation": { "repositories": [ { "commit": "d12aa25a9725e8d28c1bfc018c664b7ceb8d35c5", "fork_repository": "PalomarArchive/rwst--Subspace-Theorems--9bd9bba330e8", "source_repository": "rwst/Subspace-Theorems" } ] }, "published_at": "2026-09-25T16:31:31Z", "source": { "commit": "d12aa25a9725e8d28c1bfc018c664b7ceb8d35c5", "project_path": null, "repository": "rwst/Subspace-Theorems" }, "status": "registered", "title": "Heights, Siegel's lemma, Roth's theorem and the Subspace Theorems in Lean", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "In the sparse binary-polymer model with reversible ligation and cleavage, fix the food-length bound t, a nonnegative catalysis intensity lambda, and a linear reaction budget Cn. As the maximum polymer length n tends to infinity, the probability of a nonempty reflexively autocatalytic and food-generated reaction set with at most Cn channels tends to zero. The probability that some molecule catalyses a nonempty food-generated reaction set that constructs that molecule also tends to zero. These statements address the linear-size and single-molecule self-construction questions for the sparse catalysis law.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.PR" ], "msc2020": [ "60C05", "05C65" ] }, "formalization": { "theorem_names": [ "SparseLinearRAF.sparse_linear_raf_literature_resolution", "SparseLinearRAF.no_high_probability_linear_raf", "SparseLinearRAF.single_catalyst_probability_tendsto_zero" ] }, "id": "PALOMAR-2026-09-25-000021", "path": "entries/PALOMAR-2026-09-25-000021-v1.json", "preservation": { "repositories": [ { "commit": "324c7903021c54c3ee140637a98aef8766ec348b", "fork_repository": "PalomarArchive/mmislan--palomar-p012-sparse-raf-exclusion--812afa0bdd35", "source_repository": "mmislan/palomar-p012-sparse-raf-exclusion" } ] }, "published_at": "2026-09-25T15:40:31Z", "source": { "commit": "324c7903021c54c3ee140637a98aef8766ec348b", "project_path": null, "repository": "mmislan/palomar-p012-sparse-raf-exclusion" }, "status": "registered", "title": "Linear-size autocatalytic sets are asymptotically absent in the sparse binary polymer model", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "For every integer n >= 1, the standard sequential distributive n-site phosphorylation mechanism admits positive reaction rates and 2n-1 distinct positive stationary states with common positive kinase, phosphatase and substrate totals. The construction establishes a uniform lower bound on the number of positive equilibria of the full mass-action system.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS" ], "msc2020": [ "34C23", "92C45" ] }, "formalization": { "theorem_names": [ "PhosphorylationSharpness.sharp_lower_bound" ] }, "id": "PALOMAR-2026-09-25-000020", "path": "entries/PALOMAR-2026-09-25-000020-v1.json", "preservation": { "repositories": [ { "commit": "44cb5b18313448ede0f9e0df869c538602ffc66d", "fork_repository": "PalomarArchive/mmislan--palomar-p080-phosphorylation-attainment--ff96f90b0709", "source_repository": "mmislan/palomar-p080-phosphorylation-attainment" } ] }, "published_at": "2026-09-25T15:30:11Z", "source": { "commit": "44cb5b18313448ede0f9e0df869c538602ffc66d", "project_path": null, "repository": "mmislan/palomar-p080-phosphorylation-attainment" }, "status": "registered", "title": "Attainment of 2n-1 positive steady states in sequential distributive phosphorylation", "trust": { "level": "high" }, "version": 1, "versions": 1 }, { "abstract": "We prove that every source-minimal Type II_l autocatalytic core with l >= 3, reversible mass-action kinetics and strictly positive species degradation has at most one positive stationary state. The result covers the literal weak source grammar, including its separated cyclic realizations and the fully coincident three-species case. Any two positive concentration states satisfying the stationary equations of the same core are equal. The uniqueness theorem is proved in Lean 4 with Mathlib.", "authors": [ { "name": "Michael Mislan" } ], "classification": { "arxiv": [ "math.DS" ], "msc2020": [ "34C05", "92C42" ] }, "formalization": { "theorem_names": [ "TypeIIL.PaperRaw.paper_typeII_l_unistationarity" ] }, "id": "PALOMAR-2026-09-25-000019", "path": "entries/PALOMAR-2026-09-25-000019-v1.json", "preservation": { "repositories": [ { "commit": "29ef2baabd1c8e1d0ed659b5900e2cf4633ed541", "fork_repository": "PalomarArchive/mmislan--palomar-p006-type-ii-unistationarity--cfb655f70ee8", "source_repository": "mmislan/palomar-p006-type-ii-unistationarity" } ] }, "published_at": "2026-09-25T15:05:41Z", "source": { "commit": "29ef2baabd1c8e1d0ed659b5900e2cf4633ed541", "project_path": null, "repository": "mmislan/palomar-p006-type-ii-unistationarity" }, "status": "registered", "title": "Unistationarity of Type II_l autocatalytic cores with positive degradation", "trust": { "level": "high" }, "version": 1, "versions": 1 } ], "schema_version": 2 }