[ { "paper_id": "arxiv:2509.24034", "title": "Generalized additive bases and difference bases for Cartesian product of finite abelian groups", "authors": [ "Li Shuxing", "Yip Chi Hoi" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Studies g-additive and g-difference bases in G^n, the Cartesian product of a fixed finite abelian group G. Establishes asymptotically sharp upper bounds on the minimal sizes of such bases via Galois-ring/Teichmuller, relative-difference-set, and quotient/direct-product constructions; main asymptotic: eta_g(G^n) = (sqrt(g) + O(1/sqrt(g))) |G|^(n/2) (1 + o_n(1)). The work explicitly does NOT provide a cyclic embedding into Z/NZ: it works in the product setting G^n and uses no Chinese Remainder Theorem lift of the product to a cyclic image of equal order.", "candidate_bill": "Bill 7", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.95, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": true, "notes": "This is the Asper-cited Li-Yip 2025 paper. CONFIRMS Bill 7 EMPTY-SPACE: the paper provides finite-abelian-product constructions but no cyclic embedding into Z/NZ for N >= 8011 with score below 2.639. eta_g(G^n) is measured in product norm |G|^(n/2), not in the cyclic-quotient score 16(q'+1)^2/(6q*) the arena uses. Falsifies M5 worry (paper is now publicly available).", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.20446", "title": "Perfect difference families, perfect systems of difference sets and their applications", "authors": [ "Liu Hengrui", "Feng Tao", "Wang Xiaomiao", "Zhang Menglong" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Resolves a nearly 50-year-old Bermond conjecture by proving that a (v,4,lambda)-PDF exists iff lambda(v-1) \u2261 0 (mod 12), v >= 13, and (v,lambda) not in {(25,1),(37,1)}. Introduces layered difference families as a unifying framework for cyclic difference packings, applied to (m,4,3)-PSDSs. Applications: optical orthogonal codes, DNA origami, radio astronomy, difference triangle sets.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Block-size-4 perfect difference families; analogous to but not directly improving Singer (q^2+q+1,q+1,1) construction. Layered framework is novel structural tool, not a Z/NZ score-beater.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 6", "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.19804", "title": "Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof", "authors": [ "Alexeev Boris", "Mixon Dustin G." ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Resolves Erd\u0151s's $1000 conjecture by exhibiting {1,2,4,8,13} as a size-5 Sidon set that cannot be extended to any perfect difference set. Provides Lean 4 formal verification (LLM-assisted) of both this and Marshall Hall's overlooked 1947 counterexample. Methodology uses finite projective planes/polarities and Singer cycles.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Targets Sidon-extension question, not arena minimization. Constructive use of Singer projective-plane structure, but no new Z/NZ difference-basis bound.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.25214", "title": "Empirical evidence for size-4 counterexamples to Erd\u0151s's Sidon-extension conjecture", "authors": [ "Niu Tong" ], "date": "2026-04", "venue": "arXiv math.CO", "summary": "Provides A={0,1,3,11} and B={0,1,4,11} along with infinite families kA, kB and reflections as candidate size-4 counterexamples failing to extend for every prime power q \u2264 317 and moduli v \u2264 133. Confirms density pattern N_ne(N) = 4 floor(N/11) for small N. Singer-affine-orbit verification.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Follow-up to Alexeev-Mixon. Computational; uses Singer-orbit framework but does not address Z/NZ difference-basis minimization.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.14924", "title": "Cyclic relative difference sets and circulant weighing matrices", "authors": [ "Gordon Daniel M." ], "date": "2025-01", "venue": "Journal of Algebraic Combinatorics 63 (2026), art. 24", "summary": "Extends Lam (1977) and Pott (1995) searches for (m,n,k,lambda)-relative difference sets beyond k \u2264 64, addressing Pott's question of whether nontrivial RDS liftings exist other than classical Singer complements. Applied to existence problem for circulant weighing matrices.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cyclic relative difference sets are a Singer-adjacent structure but the parameters considered (k \u2264 64) are far below the arena regime (|B|=360, V=49109). Does not provide a Bill 7 CRT lift.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.18385", "title": "Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets", "authors": [ "Swartz Eric", "Davis James A.", "Polhill John", "Smith Ken W." ], "date": "2024-07", "venue": "arXiv math.CO", "summary": "Introduces combinatorial transfer method to construct nonabelian Denniston PDSs, Spence and McFarland difference sets in groups with non-normal Sylow subgroups, and semiregular relative difference sets with nonabelian forbidden subgroups. Includes converse to Dillon's Dihedral Trick.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Nonabelian setting. Arena problem operates in Z/NZ (cyclic abelian). M6 out of scope.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.06955", "title": "Optical orthogonal codes from a combinatorial perspective", "authors": [ "Huczynska Sophie", "Ng Siaw-Lynn" ], "date": "2024-11", "venue": "arXiv math.CO", "summary": "Treats OOCs as subsets of Z_v with decoupled auto- and cross-correlation conditions, examining bounds for correlation values >1 and structural consequences. Establishes connections to cyclic difference families and related combinatorial structures.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "OOCs are equivalent to (v,k,1) cyclic difference families. Different cost function from arena's |B|/sqrt(V) but same underlying structure.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.11736", "title": "Cardinalities of g-difference sets", "authors": [ "Schmutz Eric", "Tait Michael" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Proves the limit lim_{n\u2192\u221e} eta_g(n)/sqrt(n) exists and is finite/nonzero, answering Kravitz's question. For alpha_g(n) (largest A in [n] with at most g solutions to a_1-a_2=x for any x\u22600): alpha_g(n) = sqrt(gn)(1+o_g(1)). Extends to finite-field vector spaces.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Asymptotic existence of limit eta_g(n)/sqrt(n); does NOT provide an explicit construction at the arena's specific N=49109, only proves limit exists. Prerequisite to Li-Yip 2025.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.07750", "title": "New constructions and bounds for nonabelian Sidon sets with applications to Tur\u00e1n-type problems", "authors": [ "Byrne John", "Tait Michael" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Constructs large S_k-sets in symmetric group S_n, S_2-sets in S_n x S_n and A_n x A_n. Probabilistic constructions for A_n and S_n. Improves trivial upper bounds by constant factors. Applications to extremal digraph theory.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Nonabelian. M6 out of scope for arena Z/NZ difference-basis problem.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.12911", "title": "On large Sidon sets", "authors": [ "Czerwinski Ingo", "Pott Alexander" ], "date": "2024-11", "venue": "arXiv math.CO / cs.IT", "summary": "Constructs Sidon sets of size 192 in F_2^15 (vs prior best 152) using almost-perfect-nonlinear (APN) functions with high linearity from F_2^8 to F_2^8 (inverse and Dobbertin). Also improves upper bound for linearity of arbitrary APN functions. Yields binary linear codes with d=5 at previously unknown lengths.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": 192, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Construction in F_2^15 (size 2^15 = 32768), close to but not equal to V=49109. Not Z/NZ-cyclic and APN-derived size 192 corresponds to score ~192/sqrt(32768) = 1.061 in different normalization \u2014 appears to dominate arena's 2.639 but the score formula is incommensurable (different group, different normalization).", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2603.00174", "title": "Automated discovery of improved constant weight binary codes", "authors": [ "Rosin Christopher D." ], "date": "2026-02", "venue": "arXiv cs.IT", "summary": "CPro1 protocol (LLM-generated tabu search + random-histogram greedy heuristic) establishes improved lower bounds on A(n,d,w) for 24 (n,d,w) triples with 6 \u2264 d \u2264 18 and 18 \u2264 n \u2264 35. Examples: A(18,6,6) \u2265 133, A(22,8,9) \u2265 292.", "candidate_bill": "Bill 8", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "above_leader", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Targets larger n (18-35) than arena's support n=11 ({0,1,4,6}). Brouwer A(11,4,4) is unaffected. Bill 8 confirmed populated for larger n only, leader's specific A(11,4,4)<=35 with 30 supports is uncontested.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 8", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "Novikov A.", "Vu N.", "et al. (DeepMind)" ], "date": "2025-06", "venue": "arXiv cs.LG", "summary": "Evolutionary coding agent applied to 50+ open problems in analysis/combinatorics/geometry/number theory. Rediscovers best-known in ~75% and improves in ~20%. Improvements include Erd\u0151s minimum overlap problem (slight). Difference-basis problem is in benchmark suite; arena claim that AlphaEvolve produced 2.639027469 is unverified by community.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "needs_gate", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases", "kissing-number-d11", "heilbronn-triangles", "erdos-min-overlap", "first-autocorrelation-inequality" ], "headline_constant": 2.639027469, "leader_relationship": "matches_leader", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Bill 11 baseline. AlphaEvolve specific difference-basis output 2.639 is reportedly the arena incumbent; the construction B=R+8011*{0,1,4,6} where R is Singer (8011,90,1) is published in arena artifacts but not in a peer-reviewed venue. M5 unpublished/proprietary.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 11", "classifier_meta_cost": "M5", "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.02864", "title": "Mathematical exploration and discovery at scale", "authors": [ "Georgiev Bogdan", "G\u00f3mez-Serrano Javier", "Tao Terence", "Wagner Adam Zsolt" ], "date": "2025-11", "venue": "arXiv math.CO / cs.LG", "summary": "AlphaEvolve applied to 67 problems across analysis, combinatorics, geometry, number theory. System generalizes finite-input results into formulas, matches or improves human SOTA. Continuation/expansion of AlphaEvolve 2506.13131. Sidon-set autocorrelation included as Problem 2.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "needs_gate", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases", "kissing-number-d11", "heilbronn-triangles", "erdos-min-overlap", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Companion to AlphaEvolve 2506.13131. Does not provide a CRT cyclic embedding for difference bases \u2014 confirms Bill 7 EMPTY.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2505.16105", "title": "Sums and differences of sets (improvement over AlphaEvolve)", "authors": [ "Gerbicz Robert" ], "date": "2025-05", "venue": "arXiv math.CO", "summary": "Improves AlphaEvolve's lower bound theta = 1.1584 on sum-difference exponent to theta = 1.173050 via explicit construction of a U set with >10^43546 elements using GMP arithmetic.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M2", "verdict": "needs_gate", "confidence": 0.7, "arena_problems_relevant": [], "headline_constant": 1.17305, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Resource-unbounded (10^43546 elements). Sums-differences exponent is different metric from arena's |B|/sqrt(V) difference-basis score. M2 resource-unbounded.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M2", "classifier_version": "1.0" }, { "paper_id": "arxiv:2406.09550", "title": "New strongly regular graphs found via local search for partial difference sets", "authors": [ "Brady Andrew C." ], "date": "2024-06", "venue": "arXiv math.CO", "summary": "Local-search discovery of PDSs across 62 parameter values in 1254 nonisomorphic groups of order \u2264147. First constructions at (144,52,16,20) and (147,66,25,33). Confirms (64,18,2,6) PDS exists in 73 groups of order 64.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Strongly-regular graphs / SRGs / PDSs. Order bound \u2264147, far below arena V=49109. M6 different problem variant.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.10231", "title": "Forbidden multipliers in abelian difference sets", "authors": [ "Miller Niklas" ], "date": "2025-11", "venue": "arXiv math.CO", "summary": "Shows that group automorphisms fixing a large subgroup of an abelian group cannot be multipliers in nontrivial abelian difference sets (one Hadamard exception). Applied to exclude difference-set existence and bound numerical multiplier groups.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Negative existence theorem \u2014 narrows the construction space for abelian difference sets but does not provide a new Z/NZ difference-basis below 2.639.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2504.20959", "title": "Digraph-defined external difference families and new circular external difference families", "authors": [ "Huczynska Sophie", "Jefferson Christopher", "McCartney Struan" ], "date": "2025-04", "venue": "arXiv math.CO", "summary": "Digraph-based framework for external difference families (EDFs) and variants; constructs infinite family of (ml^2+1, m, l, 1)-CEDFs including the first such families in non-cyclic abelian groups (odd m,l) and first CEDF in a non-abelian group. Direct constructions in cyclic groups and direct products via cyclotomy.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "External difference families \u2260 difference bases. Includes non-cyclic abelian-group constructions, but parameters (ml^2+1, m, l, 1) don't match arena V=49109 regime.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2308.12406", "title": "Constructing Thick B_h-sets", "authors": [ "O'Bryant Kevin" ], "date": "2024-01", "venue": "arXiv math.NT", "summary": "Generalizes Singer and Bose-Chowla constructions of B_h sets in Z to wider commutative-semigroup setting. Provides diameter bounds for small k. Includes empirical tables of k-element B_h sets.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Singer/Bose-Chowla generalization. Does NOT improve cyclic-Z/NZ score; thicker B_h is a tradeoff (more sums to cover, larger set). No cyclic-embedding bridge.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.01986", "title": "The m-th element of a Sidon set", "authors": [ "Balasubramanian R.", "Dutta Sayan" ], "date": "2024-09", "venue": "arXiv math.NT", "summary": "For Sidon set A in [n] with |A| = n^{1/2} - L', proves a_m = m*sqrt(n) + O(n^{7/8}) + O(L^{1/2} n^{3/4}). For most n \u2264 N, sum_{a in S} a = (1/2) n^{3/2} + O(n^{11/8}), with at most O_eps(N^{4/5+eps}) exceptions.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Asymptotic position of m-th Sidon element. Density/element-bound result, not a construction at the arena's specific scale.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2503.22630", "title": "Large subsets avoiding algebraic patterns", "authors": [ "Bailleul Alexandre", "Riblet Robin" ], "date": "2025-03", "venue": "arXiv math.CO", "summary": "Transfinite-induction construction of large subsets (in cardinality/measure/dimension senses) avoiding algebraic patterns: Sidon sets in infinite abelian groups with small 2-/3-torsion, distance-avoiding sets in R^n, extending K\u00f6rner. Discusses measurability and axiom of choice.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Infinite-cardinal/measure setting. M6-adjacent: not a finite Z/NZ construction at arena scale, but tagged Bill 6 because Sidon-set extremal construction technique generalizes.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2306.11997", "title": "Cyclic relative difference families with block size four and their applications", "authors": [ "Zhao Chenya", "Zhao Binwei", "Chang Yanxun", "Feng Tao", "Wang Xiaomiao", "Zhang Menglong" ], "date": "2023-06", "venue": "arXiv math.CO", "summary": "Establishes existence of (gh,h,4,1)-CDFs for h in {2,3,6} with gh \u2261 h (mod 12), g \u2265 4 and small exceptions. Resolves existence of 1-rotational Steiner systems S(2,4,v) for v \u2261 4 (mod 12), v \u2260 28. Applications to optimal OOCs, GDDs, constant-weight codes.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cyclic difference families at block size 4, not k=90 as in arena Singer (8011,90,1). Different parameter regime.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2202.13841", "title": "On B_h[1]-sets which are asymptotic bases of order 2h", "authors": [ "Kiss S\u00e1ndor Z.", "S\u00e1ndor Csaba" ], "date": "2022-02", "venue": "arXiv math.NT", "summary": "Probabilistic existence proof: for every integer h \u2265 2, there exists a B_h[1] set (each integer expressible as h-sum at most once) that is also an asymptotic basis of order 2h.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Probabilistic existence, no explicit construction. Not a Z/NZ candidate.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.04154", "title": "On the Erd\u0151s-Tur\u00e1n conjecture and the growth of B_2[g] sequences", "authors": [ "Pliego Javier" ], "date": "2024-05", "venue": "arXiv math.NT", "summary": "For B_2[g] sequence A in N, for 0 1/eps, shows there exists such A where large n is expressible as a 3-sum with third term bounded by n^eps. Improves lower bound |A \u2229 [1,x]| \u226b x^{g/(2g+1)} over Cilleruelo, Erd\u0151s-Renyi.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "B_2[g] in N (infinite), Erd\u0151s-Tur\u00e1n density question. Asymptotic exponent x^{g/(2g+1)} doesn't directly compete with arena |B|/sqrt(V) = 360/sqrt(49109) \u2248 1.624 at fixed N=49109.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2406.16805", "title": "Beyond uniform cyclotomy", "authors": [ "Huczynska Sophie", "Johnson Laura", "Paterson Maura B." ], "date": "2024-06", "venue": "arXiv math.CO", "summary": "Direct method for evaluating cyclotomic numbers without character theory or direct field calculation, for orders dividing (q^n-1)/(q-1). Exploits connections among cyclotomy, Singer difference sets, and finite geometry.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Singer-difference-set computation toolkit. Could enable faster verification of arena candidates but no new Z/NZ construction beating 2.639.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2408.16335", "title": "A connection between unbordered partial words and sparse rulers", "authors": [ "Saarela Aleksi", "Vanhatalo Aleksi" ], "date": "2024-08", "venue": "arXiv math.CO", "summary": "Establishes bijection: domains of unbordered partial words \u2194 complete sparse rulers (restricted difference bases). Uses this to improve bounds on hole-count for unbordered partial words over alphabets of size \u2265 4. Counterexample to a previously claimed theorem.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sparse rulers = restricted difference bases on [0,n] integers, not cyclic Z/NZ. Wichmann-rulers framework. Different metric (length vs |B|/sqrt(V)).", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.01300", "title": "Note on the additive basis conjecture", "authors": [ "Yu Yang" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Proves the additive basis conjecture for p=3: the union of any four linear bases in a vector space over Z_3 is an additive basis. Alternative proof of the weak 3-flow conjecture.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Additive basis = covering via sums, in Z_3 vector space context. Different notion from arena difference basis. M6.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2504.14931", "title": "Point-transitive Steiner systems S(2,6,111/121/126), S(2,7,169/175)", "authors": [ "Hetman Ivan" ], "date": "2025-04", "venue": "arXiv math.CO", "summary": "Constructs new Steiner systems S(2,6,111), S(2,6,121), S(2,6,126), S(2,7,169), S(2,7,175) with point-transitive (mostly commutative/abelian) automorphism groups. S(2,6,111) is non-commutative exception.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Steiner-system construction at small v (\u2264175); not in arena's V=49109 regime.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.13613", "title": "Cyclotomic matrices and power difference sets", "authors": [ "[unverified]" ], "date": "2025-11", "venue": "arXiv math.CO", "summary": "Examines cyclotomic numbers via their cyclotomic matrix, relating to a basis element of a Schur ring. Application: power difference set existence via cyclotomic-matrix conditions.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.4, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Abstract only; authors not confirmed. Cyclotomic-matrix framework for power difference sets \u2014 Singer-adjacent but no published score beating 2.639.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.01949", "title": "On Kotzig's conjecture in random graphs", "authors": [ "Glock Stefan", "Sgueglia Amedeo" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Probabilistic combinatorics: Kotzig conjecture (related to PSDS / perfect-difference-family motivation) examined in random-graph setting.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.6, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Random graphs, not deterministic Z/NZ construction. M6 different variant.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2403.02904", "title": "Combinatorial designs, difference sets and bent functions as perfect colorings of graphs and multigraphs", "authors": [ "[unverified]" ], "date": "2024-03", "venue": "arXiv math.CO", "summary": "Unified framework treating difference sets, bent functions, and combinatorial designs as perfect colorings of graphs/multigraphs.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.4, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Abstract only. Structural unification; no new explicit cyclic Z/NZ construction reported.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2307.09852", "title": "A few more Hadamard partitioned difference families", "authors": [ "[unverified]" ], "date": "2023-07", "venue": "arXiv math.CO", "summary": "New Hadamard partitioned difference families constructions.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.4, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Abstract only. Hadamard family; arena Singer family is (q^2+q+1,q+1,1), not Hadamard. Different parameter family.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 6", "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2605.03274", "title": "Formalizing the prime-field Singer construction and Sidon set infrastructure in Lean 4", "authors": [ "Hulak David B.", "Ramos Arthur F.", "de Queiroz Ruy J. G. B." ], "date": "2026-05", "venue": "arXiv math.CO / cs.LO", "summary": "Lean 4 formalization (7541 lines, no unproven sorrys) of Singer's Sidon-modulo-q^2+q+1 construction of cardinality q+1 for prime power q. Includes Johnson-route shift-incidence upper bound h(N) \u2264 sqrt(N) + N^{1/4} + O(1), asymptotic two-sided h(N) = \u0398(sqrt(N)), and conditional h(N) = sqrt(N) + O_eps(N^eps) under subpolynomial prime gaps.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Formal verification of Bill 6 substrate. Singer construction at q=89 gives Sidon set of size 90 in Z/8011 \u2014 matches the R-block of arena leader. Confirms Bill 6 reference and is itself part of base bills.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": "Bill 6", "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2502.12090", "title": "On cyclotomic nearly-doubly-regular tournaments", "authors": [ "[unverified]" ], "date": "2025-02", "venue": "arXiv math.CO", "summary": "Connects cyclotomic nearly-doubly-regular tournaments to almost-difference sets from combinatorial design theory.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.4, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Abstract only. Almost-difference-set link, no explicit cyclic-difference-basis bound improvement.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.18385-v2-pdf-related", "title": "(referenced) New Spence difference sets", "authors": [ "Davis-Polhill-Smith group" ], "date": "2024", "venue": "Designs, Codes and Cryptography", "summary": "New Spence-family (3^{d+1}(3^{d+1}-1)/2, 3^d(3^{d+1}+1)/2, 3^d(3^d+1)/2) difference sets via contracted-difference-set method.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.5, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Spence parameters: v = 3^{d+1}(3^{d+1}-1)/2. For d=4 v=29646, d=5 v=265720. Doesn't hit V=49109. M6-adjacent.", "sweep_id": "sweep_001_crypto_eurocrypt_diff_bases", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:0907.1379", "title": "Improved bounds on the supremum of autoconvolutions", "authors": [ "Mate Matolcsi", "Carlos Vinuesa" ], "date": "2009-07", "venue": "J. Math. Anal. Appl. 372 (2010), no. 2, 439-447", "summary": "Anchor (pre-2020) reference. Established the still-cited upper bound 0.75496 for the supremum of autoconvolutions of nonneg functions on a compact interval, and disproved a long-standing Schinzel-Schmidt symmetry conjecture for the extremizer. Constructs an explicit asymmetric step-function that beats the symmetric Yu (2007) baseline. Sets the equiripple/PSD-autoconv frame that all subsequent 1-AC work inherits.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 1.0, "arena_problems_relevant": [ "first-autocorrelation-inequality" ], "headline_constant": 0.75496, "leader_relationship": "matches_leader", "watchlist_tier": null, "signature_construction": false, "notes": "Arena 1-AC leader at 1.5028609 matches 2 * 0.75496 = 1.50992 within 0.5%. This is the published constant the arena leader is approaching; it has NOT been formally improved as an upper bound on the supremum of autoconvolutions in the published literature 2020-2026.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1403.7988", "title": "On Suprema of Autoconvolutions with an Application to Sidon sets", "authors": [ "Alex Cloninger", "Stefan Steinerberger" ], "date": "2014-03", "venue": "arXiv math.CA", "summary": "Pre-2020 anchor. Refines the lower bound to 0.64 for the L^inf norm of autoconvolution under unit L1 mass constraint. Bridges to additive combinatorics via Sidon-set characterization. The framework still drives 2022-2025 work by White, de Dios Pont, Madrid, Boyer-Li.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "first-autocorrelation-inequality" ], "headline_constant": 0.64, "leader_relationship": "below_leader", "watchlist_tier": null, "signature_construction": false, "notes": "Pre-scope but foundational. The 0.64 LB and 0.7549 UB define the published gap that arena agents are exploring.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1903.08731", "title": "Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics", "authors": [ "Richard A. Barnard", "Stefan Steinerberger" ], "date": "2019-03", "venue": "Proc. AMS", "summary": "Pre-2020 anchor that names the three convolution inequalities used by the arena. Establishes 0.411 lower bound for the L1 autocorrelation inf-over-shift inequality (with 0.37 ruled out). All 2020-2025 'AC inequalities' papers cite this as the canonical formulation.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 0.411, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Defines the L1 inequality referenced by arena. NOT the MV-2010 supremum-of-autoconvolution problem; different normalization. Note arena 'first' problem statement is the MV-style sup-autoconv, not Barnard-Steinerberger L1 inf-shift.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2001.02326", "title": "Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics", "authors": [ "Sara Fish", "Dylan King", "Steven J. Miller" ], "date": "2020-01", "venue": "Bull. Aust. Math. Soc.", "summary": "Generalizes Barnard-Steinerberger to d-dimensional and shift-vector autocorrelations. Derives KKT-style necessary conditions for extremizers via local perturbation arguments. Does NOT improve the 0.411 constant, but extends the framework that later White/de Dios Pont/Boyer-Li exploit.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "first-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 0.411, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Variational/Hessian conditions for AC extremizers stated, but no explicit cross-grid bridge. Adjacent to Bill 4 in spirit (perturbation arguments) but does NOT apply asymmetric-extremal principle to break symmetry.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2003.06962", "title": "On optimal autocorrelation inequalities on the real line", "authors": [ "Jose Madrid", "Joao P.G. Ramos" ], "date": "2020-03", "venue": "Comm. Pure Appl. Anal.", "summary": "Improves sharp constants in the Barnard-Steinerberger family via Hausdorff-Young + dual problem reformulation. Proves existence of extremizers for several weighted variants. Connects autocorrelation to Fourier analysis but does NOT touch the MV-2010 0.7549 supremum-of-autoconvolution constant.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "first-autocorrelation-inequality" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Dual reformulation could in principle yield a Cohn-Goncalves-style LP, but the paper stays in the L1 inf-shift normalization and doesn't reach the MV regime.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2106.13873", "title": "On classical inequalities for autocorrelations and autoconvolutions", "authors": [ "Jaume de Dios Pont", "Jose Madrid" ], "date": "2021-06", "venue": "arXiv math.CA", "summary": "Establishes existence of extremizers for Gaussian-weighted and indicator-weighted AC inequalities. Uses discretization + numerical optimization to approximate optimal constants. Methodologically the closest predecessor to Boyer-Li 2025 gradient-search, but does not publish a new numerical record on the MV-2010 constant.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Existence proof + discretization framework. No explicit cross-grid bridge; uses single-resolution numerics.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2210.16437", "title": "An almost-tight L^2 autoconvolution inequality", "authors": [ "Ethan Patrick White" ], "date": "2022-10", "venue": "arXiv math.CA", "summary": "Pins inf_f ||f*f||_2 to within 0.0014% relative error via convex-optimization in Fourier space. Proves uniqueness of the minimizer. This is the L^2 cousin of the 1-AC problem, not 1-AC itself.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "DIFFERENT functional from arena 1-AC (sup-of-autoconv). White's almost-tight L^2 frame is what Rechnitzer 2026 then pushes to 128 digits.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "Canadian Math Bull 2024 / White", "title": "An optimal L^2 autoconvolution inequality", "authors": [ "Ethan Patrick White" ], "date": "2023-07", "venue": "Canad. Math. Bull. 67 (2024), 1", "summary": "Tightens the L^2 autoconvolution constant mu_2^2 to the interval [0.574636066, 0.574642912] (4e-6 width), improving prior [0.574575, 0.640733]. Proves existence and uniqueness of minimizer. Applies to B_h[g] Sidon-type sets.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.574639607, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Bh[g] connection; L^2 sup-norm framework, not L^inf-of-autoconv arena variant.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2210.01684", "title": "Sign uncertainty principles and low-degree polynomials", "authors": [ "Felipe Goncalves", "Diogo Oliveira e Silva", "Joao P.G. Ramos" ], "date": "2022-10", "venue": "arXiv math.CA", "summary": "Asymptotically sharp Bourgain-Clozel-Kahane and Cohn-Goncalves sign-uncertainty principles for sublinear-degree polynomials times Gaussian. Generalizes sign-pattern to LP bounds for sphere packing.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.7, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sign-uncertainty is the Cohn-Goncalves LP framework that could in principle dualize 1-AC, but this paper applies to sphere packing not autoconvolution. Watch-list candidate for future cross-domain transfer.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "DeepMind AlphaEvolve team (Novikov et al.)" ], "date": "2025-05", "venue": "arXiv cs.AI / Google DeepMind", "summary": "First evolutionary-LLM agent applied to autocorrelation inequalities. 50-interval nonneg step function gives 0.8962 for the L^2/(L^inf*L^1) autoconvolution ratio (i.e. the 2-AC functional). Also reports 1.5032 for the 1-AC upper bound (beating MV-2010 1.5097 by ~0.4%). Score for 3-AC noted but precision questioned by ThetaEvolve.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 1.5032, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "CRITICAL: AlphaEvolve's 1.5032 is BELOW arena leader 1.5028 (minimize) \u2014 arena agents have already beaten AE. Method = simple coordinate-descent gradient + LLM evolutionary search; NO multi-resolution bridge. Confirms Bill 3 is the active bill across 2-AC too. Bill 12 cross-grid empty-space hypothesis SURVIVES this paper.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.16750", "title": "An improved example for an autoconvolution inequality", "authors": [ "Christopher Boyer", "Zane Kun Li" ], "date": "2025-06", "venue": "arXiv math.CA (NC State)", "summary": "Achieves ||f*f||_2^2 / (||f*f||_inf * ||f*f||_1) >= 0.901564 with a 575-interval nonneg step function. Method: simulated annealing on N=23 -> gradient ascent N=115 -> 575 via coarse-to-fine upsampling. Surpasses AlphaEvolve's 0.8962. Found Nov 2024 independently of AE.", "candidate_bill": "Bill 12", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "second-autocorrelation-inequality" ], "headline_constant": 0.901564, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": true, "notes": "PARTIAL SIGNATURE HIT on Bill 12: coarse-to-fine upsampling = explicit multi-resolution bridge across grid sizes. This is the FIRST published cross-grid construction in the autoconv literature 2020-2026. Yet it stays in the 2-AC functional. Arena leader at 0.962643 (maximize) is FAR above 0.901564 \u2014 published-record is BEHIND arena. Empty-space partial falsification: cross-grid IS published, but DOES NOT cross basin boundaries that arena agents have already crossed.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 12_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2508.02803", "title": "Further Improvements to the Lower Bound for an Autoconvolution Inequality", "authors": [ "Aaron Jaech", "Alan Joseph" ], "date": "2025-08", "venue": "arXiv math.CA", "summary": "Improves Boyer-Li 0.901564 to 0.94136 using a 2399-interval nonneg step function + 4x upsampling on a 559-interval optimizer. Closes ~40% of remaining gap to 1. Method: numerical search + multi-resolution upsampling.", "candidate_bill": "Bill 12", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "second-autocorrelation-inequality" ], "headline_constant": 0.94136, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": true, "notes": "SECOND CROSS-GRID HIT on Bill 12: 4x upsampling is a true block-repeat / cross-resolution bridge across discretization levels. Yet 0.94136 < arena leader 0.962643 (maximize). Confirms that published cross-grid methods exist but DO NOT YET descend across the M4 grid-locked basins that arena agents have crossed.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": "M4", "final_bill": "Bill 12", "final_meta_cost": "M4", "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.10233", "title": "ImprovEvolve: Ask AlphaEvolve to Improve the Input Solution and Then Improvise", "authors": [ "Alexey Kravatskiy", "Valentin Khrulkov", "Ivan Oseledets" ], "date": "2026-02", "venue": "Under review KDD 2026", "summary": "Modular decomposition of evolutionary coding (initialize / improve / perturb). For the 2-AC inequality (C_2 such that ||f*f||_2^2 <= C_2 ||f*f||_1 ||f*f||_inf), evolved program reaches 0.9512; human-edited variant ImprovEvolve+E reaches 0.96258 using a progressive resolution schedule scaling up to 1.6M steps via L-BFGS. Also new hexagon-packing bests for n=11-30.", "candidate_bill": "Bill 12", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "second-autocorrelation-inequality" ], "headline_constant": 0.96258, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": true, "notes": "THIRD CROSS-GRID HIT \u2014 explicit progressive resolution schedule scaling 50,000 -> 1.6M steps. Headline 0.96258 essentially MATCHES arena leader 0.962643 (within 6e-5). Partially FALSIFIES the empty-space hypothesis on Bill 12 + 2-AC: a published progressive-resolution bridge now exists at arena-equivalent level. Arena leader ClaudeExplorer at 0.962643 might be reproducing this construction. The signature construction (Bill 12 cross-grid) IS PARTIALLY FILLED for 2-AC.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 12_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.23473", "title": "ThetaEvolve: Test-time Learning on Open Problems", "authors": [ "Yiping Wang et al." ], "date": "2025-11", "venue": "arXiv cs.LG", "summary": "Replaces AlphaEvolve's frontier-LLM ensemble with a single small LLM (Distill-Qwen3-8B) + RL on a 10K-program database. Achieves C_1=1.503133 (1-AC), C_2=0.9469 (2-AC, worse than AE-v2's 0.961), C_3=1.4930 (3-AC, claims best with caveat: AE's 3-AC implementation has discrepancies).", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 1.503133, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "1.503133 vs arena leader 1.5028609 (minimize) \u2014 arena STILL beats ThetaEvolve. 3-AC headline 1.4930 vs arena leader 1.452304 \u2014 arena STILL beats by significant margin. Caveat: ThetaEvolve flags AE's 3-AC implementation has discrepancies, which may affect arena comparison. Method does NOT use explicit cross-resolution bridge for AC problems.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 3", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.07292", "title": "The first 128 digits of an autoconvolution inequality", "authors": [ "Andrew Rechnitzer" ], "date": "2026-02", "venue": "arXiv math.CA (UBC)", "summary": "Computes nu_2^2 (inf ||f*f||_2^2 over unit L1 mass on [-1/2,1/2]) to 128 decimal digits via Kummer-series ansatz on Fourier coefficients f^(k) = (-1)^k/sqrt(k) * sum a_j k^(-j) + Bessel-function reformulation + ball arithmetic (FLINT). Bounds: 0.574639607151519... within 1.2e-129.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [], "headline_constant": 0.574639607, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "DIFFERENT functional (L^2-norm-squared minimization on unit-L1 ball) from arena's 2-AC ratio. White's L^2 problem, not arena 2-AC. Methodologically extreme: ansatz + Bessel + arbitrary-precision interval arithmetic \u2014 could in principle be ported to arena 2-AC.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.13310", "title": "Higher-Order Autocorrelations on Finite Abelian Groups", "authors": [ "Aaron Agulnick", "Toby Busick-Warner" ], "date": "2026-04", "venue": "arXiv math.CO / math.NT", "summary": "Determines signal-reconstruction order for higher-order autocorrelations on finite abelian groups. Theorem A: classifies pairs on Z_6 with matching order-5 AC data but differing at order 6. Theorem B: when S(f)^v is index-2 in , order-3v AC data is a complete invariant.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Different problem variant: higher-order AC reconstruction on finite groups vs arena's continuous 1/2/3-AC inequality constants. Arena scoring is L^inf/L^1/L^2 norm ratios on R, not abelian-group AC. Useful as boundary example.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.01658", "title": "CORAL: Towards Autonomous Multi-Agent Evolution for Open-Ended Discovery", "authors": [ "Ao Qu", "Han Zheng", "Zijian Zhou et al." ], "date": "2026-04", "venue": "arXiv cs.AI", "summary": "Multi-agent evolutionary framework with shared memory + reflection. Among 6 math optimization benchmarks, reports 3-AC score 1.4557 (lower=better, matches 'best previous SOTA') for single-agent CORAL; full multi-agent variant achieves baseline range 1.4731-1.5552.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "third-autocorrelation-inequality" ], "headline_constant": 1.4557, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Arena 3-AC leader at 1.452304 (minimize) BEATS CORAL's 1.4557. No published cross-resolution / block-repeat method for 3-AC.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 12", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 12", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2601.10481", "title": "A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies", "authors": [ "[Unspecified \u2014 math.AP / math.DG]" ], "date": "2026-01", "venue": "arXiv math.AP / math.DG", "summary": "Defines Riemannian-manifold autocorrelation and applies to non-local isoperimetric energies. Functional setup distinct from real-line AC extremal problem.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Different problem variant: Riemannian AC, not the real-line / compactly-supported L^p AC of arena.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "togethercomputer/EinsteinArena-new-SOTA", "title": "EinsteinArena-new-SOTA repository (Together AI)", "authors": [ "Together AI" ], "date": "2026", "venue": "GitHub (unpublished, post-arena-launch results)", "summary": "Reports 1-AC=1.50286286 (improving prior 1.50286290 by 4e-8), 2-AC=0.961206 (notes ImprovEvolve 0.96258 unpublished), 3-AC=1.454555 (improving prior 1.455643 by ~1e-3). All numerical, no formal proofs.", "candidate_bill": "Bill 12", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 1.50286286, "leader_relationship": "matches_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Unpublished GitHub repository \u2014 M5 (unpublished/proprietary) caveat. Mirrors arena leaderboard. No new mathematical construction; numerical-search artifacts.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": "M5", "final_bill": "DISAGREE:Bill 12_vs_Bill 3", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "Friedman compendium (autocorrelation entries, 2015 onward)", "title": "Erich Friedman's Math Magic / Packing Pages (autocorrelation entries)", "authors": [ "Erich Friedman" ], "date": "2015 onward", "venue": "math.stetson.edu (web compendium)", "summary": "Numerical compendium of best-known step-function constructions for 1-AC, 2-AC, 3-AC. Provides 0.94136 'human baseline' for 2-AC that AlphaEvolve later improved. No formal proofs; pure record-keeping.", "candidate_bill": "Bill 12", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.65, "arena_problems_relevant": [ "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 0.94136, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Friedman 0.94136 is the pre-AlphaEvolve 2-AC human baseline; cited by ImprovEvolve. Numerical record, not formal.", "sweep_id": "sweep_002_arxiv_math_NT_autocorrelation", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 12_vs_Bill 3", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2311.18753", "title": "A note on extremal constructions for the Erd\u0151s-Rademacher problem", "authors": [ "Xizhi Liu", "Oleg Pikhurko" ], "date": "2024-10", "venue": "arXiv math.CO (Combinatorics, Probability and Computing)", "summary": "Amends the Pikhurko-Razborov conjecture about extremal graphs minimizing the number of r-cliques for given edge density. The authors provide explicit formulas for the minimum count predicted by Lov\u00e1sz-Simonovits and characterize the conjectured complete set of extremal graphs for r\u22654 (complete partite + triangle-free into one part). Does not change Razborov 2008 r=3 (triangle) bound; only the r\u22654 extremal-set description.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.93, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "r=3 (triangle) case is unchanged from Razborov 2008/Liu-Pikhurko-Staden 2017 exact. The 0.21171 Razborov tight constant is NOT improved here.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2507.01596", "title": "Some exact inducibility-type results for graphs via flag algebras", "authors": [ "Levente Bodn\u00e1r", "Oleg Pikhurko" ], "date": "2025-07", "venue": "arXiv math.CO", "summary": "Resolves (\u03ba,\u2113)-edge-inducibility problem for 11 new non-trivial cases when \u03ba\u22647 via flag algebras + stability. Computes F-inducibility constants for three new 5-vertex graphs (3-edge star + isolate, C4 + isolate, C4 with pendant). Includes computer-verified flag algebra certificates.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "(\u03ba,\u2113)-inducibility \u2260 Razborov triangle-density curve, but uses same flag algebra LP. Demonstrates Bill 5 framework remains the dominant tool in 2025.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.17412", "title": "On problems of Erd\u0151s and Baumann-Briggs on minimising the density of s-cliques in graphs with forbidden subgraphs", "authors": [ "Levente Bodn\u00e1r", "Oleg Pikhurko" ], "date": "2026-02", "venue": "arXiv math.CO", "summary": "Establishes minimum density of 8-cliques in graphs without an independent set of size 3 as 491411/268435456+o(1) \u2248 0.00183, via flag algebra method with computer-verified certificates. For graphs with no independent set of size 3 and no induced C5: minimum s-clique density = 2^(1-s)+o(1) for s \u2208 {4,5,6}.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": 0.00183, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Density minimization under forbidden-subgraph constraint, not unconstrained triangle-edge curve. Shows Bill 5 actively producing new explicit constants in 2026.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2601.12741", "title": "An Introduction to Razborov's Flag Algebra as a Proof System for Extremal Graph Theory", "authors": [ "Gyeongwon Jeong", "Seonghun Park", "Hongseok Yang" ], "date": "2026-01", "venue": "arXiv cs.LO/math.CO", "summary": "Survey/introduction presenting flag algebra as a formal proof system with syntax/semantics for computer scientists. Illustrates Mantel's theorem and Goodman's Ramsey-multiplicity bound. No new bounds. Cites Razborov 2010 edge-triangle as canonical example.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": null, "signature_construction": false, "notes": "Survey \u2014 no new numerical bounds. Confirms Razborov 2010 remains canonical reference; no 2024-2026 paper improves the triangle density curve C(\u03c1).", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2601.06590", "title": "FlagAlgebraToolbox: Flag Algebra Computations in SageMath", "authors": [ "Levente Bodn\u00e1r" ], "date": "2026-01", "venue": "arXiv cs.MS/math.CO", "summary": "Software paper: SageMath extension automating flag algebra calculations and optimization. Numerical \u2192 exact certificate pipeline. No new bounds; pure tooling that lowers barrier to entry for new flag algebra results.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Tool that could enable adversary to re-run Razborov 2010 LP with larger flag sizes (n=8,9,10) \u2014 watch for surprise tight-curve refinements.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2412.12904", "title": "Forcing Graphs to be Forcing", "authors": [ "Aldo Kiem", "Olaf Parczyk", "Christoph Spiegel" ], "date": "2024-12", "venue": "arXiv math.CO", "summary": "Extends forcing-conjecture family via flag algebra to balanced blow-ups and subdivisions of Sidorenko graphs. Proves cubes are forcing. Sidorenko conjecture territory \u2014 bipartite-graph minimization, not edge-triangle.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Sidorenko forcing for bipartite graphs \u2014 not edges-vs-triangles. Flag-algebra adjacent but different problem family.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2206.04036", "title": "New Ramsey Multiplicity Bounds and Search Heuristics", "authors": [ "Olaf Parczyk", "Sebastian Pokutta", "Christoph Spiegel", "Tibor Szab\u00f3" ], "date": "2024-09", "venue": "arXiv math.CO (Found. Comp. Math.)", "summary": "Computer-assisted improvements to Ramsey-multiplicity upper bounds for K4, K5 + minimum density of size-4 independent sets in graphs with clique number \u2264 4. Constructions via search heuristics + flag-algebra-validated lower bounds.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Demonstrates 'fully computer-assisted' Bill 5 producing new clique-density bounds. Methodology transfers to edges-vs-triangles if applied; but no 2024 paper has.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.24336", "title": "Semi-inducibility of 4-vertex graphs", "authors": [ "Levente Bodn\u00e1r", "Oleg Pikhurko" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Essentially resolves H-semi-inducibility for all non-complete 4-vertex graphs except 3-edge-path BBR colouring, via Razborov flag algebra + stability arguments. Computer-generated proofs.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Semi-inducibility (red/blue edge colouring) \u2014 different problem family from edges-vs-triangles.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.09842", "title": "The semi-inducibility problem", "authors": [ "Abdul Basit", "Bertille Granet", "Daniel Horsley", "Andr\u00e9 K\u00fcndgen", "Katherine Staden" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Establishes sharp/near-sharp bounds for semi-inducibility of alternating walks, alternating C4k cycles, and 4-cycles in all colour patterns. Answers question of Liu-Mubayi-Reiher: not all extremal graphs are random across edge densities; quantum graph Q with positive coefficients and edge-density interval admits only quasirandom extremals.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Semi-inducibility \u2014 out of edges-vs-triangles scope.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2601.10548", "title": "The inducibility of Tur\u00e1n graphs", "authors": [ "Xizhi Liu", "Jie Ma", "Tianming Zhu" ], "date": "2026-01", "venue": "arXiv math.CO", "summary": "Determines inducibility of every Tur\u00e1n graph for sufficiently large n: I(F,n) uniquely attained by m-partite Tur\u00e1n graph. Confirms 1995 Bollob\u00e1s-Egawa-Harris-Jin conjecture, plus stability and analog in Kk+1-free hosts.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Inducibility of multipartite Tur\u00e1n graphs \u2260 minimum-triangle-density curve.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2503.03408", "title": "Some exact values of the inducibility and statistics constants for hypercubes", "authors": [ "Levente Bodn\u00e1r", "Oleg Pikhurko" ], "date": "2025-03", "venue": "arXiv math.CO", "summary": "Flag algebra applied to subcube densities in n-dim hypercubes. Determines 5 new configurations and 3 new (d,s) pairs. Lower bounds from blowups of small Hamming codes.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Hypercube subcube density, not graph edge-triangle density.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.24064", "title": "Edge inducibility via local directed graphs", "authors": [ "Ting-Wei Chao", "Asaf Cohen Antonir", "Anqi Li", "Hung-Hsun Hans Yu" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Introduces locally-directed graphs as common refinement of Kruskal-Katona and inducibility. Determines edge inducibility of every 4-vertex graph. Connects edge and vertex inducibility hardness.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Edge inducibility framework \u2014 closer to edges-vs-triangles than vertex inducibility, but specifically per-4-vertex-graph, not edge-vs-K3 curve.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2408.01709", "title": "A spectral Lov\u00e1sz-Simonovits theorem", "authors": [ "Yongtao Li", "Lihua Feng", "Yuejian Peng" ], "date": "2024-08", "venue": "arXiv math.CO", "summary": "Spectral counterpart of Lov\u00e1sz-Simonovits 1975 triangle-supersaturation bound. Y_{n,2,q} is unique spectral extremal graph for triangle-count constraint. Translates classical result to spectral-radius framing rather than improving the edge-based bound.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Spectral-radius framing, q=O(\u221an) tight up to constants. Does NOT improve Razborov 2010 unconstrained C(\u03c1) curve. Behavior of q\u230an/2\u230b triangles is supersaturation regime, not full curve.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2310.08081", "title": "Supersaturation beyond color-critical graphs", "authors": [ "Jie Ma", "Long-Tu Yuan" ], "date": "2024-10", "venue": "arXiv math.CO (Combinatorica 2025)", "summary": "Extends Mubayi/Pikhurko-Yilma supersaturation framework: equality h_F(n,q)=t_F(n,q) for diverse families incl. disjoint cliques, Petersen. Refutes one Mubayi conjecture for q \u2265 4. Triangle supersaturation regime, q=O(n\u00b2).", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Supersaturation when q=o(n\u00b2) is linear regime; arena leader sits at \u03c1\u22480.713, \u0394-from-Tur\u00e1n linear. Different regime than full Razborov C(\u03c1).", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2403.12016", "title": "Ordered and colored subgraph density problems", "authors": [ "Emily Cairncross", "Dhruv Mubayi" ], "date": "2024-03", "venue": "arXiv math.CO", "summary": "Corrects Reiher-Wagner error: number of k-edge stars in graphs of density x \u2208 [0,1] asymptotically maximized by clique+isolates or complement. Also new bounds for ordered k-edge stars and 3-edge-coloured density.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.78, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "k-edge stars \u2260 K3 triangles. Reiher-Wagner correction is in same problem family as Reiher 2016 clique density theorem but for star subgraphs, not K3.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.12100", "title": "On the number of triangles in K4-free graphs", "authors": [ "Jialin He", "Jie Ma", "Yan Wang", "Chunlei Zu" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Disproves Gy\u0151ri-Keszegh conjecture for triangle count in K4-free graphs with infinitely many counterexamples; establishes corrected tight lower bound. Constrained (K4-free) edge-triangle density problem.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "K4-free constraint reduces feasible region. Arena edges-vs-triangles has NO K4-free side condition \u2014 full graph class. Tight bound here is in subregion.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2410.06498", "title": "When Joints Meet Extremal Graph Theory: Hypergraph Joints", "authors": [ "Ting-Wei Chao", "Hung-Hsun Hans Yu" ], "date": "2024-10", "venue": "arXiv math.CO", "summary": "Generalizes Kruskal-Katona via hypergraph joints connection to incidence geometry. Bounds hypergraph joints; partial shadow consequence. Extends Friedgut-Kahn copying bounds.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Hypergraph joints, not graph edge-triangle density.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2209.08134", "title": "The Tur\u00e1n density of tight cycles in three-uniform hypergraphs", "authors": [ "Nina Kam\u010dev", "Shoham Letzter", "Alexey Pokrovskiy" ], "date": "2024-03", "venue": "Int. Math. Res. Notices 2024(6)", "summary": "Determines \u03c0(C_\u2113^3) = 2\u221a3 - 3 \u2248 0.464 for all large \u2113 not divisible by 3, confirming Mubayi-R\u00f6dl. First Tur\u00e1n density determined where extremal construction is iterated blow-up.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.464, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "3-uniform hypergraph Tur\u00e1n density, not graph edges-vs-triangles.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2412.21011", "title": "The Tur\u00e1n density of the tight 5-cycle minus one edge", "authors": [ "Bodn\u00e1r, S\u00e1 Le\u00f3n, Liu, Pikhurko (multi-author)" ], "date": "2024-12", "venue": "arXiv math.CO", "summary": "Proves \u03c0(C_5^{3-})=1/4 for tight 5-cycle minus one edge, settling Mubayi-Sudakov-Pikhurko 2011 conjecture for \u2113\u22655.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": 0.25, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Hypergraph tight-cycle-minus-edge Tur\u00e1n density; not graph triangle-edge.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.01782", "title": "The Tur\u00e1n density of 4-uniform tight cycles", "authors": [ "Maya Sankar" ], "date": "2024-11", "venue": "arXiv math.CO", "summary": "For 4-uniform tight cycles C_L^(4) with L not divisible by 4, Tur\u00e1n density = 1/2 for all sufficiently large L. Characterization via coloring of vertex tuples.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.5, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "4-uniform hypergraph Tur\u00e1n; far from edges-vs-triangles.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.03223", "title": "The Tur\u00e1n density of short tight cycles", "authors": [ "Bodn\u00e1r", "Le\u00f3n", "Liu", "Pikhurko" ], "date": "2025-06", "venue": "arXiv math.CO", "summary": "Determines Tur\u00e1n density of short 3-uniform tight cycles. Establishes \u03c0(C_\u2113^3) = 2\u221a3 - 3 with associated stability result for short \u2113.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": 0.464, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "3-uniform short tight cycles; hypergraph problem, not graph edge-triangle.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.07535", "title": "Chromatic Ramsey numbers and two-color Tur\u00e1n densities", "authors": [ "Maria Axenovich", "Simon Gaa", "Dingyuan Liu" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "2-color Tur\u00e1n density \u03c0^(2)(G) = 1 - (R_\u03c7(G) - 1)^{-1}. Shows \u03a9(k) distinct values for k-chromatic G; new value for 4-chromatic R_\u03c7.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "2-colour Tur\u00e1n density; not edges-vs-triangles.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2410.09617", "title": "On Tur\u00e1n-type problems and the abstract chromatic number", "authors": [ "D\u00e1niel Gerbner", "Hilal Hama Karim", "Gaurav Kucheriya" ], "date": "2024-10", "venue": "arXiv math.CO", "summary": "Combinatorial approach extending Coregliano-Razborov 2020 framework. Erd\u0151s-Stone-Simonovits-type results across stability, supersaturation, sometimes exact, in abstract-chromatic-number setting.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Extends Coregliano-Razborov continuous combinatorics framework. Tangentially-related infrastructure paper; no edges-vs-triangles improvement.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2505.17325", "title": "Uniform Tur\u00e1n density \u2014 palette classification", "authors": [ "Daniel Kr\u00e1l'", "Filip Ku\u010der\u00e1k", "Ander Lamaison", "G\u00e1bor Tardos" ], "date": "2025-05", "venue": "arXiv math.CO", "summary": "Necessary and sufficient condition for when 3-uniform hypergraph is colourable by a set of palettes; connects to uniform Tur\u00e1n density. Addresses Erd\u0151s-S\u00f3s K4^(3-) and K4^(3) uniform Tur\u00e1n density problems.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Uniform Tur\u00e1n density for 3-uniform hypergraphs.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.08771", "title": "On 3-graphs with vanishing codegree Tur\u00e1n density", "authors": [ "Laihao Ding", "Ander Lamaison", "et al." ], "date": "2024-07", "venue": "arXiv math.CO", "summary": "Answers negatively a question of Falgas-Ravry, Pikhurko, Vaughan, Volec: codegree Tur\u00e1n density not always \u2265 uniform Tur\u00e1n density. Constructs counterexamples with vanishing codegree but uniform Tur\u00e1n \u2265 4/27.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": 0.148, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Codegree vs uniform Tur\u00e1n inequality. Not relevant to edge-triangle curve.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2406.03166", "title": "Alternating paths in oriented graphs with large semidegree", "authors": [ "various, incl. Tyomkyn" ], "date": "2024-06", "venue": "arXiv math.CO", "summary": "Threshold for alternating paths in oriented graphs based on minimum semidegree.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Oriented-graph paths; not edge-triangle density.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.04950", "title": "Spectral supersaturation: Triangles and bowties", "authors": [ "Yongtao Li", "Lihua Feng", "Yuejian Peng" ], "date": "2024-07", "venue": "arXiv math.CO", "summary": "First spectral supersaturation theorem for a non-color-critical graph (bowties). Every n-vertex graph with \u03bb \u2265 \u03bb(K_\u2308n/2\u2309,\u230an/2\u230b+2) contains \u230an/2\u230b bowties. Triangle count via spectral radius.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Spectral-radius framing, not edge-density framing. Bowtie supersaturation regime.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.03418", "title": "Survey of generalized Tur\u00e1n problems \u2014 counting subgraphs", "authors": [ "D\u00e1niel Gerbner", "Cory Palmer" ], "date": "2025-06", "venue": "arXiv math.CO", "summary": "Survey of ex(n,H,F) (max H-copies in F-free n-vertex graphs). Comprehensive 2025 survey covering Alon-Shikhelman framework through current literature. Includes K3-edge-count subproblems.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Survey \u2014 generalized Tur\u00e1n counts H-copies in F-free hosts, different framing than unconstrained min-K3-given-edges. Useful reference but no new Razborov-curve result.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2106.16203", "title": "The feasible region of induced graphs", "authors": [ "Xizhi Liu", "Dhruv Mubayi", "Christian Reiher" ], "date": "2021-06", "venue": "arXiv math.CO (Adv. Math.)", "summary": "Defines \u03a9_ind(F) = limit set of (edge-density, induced-F-density) pairs. Establishes general properties of feasible region for various F. Reiher 2016 clique density framework applied to induced subgraph densities.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": null, "signature_construction": false, "notes": "Pre-2024 (2021) but cited heavily by 2024-2026 inducibility papers. Feasible region for F=K3 is exactly the Razborov triangle. Acknowledges Razborov 2010 + Reiher 2016 unchanged.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2312.08265", "title": "Clique Supersaturation", "authors": [ "Quentin Dubroff", "Benjamin Gunby", "Bhargav Narayanan", "Sam Spiro" ], "date": "2023-12", "venue": "arXiv math.CO", "summary": "If G is n-vertex with kn^(3/2) triangles, then G contains \u03a9(min{k^t n^(3/2), k^(2t\u00b2/(3t-1)) n^((5t-2)/(3t-1))}) copies of K_{2,t}, k sufficiently large. Asymptotically optimal under conditions.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Triangle-rich \u2192 K_{2,t}-rich. Connects edges-triangles to other Bill 8 (Brouwer A(n,d,w)) downstream cases.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.16561", "title": "Tur\u00e1n number of complete multipartite graphs in multipartite graphs", "authors": [ "multi-author" ], "date": "2024-05", "venue": "arXiv math.CO", "summary": "Multipartite Tur\u00e1n number for complete-multipartite forbidden patterns. Refines feasibility region of multipartite host graphs.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.7, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Multipartite host restriction \u2014 not unconstrained Razborov edge-triangle setting.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2105.09991", "title": "Stability for the Erd\u0151s-Rothschild problem", "authors": [ "multi-author" ], "date": "2021", "venue": "arXiv math.CO", "summary": "Edge-colouring stability \u2014 number of valid colourings of edges avoiding monochromatic clique. Background context only.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Erd\u0151s-Rothschild edge-colouring; distinct problem family.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2108.10406", "title": "Hypergraph Tur\u00e1n problems in \u21132-norm", "authors": [ "J\u00f3zsef Balogh", "Felix Christian Clemen", "Bernard Lidick\u00fd" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "\u21132-norm Tur\u00e1n density via computer-assisted flag algebra. Determines \u03c0_\u21132(F_{3,2}) up to 10^-9 additive error.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "\u21132-norm Tur\u00e1n in 3-uniform hypergraphs.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2202.06916", "title": "Upper tail behavior of the number of triangles in random graphs with constant average degree", "authors": [ "multi-author" ], "date": "2022-02", "venue": "arXiv math.PR/Combinatorica 2024", "summary": "Structure theorem: sub-critical upper-tail dominated by k vertex-disjoint triangles; supercritical by clique-like structure. Random-graph not extremal-graph framing.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Random-graph large-deviation problem, not deterministic extremal.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.00521", "title": "Reflection groups, Cauchy-Schwarz LP, and incidence graph extremal numbers", "authors": [ "various" ], "date": "2024-12", "venue": "arXiv math.CO", "summary": "Casts Conlon-Lee Cauchy-Schwarz inequality method in purely algebraic form using reflection groups. Yields extremal number bounds for incidence graphs.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cauchy-Schwarz LP refinement; close cousin of Razborov flag algebra. Worth tracking if applied to K3-edge curve.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2508.09841", "title": "The Brown-Erd\u0151s-S\u00f3s conjecture in dense triple systems", "authors": [ "multi-author incl. Sankar" ], "date": "2025-08", "venue": "arXiv math.CO", "summary": "BES conjecture for \u03b4>4/5: every linear 3-uniform hypergraph of size \u03b4n\u00b2 contains k edges on \u2264k+3 vertices. First bound of this kind.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.8, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "BES in dense 3-uniform; out of edges-vs-triangles scope.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": "Bill 5", "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.21571", "title": "Relative Tur\u00e1n densities for ordered graphs: all and nothing", "authors": [ "various" ], "date": "2025-11", "venue": "arXiv math.CO", "summary": "Ordered-graph Tur\u00e1n density relative to base graph. Dichotomy: relative density is 0 or 1 for various configurations.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Ordered graphs; different framing.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.12100", "title": "On the number of triangles in K4-free graphs (duplicate entry as confirmation)", "authors": [ "Jialin He", "Jie Ma", "Yan Wang", "Chunlei Zu" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "DUPLICATE \u2014 see earlier entry. Disproves Gy\u0151ri-Keszegh on K4-free triangle count; gives corrected bound.", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Duplicate; K4-free constrained.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2404.10106", "title": "Statistics for the triangle density in ERGM and its mean-field approximation", "authors": [ "multi-author" ], "date": "2024-04", "venue": "arXiv math.PR/math.ST", "summary": "Statistical mechanics of edge-triangle exponential random-graph model: studies asymptotic triangle density.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "ERGM probabilistic framing, not extremal.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.16850", "title": "Covering a Graph with Dense Subgraph Families, via Triangle-Rich Sets", "authors": [ "multi-author" ], "date": "2024-07", "venue": "arXiv cs.DS/math.CO", "summary": "Algorithmic decomposition into triangle-rich dense subgraphs.", "candidate_bill": null, "candidate_meta_cost": "M2", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Algorithmic, not extremal.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M2", "classifier_version": "1.0" }, { "paper_id": "arxiv:2401.16289", "title": "Tur\u00e1n Densities for Daisies and Hypercubes", "authors": [ "multi-author" ], "date": "2024-08", "venue": "arXiv math.CO", "summary": "Tur\u00e1n densities for daisy hypergraphs and hypercube-induced structures.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Hypergraph Tur\u00e1n; not edge-triangle.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.17224", "title": "Triangle Detection in H-Free Graphs", "authors": [ "Amir Abboud", "Ron Safier", "Nathan Wallheimer" ], "date": "2025-11", "venue": "arXiv cs.DS/math.CO", "summary": "Triangle detection in H-free hosts: ex(K3,H)=O(n^(3-1/k^2)) for 3-colourable H; ex(K3,C_{2k+1})=O(ex(K2,C_{2k})).", "candidate_bill": "Bill 5", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.75, "arena_problems_relevant": [ "edges-vs-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Generalized Tur\u00e1n ex(n,K3,H): max triangles in H-free graph. Same triangle-edge framing but constrained by host-graph forbidden subgraph.", "sweep_id": "sweep_003_arxiv_math_CO_extremal_graphs", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 5", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "Novikov A.", "Vu N.", "Eisenberger M.", "Dupont E.", "Huang P.-S.", "Wagner A. Z.", "Shirobokov S.", "Kozlovskii B.", "Ruiz F. J. R.", "Mehrabian A.", "Kumar M. P.", "See A.", "Chaudhuri S.", "Holland G.", "Davies A.", "Nowozin S.", "Kohli P.", "Balog M." ], "date": "2025-06", "venue": "arXiv cs.LG (DeepMind)", "summary": "Evolutionary coding agent on top of frontier LLMs that explores algorithm/configuration space against an objective. Among ~50 open extremal-math problems, it improved K(11) from 592 to 593 (one extra unit vector on top of Ganzhinov's 592 configuration). Also claims Heilbronn n=11 \u22480.0365, AC-1 \u22480.7549, and an improved diff-bases asymptotic \u2014 most of which only the K(11) move and Heilbronn are accompanied by reproducible coordinate dumps.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.97, "arena_problems_relevant": [ "kissing-number-d11", "heilbronn-triangles", "first-autocorrelation-inequality", "difference-bases" ], "headline_constant": 593, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Headline K(11)=593 is THE published baseline; arena's 594 basin is one above this. K(11)>=594 is NOT in this paper. Same paper is bill source for ~11 of 15 arena problems.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2207.08266", "title": "Highly symmetric lines", "authors": [ "Ganzhinov M." ], "date": "2022-07", "venue": "Linear Algebra Appl. 722 (2025), 12-37", "summary": "Constructs highly symmetric line systems via twisted spherical functions on finite groups, yielding 3 new kissing configurations improving K(10) to 510, K(11) to 592, K(14) to 1932. Pre-AlphaEvolve baseline for K(11); AlphaEvolve added 1 vector. Author publicly stated K(11) lower bound is 'still quite weak' and could extend 'well beyond 600'.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.96, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 592, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Ganzhinov's 592 sits inside Bachoc-Vallentin SDP feasibility (BV upper ~870); engages Bill 2's spherical-code framework via 3-point SDP-compatible symmetric structure. Direct comparator to arena basin 610's 594.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 2", "classifier_meta_cost": null, "final_bill": "Bill 2", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.13391", "title": "Finding Kissing Numbers with Game-theoretic Reinforcement Learning", "authors": [ "Ma C.", "Zhaowei T. T.", "Li P.", "Liu M.", "Chen H.", "Mao Z.", "Cheng Y.", "Qi Y.", "Yang Y." ], "date": "2025-11", "venue": "arXiv cs.LG", "summary": "PackingStar: two-player matrix-completion game whose entries are pairwise cosines; trained from scratch (no human prior). First breakthrough beyond Zinoviev-Ericson's 1971 rational K(13)=1154 structure, surpasses all human records in d=25-31, and discovers ~6000 new configurations in d=14 and others. Reproduces all known kissing configs up to d=31.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 1154, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "CRITICAL: paper explicitly reproduces all d<=31 records \u2014 implicitly confirms NO improvement on K(11)>=593 found by their RL. PackingStar in 11D matches AlphaEvolve's 593, no higher. Closes one path to 594.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.04916", "title": "Improved kissing numbers in seventeen through twenty-one dimensions", "authors": [ "Cohn H.", "Li A." ], "date": "2024-11", "venue": "arXiv math.MG", "summary": "First improvement on Leech-1967 kissing constructions in d=17-21 since the 1960s, obtaining K(17)>=5730, K(18)>=7654, K(19)>=11692, K(20)>=19448, K(21)>=29768. Constructions modify signs in Leech-lattice cross-section vectors to open space for additional spheres, breaking the cross-section-only paradigm.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 5730, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sign-flip-in-lattice-vectors technique is methodologically analogous to D_n+ glue moves; if transferred to d=11 (where the analog substrate is the K_11 cross-section), it could plausibly produce additional vectors beyond 593. Method-transfer candidate.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2603.10425", "title": "A new lower bound for the kissing number in 19 dimensions", "authors": [ "Ho B. S." ], "date": "2026-03", "venue": "arXiv math.MG", "summary": "Combines Cohn-Li's odd-sign construction with an explicit non-linear binary code of length 19, size 1280, min distance 5 inside a 5-punctured extended Golay code, to prove K(19)>=11948 (+256 over Cohn-Li). Construction uses nested code structure (Clebsch-graph quotient, lifted to 320-word code, four cosets).", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 11948, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Non-linear-code-augmented Cohn-Li construction is exactly the kind of D_n+/code-augmentation move that could lift K(11) above 593. Strong method-transfer candidate for d=11.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2403.16874", "title": "Optimality of spherical codes via exact semidefinite programming bounds", "authors": [ "Cohn H.", "de Laat D.", "Leijenhorst N." ], "date": "2024-03", "venue": "arXiv math.MG / J. AMS", "summary": "Develops exact rounding of 3-point Bachoc-Vallentin SDP bounds to certify universal optimality of spherical codes: triangle-free SRG embeddings (56pts/d=20, 50pts/d=21, 77pts/d=21), Kerdock-MUB codes in d up to 1024, and 288 pts on S^15. New sharp cases for the BV SDP.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.96, "arena_problems_relevant": [ "kissing-number-d11", "tammes-problem" ], "headline_constant": 288, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Critical direct hit on Bill 2 \u2014 provides the rounding methodology that turns numerical BV bounds into rigorous certificates. Doesn't certify K(11) but is the toolchain a hypothetical K(11) optimality proof would use.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 2", "classifier_meta_cost": null, "final_bill": "Bill 2", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2512.04829", "title": "Model-Based and Sample-Efficient AI-Assisted Math Discovery in Sphere Packing", "authors": [ "[DeepMind/Imperial team]" ], "date": "2025-12", "venue": "arXiv cs.LG", "summary": "Formulates 3-point SDP construction as a sequential decision process ('SDP game'): policy assembles SDP formulations from admissible components; Bayesian optimization + MCTS guides search. Obtains new state-of-the-art UPPER bounds in d=4-16, surpassing all prior known SDP bounds in that band.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.94, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Tightens the BV SDP upper bound for K(11) (previously ~870 from Mittelmann-Vallentin / Machado-Filho 2018). The new upper does NOT cross 593 \u2014 confirms BV gap remains wide and K(11)>=594 is consistent with BV. Bill 2 status update.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 2", "classifier_meta_cost": null, "final_bill": "Bill 2", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.06960", "title": "Conic optimization for extremal geometry", "authors": [ "Vallentin F." ], "date": "2025-10", "venue": "arXiv math.OC (survey)", "summary": "Survey of conic-optimization tools for kissing, \u03c0/2-avoiding sets, sphere packing, and 1-avoiding sets. Three pillars: SDP, symmetry reduction via harmonic analysis, rigorous rounding of numerical SDP solutions. Frames the post-Viazovska state of the art including BV-SDP, three-point methods, and recent rounding advances.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.93, "arena_problems_relevant": [ "kissing-number-d11", "tammes-problem", "thomson-problem" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Canonical 2025 reference for Bill 2 (BV-SDP) framework and Bill 1 (Cohn-Elkies LP); cites all recent improvements. Useful as the survey baseline against which arena leaders are compared.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 2_vs_Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2403.07457", "title": "Energy bounds for weighted spherical codes and designs via linear programming", "authors": [ "Borodachov S.", "Boyvalenkov P.", "Dragnev P.", "Hardin D.", "Saff E.", "Stoyanova M." ], "date": "2024-03", "venue": "arXiv math.MG", "summary": "Universal LP energy bounds for weighted spherical codes in the Cohn-Kumar / Levenshtein sense: every code attaining the bound is universally optimal across all absolutely-monotone potentials. Provides both necessary and (under strict-monotonicity) sufficient optimality conditions, plus bounds for weighted spherical designs.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "thomson-problem", "tammes-problem" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Direct Bill 10 extension to weighted codes. Thomson-N (arena leader 37147.294 at N=200) sits within this LP-bound framework, but the paper does not give a numerically tight bound for arbitrary N \u2014 gap is acknowledged.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2412.07442", "title": "Bounds on energy and potentials of discrete measures on the sphere", "authors": [ "Borodachov S.", "Boyvalenkov P.", "Dragnev P.", "Hardin D.", "Saff E.", "Stoyanova M." ], "date": "2024-12", "venue": "arXiv math.MG", "summary": "Universal upper and lower bounds on potentials of weighted spherical designs depending only on quadrature nodes/weights. Constrained energy minimization for Borel probability measures on S^{n-1}; applies to optimal charge distribution at fixed point count; p-frame energy results.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "thomson-problem" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Direct extension of Cohn-Kumar universal LB framework with explicit constraints. Methodologically applicable to Thomson but does not produce a closed-form bound at N=200 that would falsify the arena leader.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.16038", "title": "The Tammes Problem in R^n and Linear Programming Method", "authors": [ "Lian Y.", "Mo Q.", "Xia Y." ], "date": "2024-11", "venue": "arXiv math.MG", "summary": "Articulates sufficient conditions for attaining the optimal value in the Tammes problem for general n, N via the Delsarte LP framework. Provides illustrative examples in several dimensions and N values where LP yields optimal configurations. Pure analytical framework, no high-precision SDP.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "tammes-problem" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Bill 1 (Cohn-Elkies / Delsarte LP) applied directly to Tammes. Examples don't beat KawaiiCorgi 0.513472 at specific N \u2014 illustrative not breakthrough.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2309.13854", "title": "Semidefinite programming bounds for distance distribution of spherical codes", "authors": [ "Musin O. R." ], "date": "2023-09", "venue": "arXiv math.MG", "summary": "Extension of SDP and LP upper bounds for spherical codes, applied to the distance distribution of the code. Effective when standard BV-SDP plateaus. Shorter solution to the kissing number problem in dimension 4.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.88, "arena_problems_relevant": [ "kissing-number-d11", "tammes-problem" ], "headline_constant": 24, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Methodologically interesting variant of Bill 2 that conditions on the distance distribution rather than the full code structure \u2014 potentially relevant for asymmetric arena basins like the 594 construction.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.13906", "title": "Universal optimality of T-avoiding spherical codes and designs", "authors": [ "Boyvalenkov P. G.", "Cherkashin D. D.", "Dragnev P. D." ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Spherical codes whose inner products avoid a forbidden set T. Minimal vectors of certain Leech-derived lattices, Barnes-Wall lattice minimal vectors, and codes from strongly regular graphs are shown universally optimal in their T-avoiding class. Extends Delsarte-Goethals-Seidel three-distance theorem.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "T-avoiding framing maps onto the arena's 'forbidden cosine' regime. Universal-optimality results apply to Leech/Barnes-Wall \u2014 not d=11. K_11 does not naturally fit the strongly-regular-graph paradigm here.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 2_vs_Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2312.05121", "title": "The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000", "authors": [ "Boyvalenkov P.", "Cherkashin D." ], "date": "2023-12", "venue": "arXiv math.MG / Results Math.", "summary": "Modifies Delsarte-Goethals-Seidel LP bounds to a forbidden-inner-product variant. Proves K(48) restricted to antipodal codes with certain forbidden inner products is exactly 52,416,000, attained by minimal vectors of even-unimodular extremal lattices. Also shows the corresponding spherical 11-designs are minimal.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 52416000, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Forbidden-distance LP refinement is method-transferable to d=11 (where K_11 minimal vectors have a specific inner-product spectrum). Useful framing for proving optimality on a restricted subclass.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2410.04800", "title": "A New Linear Programming Method in Sphere Packing", "authors": [ "Mo Q.", "Wen J.", "Xia Y." ], "date": "2024-10", "venue": "arXiv math.MG", "summary": "Generalizes Cohn-Elkies LP: sequence of m\u039b-periodic auxiliary functions {g_m} on R^n rather than a single function. Greater flexibility for the dual LP. Constructs effective auxiliary functions in d=1,2,3 with view toward higher d.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Direct Bill 1 extension. Currently demonstrated only in low d but is a candidate for tightening Bill 1 in d=11 via lattice-periodic auxiliaries (D_11+ etc.).", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2505.23572", "title": "Linear programming bounds in homogeneous spaces, I: Optimal packing density", "authors": [ "Wackenhuth M." ], "date": "2025-05", "venue": "arXiv math.MG", "summary": "LP bounds for sphere packing density on commutative homogeneous spaces. Solves Cohn-Zhao conjecture on LP bounds in hyperbolic space as a special case. Generalizes Cohn-Elkies LP from R^n to broader symmetric spaces.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Bill 1 in non-Euclidean settings. Does not engage arena targets directly (arena is Euclidean) but is the canonical 2025 LP-bounds extension paper.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.07139", "title": "Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao", "authors": [ "Wackenhuth M." ], "date": "2024-11", "venue": "arXiv math.MG", "summary": "Proves sphere-packing density bounds in hyperbolic space and irreducible symmetric spaces of non-compact type, generalizing Euclidean Cohn-Elkies. Uses Bowen-Radin packing density framework with quasicrystal-style Poisson formulas in place of the standard Poisson summation.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Companion to 2505.23572. Out of arena scope (hyperbolic) but methodologically a key Bill 1 reference.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2412.00937", "title": "Variations on five-dimensional sphere packings", "authors": [ "Cohn H.", "Rajagopal I." ], "date": "2024-12", "venue": "arXiv math.MG", "summary": "Analyzes Sz\u00f6ll\u0151si's conjecturally-optimal 5D kissing configuration and produces a new (4th) such configuration, plus a new 9D kissing configuration. None of these improves on known records but they are geometrically distinct optimal-record configurations.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Methodologically suggestive: distinct optimal configurations exist at the same record cardinality. Implies that, even if K(11)=593 is optimal, multiple distinct configurations may attain it \u2014 relevant for arena basin classification.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2410.16660", "title": "Difficulties Constructing Lattices with Exponential Kissing Number from Codes", "authors": [ "Bennett H.", "Golovnev A.", "Stephens-Davidowitz N." ], "date": "2024-10", "venue": "arXiv math.MG / IEEE Trans. Inf. Theory", "summary": "Shows the standard code-to-lattice constructions (Constructions A/B/D and refinements) do NOT in general give a correspondence between min-weight codewords and shortest lattice vectors. Invalidates Vladut's 2018/2021/2024 claims of exponential-kissing-number lattices. Open problem reopens.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.92, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cautionary tale: code-to-lattice constructions need careful verification of the kissing-set correspondence. Directly impacts any code-augmented K(11) construction (e.g., arena basin A's (11,30,4,>=4) constant-weight code structure).", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 8", "classifier_meta_cost": null, "final_bill": "Bill 8", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.07371", "title": "Lattices with exponentially large kissing numbers do exist (WITHDRAWN)", "authors": [ "Vladut S." ], "date": "2024-11", "venue": "arXiv math.MG", "summary": "Author's own correction/retraction note attempting to fix the 2018/2021 papers. Withdrawn after Bennett-Golovnev-Stephens-Davidowitz identified the construction is still flawed. As of 2025, no published family with exponential kissing number.", "candidate_bill": null, "candidate_meta_cost": "M5", "verdict": "out_of_scope", "confidence": 0.95, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Withdrawn/invalid. M5 (unpublished claim) properly applied \u2014 paper exists but the claim does not.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": "M5", "final_bill": null, "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2312.10026", "title": "A new lower bound for sphere packing", "authors": [ "Campos M.", "Jenssen M.", "Michelen M.", "Sahasrabudhe J." ], "date": "2023-12", "venue": "arXiv math.MG", "summary": "First asymptotically-growing improvement on Rogers' 1947 sphere-packing bound: density >= (1-o(1)) d log(d) / 2^(d+1) as d->inf. Non-lattice graph-theoretic method with similar constructions for spherical codes and kissing numbers.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.88, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Method is asymptotic (d->inf), not directly applicable at d=11. But the graph-theoretic kissing-number side could plausibly yield concrete d=11 bounds with constants \u2014 flag for new-bill consideration (graph-theoretic / probabilistic kissing).", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2504.05042", "title": "Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid", "authors": [ "Klartag B." ], "date": "2025-04", "venue": "arXiv math.MG", "summary": "Sphere packing density at least c n^2 / 2^n in R^n via a stochastically-evolving origin-symmetric ellipsoid containing no Z^n point except origin. Improves Campos et al's n log(n) to n^2. Lattice construction (vs. non-lattice in 2312.10026).", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.88, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Asymptotic (d->inf) lattice construction. Method could in principle be adapted to give concrete d=11 bounds via ellipsoid + Z^11 \u2014 flag for new-bill consideration (stochastic-ellipsoid kissing).", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2406.07479", "title": "Lower bounds for sphere packing in arbitrary norms", "authors": [ "Schildkraut C." ], "date": "2024-06", "venue": "arXiv math.MG", "summary": "Generalizes Campos-Jenssen-Michelen-Sahasrabudhe ell_2 result to arbitrary norms: any d-dim real normed space admits unit-ball packing with density >= (1-o(1)) d log(d) / 2^(d+1). Improves Schmidt 1958 by log factor. Uses graph theory plus Petty/Schmuckenschlager volume bounds.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Arbitrary-norm generalization is out of arena scope (arena is ell_2). Companion to 2312.10026.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2303.07908", "title": "A quantitative stability result for the sphere packing problem in dimensions 8 and 24", "authors": [ "Boroczky K. J.", "Radchenko D.", "Ramos J. P. G." ], "date": "2023-03", "venue": "Crelle (Reine Angew. Math.) 2024", "summary": "Explicit stability estimates: any lattice / periodic packing within epsilon of optimal density in d=8 or 24 is within O(epsilon^{1/2}) of E_8 or Leech, respectively. Uses Viazovska + Cohn-Kumar-Miller-Radchenko-Viazovska magic functions.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.88, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "d=8, 24 only \u2014 out of arena scope. Methodology (magic-function stability) is the cleanest version of Bill 1 in the solved dimensions.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 1_vs_Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2308.03925", "title": "Sphere Packings in Euclidean Space with Forbidden Distances", "authors": [ "Goncalves F.", "Vedana G." ], "date": "2023-08", "venue": "Forum Math. Sigma 2025", "summary": "Sphere packing with forbidden inter-center distances. Proves d=48 packings avoiding centers at distance in (sqrt(4/3), sqrt(5/3)) * 2r have center density <= (3/2)^24, attained iff packing is 48D even unimodular extremal lattice. Results extend to d <= 1200.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Forbidden-distance regime \u2014 methodologically interesting and same family as 2312.05121. Direct d=11 application would require a different magic function.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2206.09876", "title": "Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions", "authors": [ "Li R." ], "date": "2022-06", "venue": "Adv. Math. 2024", "summary": "Maps the infinite-dim Cohn-Elkies LP into a finite-dim LP via discrete reduction. Proves the Cohn-Elkies LP CANNOT come close to known best packing densities in d=3..13 except d=8. Falsifies the 'Cohn-Elkies is tight' optimism in dimensions 3-13 besides 8.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.94, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Critical Bill 1 status update: Cohn-Elkies LP is provably NOT sharp at d=11 (and d=3-13 except 8). Implies that K(11) optimality, if attainable, will require Bill 2 (BV-SDP) or higher-point methods, not pure Bill 1.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2502.06002", "title": "Fixed-strength spherical designs", "authors": [ "Dillon T." ], "date": "2025-02", "venue": "arXiv math.CO", "summary": "Fixed t, d->infinity regime for spherical t-designs. Constructs smaller t-designs via Gaussian-design connection; exact order of magnitude for signed t-designs, smaller than the degrees-of-freedom heuristic predicts. Projection results across dimensions.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Tangential to arena targets but relevant to weighted-spherical-design analogs of Bill 10.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2404.15544", "title": "Constructions of Spherical 3-Designs", "authors": [ "[multi-author]" ], "date": "2024-04", "venue": "arXiv math.CO", "summary": "Explicit constructions of spherical 3-designs with specific cardinality constraints across dimensions. Combinatorial 3-design framework.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.7, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Low-strength designs; tangential to arena.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.11437", "title": "More on the corner-vector construction for spherical designs", "authors": [ "Tanino K.", "Sawa M.", "[al.]" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Generalized corner-vector method for weighted spherical designs. Uniform upper bound on the degree via hybrid combinatorial-analytic argument including Xu 1998 and Nozaki-Sawa 2013 cross-ratio techniques. Examples of degree 7-13 designs.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.78, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Corner-vector constructions relate to integer-lattice kissing structures (D_n) but are out of immediate arena scope.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2412.05800", "title": "Lower and upper bounds for configurations of points on a sphere", "authors": [ "Amore P.", "Saenz R. A." ], "date": "2024-12", "venue": "arXiv math-ph", "summary": "Spectral-decomposition proof of Sidelnikov's frame-potential bound on S^{d-1}; sharp upper bound in 3D for a specific sum involving cross/dot products. Applied to local-minimum configurations of Thomson at N=972, plus 3D volume distributions.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.82, "arena_problems_relevant": [ "thomson-problem" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Thomson at N=972 specifically \u2014 arena Thomson is at smaller N. Direct Bill 10 (Cohn-Kumar universal LB) variant via spectral decomposition.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.08398", "title": "Exploring the energy landscape of the Thomson problem: local minima and stationary states", "authors": [ "[multi-author Spain/Italy team]" ], "date": "2025-06", "venue": "arXiv cond-mat", "summary": "Comprehensive numerical investigation of Thomson energy landscape for N <= 150 plus stationary-point exploration for N <= 24. Shows number of distinct configurations grows exponentially in N, FASTER than previously reported. Reformulates stationary-point search as minimization with novel potential.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "thomson-problem" ], "headline_constant": null, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Thomson landscape characterization for N <= 150 \u2014 covers arena range. Faster-than-expected exponential growth of basins implies arena leader at 37147.294 may have many distinct optima at the same energy; basin classification matters.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2508.17625", "title": "Thomson problem on a spherical cap", "authors": [ "[multi-author]" ], "date": "2025-08", "venue": "arXiv cond-mat", "summary": "Low-energy configurations of N repelling charges on a spherical cap (vs. full sphere) with Coulomb potential. Topological-defect emergence as function of N and cap angular width. Efficient numerical method.", "candidate_bill": "Bill 10", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "M6 (different problem variant) \u2014 spherical cap not full sphere.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.16508", "title": "Geodesic distance Riesz energy on projective spaces", "authors": [ "[multi-author]" ], "date": "2024-09", "venue": "arXiv math.MG", "summary": "Riesz energy minimizers on projective spaces using chordal and geodesic distances. Discovers multiple phase transitions in the minimizers as s varies (contrast with classical results having one transition).", "candidate_bill": "Bill 10", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Projective-space variant \u2014 out of arena scope (M6). Riesz framework however is the natural Bill 10 setting.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.00424", "title": "Minimal Riesz and logarithmic energies on the Grassmannian Gr_{2,4}", "authors": [ "[multi-author]" ], "date": "2024-12", "venue": "arXiv math.MG", "summary": "Continuous Riesz/log energies on Grassmannian Gr_{2,4} uniquely minimized by uniform measure. Asymptotic upper/lower bounds for the discrete minimal Riesz/log energies as N->inf.", "candidate_bill": "Bill 10", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Grassmannian variant \u2014 out of arena scope.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.00120", "title": "Riesz energy with a radial external field: when is the equilibrium support a sphere?", "authors": [ "[multi-author]" ], "date": "2024-04", "venue": "arXiv math.MG", "summary": "Continuous Riesz energy with radial external field. Studies when the equilibrium support is a sphere (uniform distribution).", "candidate_bill": "Bill 10", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Continuous variant \u2014 out of arena scope.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.00290", "title": "Bounds on Discrete Potentials of Spherical (k,k)-Designs", "authors": [ "[multi-author]" ], "date": "2024-11", "venue": "arXiv math.MG", "summary": "Discrete potential bounds for spherical (k,k)-designs, intermediate between standard spherical designs and weighted designs. Universal-LB-style results.", "candidate_bill": "Bill 10", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.78, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Bill 10 variant for (k,k)-designs.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2502.07720", "title": "Hybrid spherical designs", "authors": [ "[multi-author]" ], "date": "2025-02", "venue": "arXiv math.CO", "summary": "Hybrid spherical designs interpolating between Chebyshev-type and standard designs. Combinatorial constructions.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.75, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Hybrid-design framework \u2014 out of arena scope.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.07452", "title": "Real and complex spherical designs and their Gramian", "authors": [ "[multi-author]" ], "date": "2025-11", "venue": "arXiv math.CO", "summary": "Real and complex spherical designs characterized via Gram-matrix structure. Includes connections to MUBs and equiangular line systems.", "candidate_bill": "Bill 2", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.7, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Gram-matrix framing of spherical designs \u2014 methodologically related to PackingStar's matrix-completion game.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2111.01255", "title": "New lower bounds on kissing numbers and spherical codes in high dimensions", "authors": [ "Jenssen M.", "Joos F.", "Perkins W." ], "date": "2021-11", "venue": "Amer. J. Math.", "summary": "Probabilistic / graph-theoretic kissing number lower bounds. K(d) >= (1+o(1)) c d^{3/2} (2/sqrt(3))^d as d->inf \u2014 improves earlier asymptotic by factor of 3.442. Precursor to Campos-Jenssen-Michelen-Sahasrabudhe 2023.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.88, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Asymptotic \u2014 does not give concrete K(11). Flagged for new-bill consideration (graph-theoretic / probabilistic kissing).", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2207.09030", "title": "On the lower bound for kissing numbers of l_p-spheres in high dimensions", "authors": [ "[multi-author]" ], "date": "2022-07", "venue": "arXiv math.MG", "summary": "Lower bounds for kissing numbers in l_p (non-Euclidean) norms in high dimensions.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "out_of_scope", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "l_p non-Euclidean \u2014 out of arena scope.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2001.00185", "title": "New upper bounds for spherical codes and packings", "authors": [ "de Laat D.", "Leijenhorst N. M." ], "date": "2020-01", "venue": "Math. Ann. 2023", "summary": "Strengthens Cohn-Elkies LP and Levenshtein LP bounds in d=4-23. Computational SDP upper bounds for K(d). Pre-2024 SDP baseline for d=11.", "candidate_bill": "Bill 2", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "kissing-number-d11", "tammes-problem" ], "headline_constant": 870, "leader_relationship": "above_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Provides upper bound for K(11) around 870 (well above 593). Bill 2 reference for the K(11) BV-SDP feasibility gap.", "sweep_id": "sweep_004_arxiv_math_MG_sphere_packing", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 2_vs_Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "Novikov Alexander", "Vu Ngan", "Eisenberger Marvin", "Dupont Emilien", "Huang Po-Sen", "Wagner Adam Zsolt", "Shirobokov Sergei", "Pioger Borislav", "Ruiz Francisco J. R.", "Mankowitz Daniel J.", "Hassabis Demis", "Fawzi Alhussein", "Barekatain Mohammadamin", "Balog Matej" ], "date": "2025-06", "venue": "arXiv cs.LG", "summary": "Evolutionary coding agent (Gemini-powered) that improves point configurations for the Heilbronn-in-unit-triangle problem, reporting n=11 minimum-area >0.0365 (arena leader 0.036530 traces here). Section 3 lists open-problem advances but full Heilbronn coordinates are deferred to the companion notebook in google-deepmind/alphaevolve_results. Paper claims improvement but does not publish explicit n=11 coordinates inline.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "heilbronn-triangles", "kissing-number-d11", "thomson-problem", "difference-bases", "first-autocorrelation-inequality" ], "headline_constant": 0.03653, "leader_relationship": "matches_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "AlphaEvolve construction is Z/2 reflection-symmetric (atlas finding via arena verifier inspection). Paper itself is M5-adjacent for n=11: score reported but full configuration is in companion repo only (not in main PDF body). Bill 4 (asymmetric extremal) NOT engaged \u2014 AlphaEvolve produced \u03c3-equivariant config, same symmetry class as prior Cantrell 2006 leader.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": "Bill 4", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 4", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2603.11107", "title": "From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization", "authors": [ "Sudermann-Merx Nathan" ], "date": "2026-03", "venue": "arXiv math.OC", "summary": "MINLP-based optimize-then-refine framework that certifies global optimality of n=5..9 Heilbronn configurations on the unit square in minutes (vs Chen-Xu-Zeng 2017 GPGPU's ~1 day for n=9). First rigorous proof that Comellas-Yebra 2002 n=5..9 unit-square configs are optimal. Symmetry-breaking is used as a *solver heuristic*, not as a constructive principle \u2014 paper assumes symmetric optimum and verifies.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.88, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.0548767, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Unit square, not unit triangle (arena metric = unit-triangle min_area / hull_area). Does not reach n=11. n=9 is invariant under (x,y)\u2192(1-y,1-x) anti-diagonal reflection, confirming symmetric optimum hypothesis for small n in square \u2014 does NOT help Bill 4 since it confirms symmetry is optimal for n\u22649 (rather than refuting it for n=11).", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2512.14505", "title": "Solving the Heilbronn Triangle Problem using Global Optimization Methods", "authors": [ "Monji Mohammad", "Modir Mohammad", "Kocuk Burak" ], "date": "2025-12", "venue": "arXiv math.OC", "summary": "Global optimization formulation with bound tightening, symmetry-breaking inequalities, and boundary occupancy constraints. Certifies n=9 in ~10 minutes, addresses n\u22648 in seconds. No n=11 construction. Symmetry breaking is enforced (not relaxed).", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.87, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.0548767, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Unit square; n\u22649 only. Treats symmetry as solver tool (assumed optimum), not as variational signal to falsify. Provides no n=11 frontier movement.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2305.18253", "title": "A new upper bound for the Heilbronn triangle problem", "authors": [ "Cohen Alex", "Pohoata Cosmin", "Zakharov Dmitrii" ], "date": "2023-05", "venue": "arXiv math.CO", "summary": "Establishes upper bound n^{-8/7-1/2000} for the Heilbronn triangle area in the unit square for large n, first asymptotic improvement over Koml\u00f3s-Pintz-Szemer\u00e9di 1982. Connects Heilbronn to projection theory and discretized sum-product. Strictly asymptotic \u2014 vacuous for n=11.", "candidate_bill": null, "candidate_meta_cost": "M2", "verdict": "out_of_scope", "confidence": 0.93, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "above_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Asymptotic upper-bound theorem; constant is too loose at n=11 to bound any specific configuration. Does not engage construction-side bills. Important context but not a candidate for arena attack.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M2", "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.07658", "title": "Lower bounds for incidences", "authors": [ "Cohen Alex", "Pohoata Cosmin", "Zakharov Dmitrii" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "Follow-up to 2305.18253 attaining n^{-7/6+o(1)} upper bound for Heilbronn via incidence-geometry lower bounds (high-low limit). Pure asymptotic upper-bound result, no constructive content.", "candidate_bill": null, "candidate_meta_cost": "M2", "verdict": "out_of_scope", "confidence": 0.94, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "above_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Same group, sharper exponent. Bounds entire family at large n; uninformative for n=11. Same exclusion as 2305.18253.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M2", "classifier_version": "1.0" }, { "paper_id": "arxiv:1703.03297", "title": "On Heilbronn's Triangle-Problem (a lower bound)", "authors": [ "Ellmann Gabor" ], "date": "2017-03", "venue": "arXiv math.CO", "summary": "Construction of n points on unit circle giving every triangle area \u2265 \u03a9(n^{-3/2} (log n)^{-7/2}), polynomial improvement on classical lower bound. Construction uses prime-indexed regular polygons on concentric circles \u2014 highly symmetric (rotationally invariant by construction). v12 revision November 2025.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Asymptotic lower-bound construction on the unit *circle*, not in-triangle. Symmetric construction. The construction's penalty factor makes it sub-leader at n=11. Confirms direction: improvements at small n come from numerical search, not analytic constructions.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.26644", "title": "Heilbronn's triangle problem in three dimensions", "authors": [ "Maldague Dominique", "Wang Hong", "Zakharov Dmitrii" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "First non-trivial upper bound for the 3D Heilbronn problem: among n points in the unit cube one can find a triangle of area \u2264 n^{-2/3-c}. Pure upper-bound result on a different domain.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.96, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "3D variant \u2014 different problem, not applicable to arena n=11 in-triangle metric.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.12945", "title": "Improved upper bounds for the Heilbronn's Problem for k-gons", "authors": [ "Gajjala Rishikesh" ], "date": "2024-05", "venue": "arXiv math.CO", "summary": "\u0394_k(n) \u2264 (k-2)/n + o(1/n) for convex k-gons (in the small-min-area = Heilbronn's-Q sense). First 50-year asymptotic improvement on the trivial 3/n bound for the dual problem.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Different variant (min over k-gons rather than triangles). Asymptotic. Doesn't engage Bill 4.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "URL:https://erich-friedman.github.io/packing/heiltri/", "title": "The Heilbronn Problem for Triangles (Erich Friedman's Packing Center)", "authors": [ "Friedman Erich" ], "date": "2007-06", "venue": "Erich's Packing Center", "summary": "Curated repository of best-known Heilbronn-in-unit-triangle configurations. For n=11: \u0394 \u2248 0.0360+, Cantrell 2006, HORIZONTALLY SYMMETRIC (Z/2 reflection across vertical axis). For n=8 (Cantrell 2006), n=10 (Cantrell 2007), n=13 (Karpov 2015), n=14 (Cantrell 2007): NOT symmetric. So asymmetric configurations are *known and accepted* for adjacent n, but n=11 has only had symmetric improvements.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "known_bill", "confidence": 0.95, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.036, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "CRITICAL Bill 4 evidence. n=11 prior leader (Cantrell 2006) is horizontally symmetric \u2014 same Z/2 reflection class as AlphaEvolve 2025's 0.036530. Asymmetric configurations exist in literature for n=8, 10, 13, 14 in unit-triangle but NOT n=11. Asymmetry has not been published for n=11 in the canonical curated source. Confirms Bill 4 empty-space prediction at the construction-curation layer.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "URL:https://erich-friedman.github.io/packing/heilconvex/", "title": "The Heilbronn Problem for Convex Regions (Erich Friedman's Packing Center)", "authors": [ "Friedman Erich", "Cantrell David" ], "date": "2007-06", "venue": "Erich's Packing Center", "summary": "Best-known configs in convex region of unit area. n=11: \u0394=2/47\u22480.0425, n=12: \u0394=2/51\u22480.0392, n=14: \u0394\u22480.0277. Most due to Cantrell 2007. Atlas anchor 'heilbronn_convex_14_arena_frontier' references this variant.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.0425, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Different domain (convex region) than arena's unit-triangle. Convex-region values are systematically larger because the constraint is weaker. Atlas anchor `heilbronn_convex_14_arena_frontier` lives here but is M6-out-of-scope for arena verifier.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "URL:https://erich-friedman.github.io/packing/heilbronn/", "title": "The Heilbronn Problem for Squares (Erich Friedman's Packing Center)", "authors": [ "Friedman Erich" ], "date": "2007-06", "venue": "Erich's Packing Center", "summary": "Curated repository of best-known Heilbronn-in-unit-square configurations. n=11: \u0394=1/27\u22480.0370 (Goldberg 1972), horizontally symmetric. n=12: \u0394\u22480.0325 (Comellas-Yebra 2001), completely symmetric. n=13: Karpov 2011, not symmetric. n=14: Beyleveld 2006, rectangle symmetries.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.93, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.037, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Unit square, not unit triangle (arena metric). Goldberg 1972 n=11 in-square is also horizontally symmetric \u2014 independent corroboration that the n=11 Heilbronn problem (both square AND triangle variants) has only symmetric published constructions. Bill 4 is dual-empty across both Heilbronn variants for n=11.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "doi:10.37236/1623", "title": "New Lower Bounds for Heilbronn Numbers", "authors": [ "Comellas Francesc", "Yebra J. Luis A." ], "date": "2002-01", "venue": "Electronic J. Combinatorics", "summary": "Simulated annealing + analytical refinement to obtain best-known unit-square configurations for n=7..16. The n=11 configuration is attributed to Goldberg 1972 (this paper covers n=10, 12, 13, 14, 15, 16). Symmetric configurations.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.88, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.037, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Unit square. Coordinates archived in spiralulam/heilbronn config_n11.json (\u0394 \u2248 1/27, horizontally symmetric). Establishes the symmetric-optimum tradition that Bill 4 challenges.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "doi:10.1007/s10898-016-0453-1", "title": "Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing", "authors": [ "Chen Liangyu", "Xu Yong", "Zeng Zhenbing" ], "date": "2017-01", "venue": "J. Global Optimization", "summary": "GPGPU-accelerated branch-and-bound for n=9 Heilbronn-in-square, taking ~1 day. Predecessor to Sudermann-Merx 2026 MINLP framework. Confirms Comellas-Yebra n=9 as locally optimal.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "n=9 only. Unit square. Symmetric configuration confirmed.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "doi:10.1007/s11424-022-1220-7", "title": "Heilbronn's Problem of Eight Points in the Square", "authors": [ "Dehbi Lyes", "Zeng Zhenbing" ], "date": "2022-01", "venue": "J. Systems Sci. Complexity", "summary": "Certified exact optimum for n=8 Heilbronn-in-square: h(8) = 0.0723764... Local search + algebraic verification.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.92, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.0723764, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "n=8 only; unit square. The n=8 configuration is one of the few Cantrell unit-triangle entries marked 'not symmetric' \u2014 but this paper is unit-square, so doesn't bear on Bill 4 unit-triangle empty-space.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "doi:10.1007/s11424-024-3366-y", "title": "Global Optimization for Heilbronn Problem of Convex Polygons Based on Bilinear Matrix Inequalities Solving", "authors": [ "Liu Jiawang", "Zeng Zhenbing" ], "date": "2024-01", "venue": "J. Systems Sci. Complexity", "summary": "Reformulates Heilbronn via BMI (bilinear matrix inequalities), applied to small-n configurations. Solver-style paper, no new lower-bound construction at the arena n=11 scale.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.78, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Methodology paper; addresses small n in polygons. Does not provide new n=11 in-triangle construction.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "doi:10.1007/978-3-030-41258-6_23", "title": "A Heilbronn Type Inequality for Plane Nonagons", "authors": [ "Yang Lu", "Zhang Jingzhong", "Zeng Zhenbing" ], "date": "2020-01", "venue": "Springer (book chapter)", "summary": "Heilbronn-type inequality for n=9 in convex nonagon. Theoretical, no construction beating arena n=11 in-triangle metric.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Plane nonagon variant, n=9 only. M6-different problem.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "URL:https://github.com/google-deepmind/alphaevolve_results", "title": "AlphaEvolve Results Repository (companion to arXiv:2506.13131)", "authors": [ "Google DeepMind" ], "date": "2025-06", "venue": "GitHub repository", "summary": "Companion Colab notebook to AlphaEvolve paper. Lists problems where AE outperforms SOTA, including Heilbronn-in-unit-triangle. Verification code present; whether explicit n=11 coordinates are inlined is verifier-checkable in the notebook (mathematical_results.ipynb). Repo description states only problems where AE outperformed are included.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "needs_gate", "confidence": 0.65, "arena_problems_relevant": [ "heilbronn-triangles", "kissing-number-d11" ], "headline_constant": 0.03653, "leader_relationship": "matches_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "GATE: Verify whether mathematical_results.ipynb explicitly publishes the 11-point coordinates in textual form (vs only the score). If coordinates are in the notebook \u2192 Bill 11 trigger, M5 retracted for Heilbronn. If only score is shown \u2192 M5 (unpublished construction beyond preprint claim). Two arena agents (CHRONOS, alpha_omega_agents) tied at 0.036530 suggests reproducibility \u2014 implying coordinates are extractable. Atlas finding confirms Z/2 reflection-symmetric.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": "Bill 11", "classifier_meta_cost": "M5", "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "URL:https://github.com/google-deepmind/alphaevolve_repository_of_problems", "title": "AlphaEvolve Repository of Problems", "authors": [ "Google DeepMind" ], "date": "2026-01", "venue": "GitHub repository", "summary": "67-problem catalog with evolution harness. Likely contains Heilbronn-in-triangle as a benchmark problem with evaluator. Each problem at problems/{n}.html.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.55, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Watchlist: poll this repo for any update including new n=11 best-or-different configurations. The repo enables third-party agents (incl. arena agents) to reproduce AE-style search.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "URL:https://github.com/spiralulam/heilbronn", "title": "Heilbronn certified-configuration repository (Sudermann-Merx)", "authors": [ "Sudermann-Merx Nathan" ], "date": "2026-03", "venue": "GitHub repository", "summary": "JSON configurations for n=3..16 in unit square, with certified status flags. n=11 stored config (Comellas-Yebra 2002) gives \u03b4 \u2248 1/27 = 0.0370 with 11 points symmetric under x\u21921-x reflection: e.g. (0.333,0)/(0.667,0), (0.167,1)/(0.833,1), (0.5,0.778).", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.92, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": 0.037, "leader_relationship": "different_metric", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Stored n=11 unit-square config is exactly Z/2 reflection-symmetric \u2014 independent confirmation that both Heilbronn variants (square AND triangle) have only symmetric published n=11 constructions. The 11 coordinates pair up under x\u21921-x except (0.5, 0.778) which is on the axis. Empty-space prediction for Bill 4 corroborated.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "URL:arena/n11_atlas_thought_041d125d", "title": "Asymmetric extremal principle for Heilbronn n=11 (atlas thought formalization)", "authors": [ "Cultural Soliton Observatory" ], "date": "2026-05", "venue": "atlas2.db internal", "summary": "Formalization of Bill 4: for T-involution-invariant variational problem (min triangle area, T = reflection across hull-fixed axis), inf on T-fixed subspace \u2265 inf on full space; strict iff extremizer is not T-invariant. Predicts: for n=11 Heilbronn-in-triangle, an asymmetric configuration should beat the \u03c3-equivariant arena leader 0.036530.", "candidate_bill": "Bill 4", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.9, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "triggered", "signature_construction": true, "notes": "Bill 4 SIGNATURE \u2014 the variational principle is formalized internally but not yet realized as a published construction. Atlas had tested 48-candidate \u03c3-pair surgery and 559-seed asymmetric multistart, all bounded above leader. Next: 10^4+ asymmetric DGX multistart (Route 1). If any seed beats 0.036530 + 2 \u00d7 minImprovement, Bill 4 becomes 'filled' by an arena artifact and the empty-space prediction is partially falsified at the empirical level even though no peer-reviewed paper yet engages the principle.", "sweep_id": "sweep_005_heilbronn_constructions", "classifier_bill": "Bill 4", "classifier_meta_cost": null, "final_bill": "Bill 4", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "Novikov Alexander", "Vu Ngan", "Eisenberger Marvin", "Dupont Emilien", "Huang Po-Sen", "Wagner Adam Zsolt", "Shirobokov Sergey", "Kozlovskii Borislav", "Ruiz Francisco J. R.", "Mehrabian Abbas", "Kumar M. Pawan", "See Abigail", "Chaudhuri Swarat", "Holland George", "Davies Alex", "Nowozin Sebastian", "Kohli Pushmeet", "Balog Matej" ], "date": "2025-06", "venue": "arXiv cs.LG (whitepaper)", "summary": "Original AlphaEvolve whitepaper. Describes an LLM-driven evolutionary coding agent that evolves programs (not raw vectors) against domain-specific verifiers. Headline math claims: kissing K(11) >= 593, Heilbronn-tri n=11 >= 0.0365, autocorrelation constants pushed, plus matrix-multiplication 4x4 in 48 scalar mults. Constructions delivered as evolved Python programs; numerical artifacts and verifier exploits acknowledged by Tao follow-up. Bill 11 SEED paper.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "needs_gate", "confidence": 0.95, "arena_problems_relevant": [ "kissing-number-d11", "heilbronn-triangles", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality", "difference-bases", "thomson-problem", "tammes-problem", "circle-packing", "circles-rectangle", "min-distance-ratio-2d" ], "headline_constant": 593, "leader_relationship": "matches_leader", "watchlist_tier": "quarterly", "signature_construction": true, "notes": "WHITEPAPER ONLY in initial form (May 2025); arXiv v1 posted 2025-06 as a 'whitepaper PDF', not a structured paper. Davis 2025 critique: only 13 of 50 problems had improved-solution statements published; 37+ silent. Cohn kissing-numbers table cites it as the K(11)>=593 source but provides no independent rigor statement. Per Tao co-authored follow-up 2511.02864: 'blindly trusting AE values can be risky as they may be a consequence of verifier exploits rather than any true progress.' Aligns with atlas2 ARENA_K11_ALPHAEVOLVE_RIGOR \u2014 rigor is at the level of verifier-passing, not formal lattice-kissing proof.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.02864", "title": "Mathematical exploration and discovery at scale", "authors": [ "Georgiev Bogdan", "Gomez-Serrano Javier", "Tao Terence", "Wagner Adam Zsolt" ], "date": "2025-11", "venue": "arXiv math.HO", "summary": "The OFFICIAL math follow-up to AlphaEvolve (81pp, 35 figures, CC-BY 4.0, with Tao co-authored). Surveys 67 problems across analysis, combinatorics, geometry, number theory; rediscovers SOTA in most, improves several. Companion repo `alphaevolve_repository_of_problems` provides per-problem HTML pages and Colab notebooks (prompts, verifier code, initial+final evolved programs). Explicitly warns that verifier-design is load-bearing and AE's outputs can exploit float64 rounding.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M3", "verdict": "needs_gate", "confidence": 0.95, "arena_problems_relevant": [ "kissing-number-d11", "heilbronn-triangles", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality", "circle-packing", "circles-rectangle", "min-distance-ratio-2d", "thomson-problem" ], "headline_constant": 0.0365, "leader_relationship": "matches_leader", "watchlist_tier": "quarterly", "signature_construction": true, "notes": "Reports Heilbronn n=11 >= 0.0365 (matches arena 0.036530), first-AC bound 1.50992 -> 1.5032 (LOWER than arena leader's 1.502861), kissing K(11)=593 (BELOW arena's 604 since 2026-04). The repo publishes EVOLVED PROGRAMS \u2014 not the raw numerical artifacts (coordinates/vectors) \u2014 for many problems, which is precisely the reproducibility gap exploited by the arena attack: an evolved program may not reproduce the headline score on a fresh GPU/CPU due to nondeterminism in scipy.optimize. Critical Tao quote: 'a non-trivial amount of human effort needs to go into designing a non-exploitable verifier.' Wagner co-authored = same author as AE whitepaper. M3 flag: verifier-exploit warnings are explicit.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M3", "classifier_version": "1.0" }, { "paper_id": "URL:github.com/google-deepmind/alphaevolve_results", "title": "AlphaEvolve Results Repository (Google DeepMind official)", "authors": [ "Google DeepMind" ], "date": "2025-05", "venue": "GitHub (companion repo)", "summary": "Companion notebook `mathematical_results.ipynb` for the original AE whitepaper. Contains verification code only for problems where AE OUTPERFORMED state-of-the-art (matched results explicitly excluded 'to avoid clutter'). Does NOT contain the code to RUN AlphaEvolve. Per Davis 2025: provides input/output for ~13 problems with improved solutions, silent on the remaining 37+ in the 50-problem suite.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "needs_gate", "confidence": 0.92, "arena_problems_relevant": [ "kissing-number-d11", "heilbronn-triangles", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "thomson-problem", "difference-bases" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": true, "notes": "Repository publishes evolved Python functions and verification code; for K(11) specifically the notebook reportedly contains a configuration array, but THE PROGRAM that generated it is the published artifact, not a cleaned set of integer/rational coordinates. Arena verifier on KawaiiCorgi's score=0 path replicates the AE configuration only under specific float64 rounding (atlas2 finding). M3+M5 interaction: published-as-program-only + verifier-exploit-risk = sig_construction confirmed.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "URL:github.com/google-deepmind/alphaevolve_repository_of_problems", "title": "AlphaEvolve Repository of Problems (companion to 2511.02864)", "authors": [ "Georgiev Bogdan", "Gomez-Serrano Javier", "Tao Terence", "Wagner Adam Zsolt" ], "date": "2025-11", "venue": "GitHub (companion repo)", "summary": "67 HTML pages (problems/1.html .. 68.html) plus Colab notebooks for many problems. Each notebook contains the prompt, verification code, initial program, and final evolved program. Numbered indices; no semantic problem-name index in the top-level README, requires opening individual pages to find K(11), Heilbronn, autocorrelation, etc.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5", "verdict": "needs_gate", "confidence": 0.9, "arena_problems_relevant": [ "kissing-number-d11", "heilbronn-triangles", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality", "thomson-problem", "tammes-problem", "circle-packing", "circles-rectangle", "min-distance-ratio-2d", "difference-bases" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": true, "notes": "Numbered (1.html .. 68.html) without README cross-index = papers-cite-it-then-cant-find-it problem. Critical for arena attack: each .html page is the only reproducibility artifact for the corresponding arena problem. For arena Heilbronn n=11 = 0.036530, the AE construction is in one of these pages but the page-to-problem mapping is opaque.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "doi:10.1038/s41586-023-06924-6", "title": "Mathematical discoveries from program search with large language models (FunSearch)", "authors": [ "Romera-Paredes Bernardino", "Barekatain Mohammadamin", "Novikov Alexander", "Balog Matej", "Kumar M. Pawan", "Dupont Emilien", "Ruiz Francisco J. R.", "Ellenberg Jordan", "Wang Pengming", "Fawzi Omar", "Kohli Pushmeet", "Fawzi Alhussein" ], "date": "2023-12", "venue": "Nature (DeepMind)", "summary": "Direct ancestor of AlphaEvolve. Pairs a pretrained LLM with a systematic evaluator to evolve programs that GENERATE mathematical objects rather than direct solutions. Headline: cap set n=8 size 512 (vs prior best 496); largest cap-set asymptotic improvement in 20 years. Publishes BOTH the program and the numerical artifact (cap_set.ipynb has the 512-vector set explicitly).", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": 512, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "FunSearch lineage: Nature 2023 \u2192 AlphaEvolve whitepaper May 2025 (= arxiv:2506.13131) \u2192 2511.02864 Tao-coauthored math paper Nov 2025. PRECEDENT for arena attack: FunSearch DID publish the numerical artifact (cap_set vectors) alongside the program. AE 2025 retreated from this \u2014 publishing only programs + verifiers, not the numerical artifacts. This is the regression that creates the arena's M3/M5 gap.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.16750", "title": "An improved example for an autoconvolution inequality", "authors": [ "Boyer Christopher", "Li Zane Kun" ], "date": "2025-08", "venue": "arXiv math.CA", "summary": "Arm's-length improvement over AE's second-autocorrelation result. Boyer-Li construct a 575-step nonnegative function via simulated annealing + gradient-based refinement, achieving c >= 0.901564 (vs AE's 0.8962). Exact coefficients in Appendix A, integer arithmetic verification. Cites AE only as blog post [1], not arXiv ID.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "second-autocorrelation-inequality" ], "headline_constant": 0.901564, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Arm's-length verification of AE's 0.8962 claim (not 'verification of construction' but independent improvement). Construction is COMPLETELY published (575-step function coefficients in Appendix A). Bill 3 (Matolcsi-Vinuesa autocorrelation). Arena leader on second-AC inequality is ClaudeExplorer 0.962643; Boyer-Li's 0.9016 is below leader. Demonstrates that the autocorrelation extremal problem is engaged by arm's-length human work, not just AE.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2508.02803", "title": "Further Improvements to the Lower Bound for an Autoconvolution Inequality", "authors": [ "Jaech Aaron", "Joseph Alan" ], "date": "2025-08", "venue": "arXiv math.CA", "summary": "Further independent improvement on the autoconvolution lower bound, post-AE. Includes appendices with source-code listings, GitHub repo (ajaech/autocorrelation_inequality), and citation to Novikov et al. 2025 (AlphaEvolve). Construction is fully published with reproducible numerical search algorithm.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.88, "arena_problems_relevant": [ "second-autocorrelation-inequality" ], "headline_constant": null, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cites cite.novikov2025alphaevolve (AE), uses gradient-based methods (not LLM-driven). Code published. Bill 3 (Matolcsi-Vinuesa). Independent of AE methodology, confirms AE constructions improvable by classical numerical optimization. Demonstrates the autoconvolution constant is not yet a 'tight' frontier.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.14150", "title": "CodeEvolve: An open source evolutionary coding agent for algorithm discovery and optimization", "authors": [ "Assumpcao Henrique", "Ferreira Diego", "Campos Leandro", "Murai Fabricio" ], "date": "2025-10", "venue": "arXiv cs.LG", "summary": "Open-source evolutionary coding agent inspired by AE; reproducibility-focused. Achieves 0.93768 on second autocorrelation inequality (vs AE 0.89627). Also tackles min-distance-ratio and circle-packing variants matching/approaching AE bounds. Authors explicitly call out AE as 'closed-source whitepaper that provides few technical implementation details.'", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.88, "arena_problems_relevant": [ "second-autocorrelation-inequality", "circle-packing", "circles-rectangle", "min-distance-ratio-2d" ], "headline_constant": 0.93768, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Open-source CLONE-AND-IMPROVE of AE methodology. Same lineage = Bill 11. Confirms AE bounds are surmountable by smaller open systems on autocorrelation. Authors flag the closed-source nature of AE itself as a reproducibility issue. Arena-relevant: CodeEvolve's autocorrelation 0.93768 < ClaudeExplorer arena 0.962643.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.23473", "title": "ThetaEvolve: Test-time Learning on Open Problems", "authors": [ "Wang Y." ], "date": "2025-11", "venue": "arXiv cs.LG", "summary": "Open-source test-time RL extension of AE/OpenEvolve. Achieves circle-packing n=26 sum-of-radii 2.63598308 (vs AE 2.63586276) with an 8B parameter model. First-autocorrelation inequality 1.503133. Simplifies AE (single LLM, no ensemble), scales program database 70->10000, batch sampling 32x16, lazy-penalty mechanism.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.87, "arena_problems_relevant": [ "circle-packing", "first-autocorrelation-inequality", "second-autocorrelation-inequality", "third-autocorrelation-inequality" ], "headline_constant": 2.63598308, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Arena JSAgent circle-packing leader is 2.635983, which MATCHES ThetaEvolve's 2.63598308 \u2014 same float64 basin. Arena's first-AC leader 1.502861 < ThetaEvolve's 1.503133. Engineering, not theoretical-bound, work. Bill 11 lineage. Compute-scaling argument: 8B + test-time RL ~= ensemble-of-frontier-models for code evolution.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.18057", "title": "Reinforced Generation of Combinatorial Structures: Hardness of Approximation", "authors": [ "Nagda Ansh", "Raghavan Prabhakar", "Thakurta Abhradeep" ], "date": "2025-09", "venue": "arXiv cs.CC", "summary": "AE applied to complexity theory: constructs nearly-extremal Ramanujan graphs (up to 163 vertices) for MAX-CUT / MAX-IS certification, gadget reductions for MAX-3-CUT (0.9649) / MAX-4-CUT (0.987) hardness, metric TSP hardness 111/110 (vs prior 117/116). Uses AE itself to evolve faster verifiers (~10000x speedup).", "candidate_bill": "Bill 11", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": 1.009091, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Different problem class (complexity theory, not extremal geometry/combinatorics in arena's 15-problem space). Listed because the recursive-verifier-evolution trick (AE evolving its own verifier 10000x) is a Bill 11 reproducibility methodology \u2014 relevant if arena verifiers themselves become a research target. M6 flag: out of arena scope but documents AE's verifier-design pattern.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.13391", "title": "Finding Kissing Numbers with Game-theoretic Reinforcement Learning (PackingStar)", "authors": [ "Ma Chengdong", "et al." ], "date": "2025-11", "venue": "arXiv cs.LG", "summary": "Reformulates kissing-number problem as two-player matrix-completion game on Gram matrices. Improves SOTA in dimensions 25-31 (8 to 5204 additional spheres). Dim-13 new rational config 1146 spheres (vs prior 1130, 1971). Critically: discusses AE K(11)=593 explicitly but does NOT independently verify it.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.86, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 1146, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "ARM'S-LENGTH PARTIAL evaluation of AE K(11). Quote: AE 'optimizes explicit coordinates in R^n (numerically unstable)', 'lacks clear geometric structure', 'does not generalize'. PackingStar operates in Gram-matrix domain (geometrically constrained). Does not independently produce K(11)>=593; characterizes the difference as METHODOLOGICAL rather than verification. Bill 1 Cohn-Elkies engaged.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "doi:10.1016/j.laa.2025.xxxx", "title": "Improvements on kissing numbers in dimensions 10, 11, 14 (Ganzhinov)", "authors": [ "Ganzhinov Mikhail" ], "date": "2025-10", "venue": "Linear Algebra and its Applications", "summary": "Aalto PhD thesis result: K(10) >= 510, K(11) >= 592, K(14) >= 1932 via high-symmetry arrangements. K(11) 592 is ONE BELOW AE's 593. Construction is symmetry-reduced and fully published (Linear Algebra and its Applications). Tightens K(10) and K(14) above any AE-reported bound.", "candidate_bill": "Bill 1", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "kissing-number-d11" ], "headline_constant": 592, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "FULLY PUBLISHED human/symmetry-based construction at 592 in d=11. AE's claim of 593 is exactly 1 above Ganzhinov's verified construction = the kind of differential that under arena's float64 verifier is suspicious (atlas2 K(11) basin 610 at 594). Publication peer-reviewed in LAA. Bill 1 (Cohn-Elkies LP / Delsarte adjacent). Arena now claims K(11) >= 604 \u2014 both AE and Ganzhinov are now below arena leader.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 1", "classifier_meta_cost": null, "final_bill": "Bill 1", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2603.11107", "title": "From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization", "authors": [ "Sudermann-Merx Nathan" ], "date": "2026-03", "venue": "arXiv math.OC", "summary": "Independent mixed-integer optimization framework producing EXACT (symbolic) coordinates for Heilbronn's triangle on unit square, n=5..9. Confirms Comellas-Yebra 2002 for n=9. Does NOT address n=11. Symmetry-breaking via reflection-anti-diagonal invariance. All configurations + code public.", "candidate_bill": "Bill 4", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "heilbronn-triangles" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "DOES NOT REACH n=11 (the arena target). Bill 4 (asymmetric extremal principle) relevant: the n=9 optimum is invariant under anti-diagonal reflection \u2014 but the framework doesn't apply to n=11. Arena leader 0.036530 (AE) is in the EMPTY ZONE Bill 4 predicts: no published n=11 asymmetric Heilbronn config beats AE in 2024-2026 corpus. EMPTY-SPACE HYPOTHESIS BILL 4 REMAINS HOLDING.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 4", "classifier_meta_cost": null, "final_bill": "Bill 4", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2505.16105", "title": "Sums and differences of sets (improvement over AlphaEvolve)", "authors": [ "Gerbicz Robert" ], "date": "2025-05", "venue": "arXiv math.NT", "summary": "Direct improvement over AE's sum/difference-set bound: theta = 1.173050 (vs AE theta = 1.1584 with 54265 integers). Construction explicit (>10^43546 elements), GMP arithmetic. 4-page note. Demonstrates AE's number-theoretic bounds are not tight; arm's-length human bound directly beats AE.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": 1.17305, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sum/difference-set is a sibling-but-different problem to arena's difference-bases. Listed because: (a) Gerbicz published the FULL CONSTRUCTION (vectors in GMP), (b) directly beats AE bound, (c) demonstrates AE's number-theoretic numerical work is NOT tight to arena-quality construction. Suggests arena difference-bases (leader AE 2.639027) is similarly bounded-not-tight. Bill 6 (Singer + heights) likely engaged but not explicitly cited. Cf. atlas2 thought 9fa54075.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.06056", "title": "Scientific Algorithm Discovery by Augmenting AlphaEvolve with Deep Research (DeepEvolve)", "authors": [ "Liu Gang", "Zhu Yihan", "Chen Jie" ], "date": "2025-10", "venue": "arXiv cs.AI", "summary": "DeepEvolve adds web-retrieval + cross-file editing + automatic debugging to AE. Includes circle packing 666% improvement (over what baseline unclear). Focuses on chemistry/biology/materials, not arena's extremal geometry/combinatorics.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.7, "arena_problems_relevant": [ "circle-packing" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Circle packing claim 666% is likely a different metric (sum-of-radii vs density vs num-circles), not directly comparable to arena's JSAgent 2.635983. Bill 11 lineage. M6 because dominant focus is chemistry/biology benchmarks.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.17592", "title": "GigaEvo: An Open Source Optimization Framework Powered by LLMs and Evolution Algorithms", "authors": [ "Khrulkov Valentin", "Galichin Andrey", "Bashkirov", "et al." ], "date": "2025-11", "venue": "arXiv cs.LG", "summary": "AIRI/Sber open-source AE-style framework. Heilbronn triangle area 0.0364 (matching AE 0.0365); circle packing n=26 sum-of-radii 2.636 (vs AE 2.635); kissing recovers established lower bounds; bin-packing new SOTA. Hydra config, Redis storage, MAP-Elites.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "heilbronn-triangles", "circle-packing", "kissing-number-d11" ], "headline_constant": 0.0364, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Heilbronn 0.0364 < arena 0.036530 (AE) \u2014 independent open-source AE clone falls SHORT of arena leader by margin > minImprovement. Circle packing 2.636 ABOVE arena JSAgent 2.635983 by ~0.0000017 = within float64 noise. Bill 11. Confirms there is no arm's-length-reproduced numerical artifact above AE's Heilbronn 0.0365 (= empty-space Bill 4 still holds).", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "URL:github.com/algorithmicsuperintelligence/openevolve", "title": "OpenEvolve: Open Source Implementation of AlphaEvolve", "authors": [ "Sharma Asankhaya", "et al." ], "date": "2025-06", "venue": "GitHub", "summary": "Open-source implementation explicitly claiming 'perfect 2.635 match on circle packing' replicating AE's circle-packing 26-disk sum-of-radii. Mathematical optimization via scipy.optimize as breakthrough. Includes best_program.py for circle packing.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.78, "arena_problems_relevant": [ "circle-packing", "circles-rectangle" ], "headline_constant": 2.635, "leader_relationship": "matches_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Demonstrates that AE's circle-packing 2.635 IS reproducible from a clean-room open-source AE clone, but only because the result is a scipy.optimize basin attractor \u2014 NOT because the AE-claimed numerical artifact was published. Reproducibility-via-rerun, not reproducibility-via-publication. Bill 11. Arena JSAgent 2.635983 matches.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "URL:cs.nyu.edu/~davise/papers/AlphaEvolveNotes.pdf", "title": "Some comments on AlphaEvolve (Davis)", "authors": [ "Davis Ernest" ], "date": "2025-05", "venue": "NYU Courant (preprint)", "summary": "Critical analysis by computer scientist Ernest Davis. Notes: only 13 of ~50 problems have published 'improved solution' statements (37+ silent); 'highly user-configurable systems are a mixed blessing'; 7 of 13 are elementary geometry, 5 real analysis, 1 combinatorics; 'grounds for doubt about their significance' despite distinguished suggesters.", "candidate_bill": null, "candidate_meta_cost": "M5", "verdict": "needs_gate_declaration", "confidence": 0.78, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "triggered", "signature_construction": false, "notes": "External critique establishing the publication-completeness gap. ~13/50 published = 26% publication rate of AE's claimed improvements; the remaining 37 are statistically asymmetric. Marcus shared on Substack as 'wildly oversold.' Needs_gate_declaration: the framework should consider a Bill for 'differential publication asymmetry' \u2014 claiming N improvements but publishing details on only k 1.952659676624688 ONLY AFTER fixing verifier exploits. Implies any AE-numerical-floor on a float64-scored arena problem is suspect. Triggers gate-declaration for M3 expansion: 'verifier-exploitation pattern' should be formalized as a sub-meta-cost.", "sweep_id": "sweep_006_alphaevolve_deepmind_extremal", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M3", "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.24034", "title": "Generalized additive bases and difference bases for Cartesian product of finite abelian groups", "authors": [ "Shuxing Li", "Chi Hoi Yip" ], "date": "2025-09", "venue": "arXiv math.CO / math.NT", "summary": "Constructs g-additive and g-difference bases inside Cartesian products G^n of a finite abelian group G using Galois ring GR(p^s,n) Teichmuller systems and relative-difference-set constructions. Main theorem (Thm 1.5): limsup_{n} eta_g(G^n)/(sqrt(g)|G|^{n/2}) = 1 + O(1/sqrt(g)), asymptotically sharp upper bound INSIDE the product group. The constructions remain strictly in G^n; no CRT-style cyclic embedding into Z/NZ with score preservation is produced.", "candidate_bill": "Bill 7", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.92, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": 1.0, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": true, "notes": "\u2605 THE SIGNATURE PAPER. Asper-cited in reply #917 thread 213. Confirmed empty-space bridge: theorems are inside G^n, never lifted to Z/(|G|^n)Z. 'Embedding' only appears for Galois-ring internal structure. No Chinese Remainder Theorem, no Singer comparison, no |B|^2/(V-1) ~ 2.639 evaluation. Asper's flag is correct: the missing bridge is an explicit cyclic embedding with score preservation, and Li-Yip do not supply it.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "Bill 7", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.11736", "title": "Cardinalities of g-difference sets", "authors": [ "Eric Schmutz", "Michael Tait" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Proves the limit lim_{n} eta_g(n)/sqrt(n) exists as a finite non-zero constant, answering Kravitz's question for g-difference bases on intervals [n]. Also alpha_g(n) = sqrt(gn)(1+o_g(1)) for relaxed g-Sidon counts. Existence/asymptotic result, no explicit construction in Z/NZ.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Companion to Li-Yip in the same research thread. Limit exists but its explicit value is not computed; cannot be compared to 2.639 leader score directly. No cyclic embedding from G^n.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.14924", "title": "Cyclic relative difference sets and circulant weighing matrices", "authors": [ "Daniel M. Gordon" ], "date": "2025-01", "venue": "arXiv math.CO", "summary": "Computationally extends Lam (1977) and Pott (1995) searches for relative difference sets that are liftings of classical Singer-parameter difference sets, applying findings to circulant weighing matrices. The objects of study are already cyclic; the paper investigates which Singer DS admit further relative-DS lifts, not a non-cyclic to cyclic CRT bridge for difference bases.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Closest in spirit to Bill 7 but the 'lifting' here is from Singer cyclic to a cyclic relative DS, not from a non-cyclic finite abelian group to cyclic. No score evaluation in arena units.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 6", "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.20446", "title": "Perfect difference families, perfect systems of difference sets and their applications", "authors": [ "Hengrui Liu", "Tao Feng", "Xiaomiao Wang", "Menglong Zhang" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Resolves Bermond's nearly 50-year-old conjecture: (v,4,lambda)-PDF exists iff lambda(v-1) \u2261 0 (mod 12), v >= 13, and (v,lambda) not in {(25,1),(37,1)}. Introduces 'layered difference families' to unify existence proofs. Block size 4 only; produces cyclic difference families but on small block sizes far from arena difference-bases regime.", "candidate_bill": "Bill 6", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Major existence result for fixed block size 4; the arena difference-bases metric |B|^2/(V-1) with V >= 8011 needs k = O(sqrt(V)) ~ 90+, so block-size-4 PDFs do not engage.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.19804", "title": "Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof", "authors": [ "B. Alexeev", "D. G. Mixon" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Disproves Erd\u0151s's $1000 Sidon-extension conjecture: exhibits size-5 Sidon set {1,2,4,8,13} not extending to any finite perfect difference set, with formally verified proof. Shows forbidden Sidon subsets exist in cyclic-projective-plane PDS. Negative-direction result for cyclic PDS embedding.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Strengthens the empty-space hypothesis: not every Sidon configuration lifts to a cyclic PDS even abstractly. Suggests the CRT-cyclic-embedding bridge has structural obstructions in some cases.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.25214", "title": "Empirical evidence for size-4 counterexamples to Erd\u0151s's Sidon-extension conjecture", "authors": [ "B. Alexeev", "D. G. Mixon" ], "date": "2026-04", "venue": "arXiv math.CO", "summary": "Follow-up to 2510.19804: provides empirical/computational evidence that size-4 Sidon sets can also fail to extend to perfect difference sets, modulo Desargues conjecture for cyclic projective planes. Negative result for cyclic embedding of small Sidon sets.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.7, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Same authors as 2510.19804. Reinforces the empty-space picture but conditional on a deep conjecture.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.12911", "title": "On large Sidon sets", "authors": [ "Ingo Czerwinski", "Alexander Pott" ], "date": "2024-11", "venue": "arXiv math.CO / cs.IT", "summary": "Constructs large Sidon sets in F_2^t using graphs of APN functions with high linearity. Achieves |B| = 192 in F_2^15, improving prior maximum of 152. Provides binary linear codes with new parameters. Construction is in non-cyclic F_2^t, no cyclic embedding into Z/NZ is shown.", "candidate_bill": "Bill 7", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.88, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": 192.0, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Non-cyclic Sidon construction in elementary abelian 2-group. Exactly the input where a CRT-cyclic embedding would apply IF Li-Yip's bridge existed \u2014 but it doesn't. So this paper sits stranded on the non-cyclic side of the gap.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "Bill 7", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2308.12406", "title": "Constructing thick B_h-sets", "authors": [ "Kevin O'Bryant" ], "date": "2024-01", "venue": "arXiv math.NT", "summary": "New parameterized constructions of B_h-sets for h >= 3 that subsume Bose-Chowla and Singer. Provides diameter bounds for k-element B_h-sets in Z. Working in integer interval [N], not in cyclic Z/NZ; relates to Sidon (h=2) constructions but doesn't address arena |B|^2/(V-1) score.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.78, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Extends Singer/Bose-Chowla for higher-order B_h, but stays inside the cyclic Singer family on the cyclic side and inside Z on the integer-ruler side. No non-cyclic to cyclic bridge.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2408.16721", "title": "Modular Golomb rulers and almost difference sets", "authors": [ "Daniel M. Gordon" ], "date": "2024-08", "venue": "arXiv math.CO", "summary": "Existence questions for modular Golomb rulers via almost difference sets (lambda and lambda+1 difference multiplicities). Extends constructions by adding/removing an element from a known difference set; octic-residue construction. Operates in cyclic Z/vZ already; does not embed from non-cyclic.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Modular Golomb rulers ARE the arena objective. But this paper's constructive moves (add/remove element, residue-class build) stay within cyclic-residue framework \u2014 not the CRT bridge Asper asks for.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.14409", "title": "Some constructive results on Disjoint Golomb Rulers", "authors": [ "Xiaodong Xu", "Baoxin Xiu", "Changjun Fan", "Meilian Liang" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "Improves and generalizes Kl\u00f8ve inequalities for disjoint Golomb rulers (collections of GRs with pairwise-disjoint mark sets). Proposes structural conjectures and a partition-based generalization of Koml\u00f3s-Sulyok-Szemer\u00e9di. Multi-ruler setting; does not engage |B|^2/(V-1) scoring or CRT cyclic embedding.", "candidate_bill": "Bill 6", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Disjoint-rulers variant, not single-ruler optimal score problem.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2402.09999", "title": "Davenport constant for finite abelian groups with higher rank", "authors": [ "[authors not extracted]" ], "date": "2024-02", "venue": "arXiv math.CO", "summary": "Studies Davenport constant D(G) for G = (C_p)^{d-1} x C_{pq} with prime p and integer q, deriving sufficient conditions for Davenport-style conjectures. Pure additive-combinatorics in non-cyclic product groups; no relation to difference-basis score.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Davenport constant is generalized-additive but its objective (longest zero-sum-free sequence) is disjoint from |B|^2/(V-1).", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.01148", "title": "An algebraic approach towards a conjecture on the Davenport constant", "authors": [ "[authors not extracted]" ], "date": "2024-07", "venue": "arXiv math.CO", "summary": "Algebraic approach (Loewy length of F_p[G]) to bound D(G) for finite p-groups. Computes D(G) for metacyclic and certain non-abelian p-groups. Not relevant to difference-basis embedding.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Davenport-conjecture algebra, not difference-basis construction.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2503.21231", "title": "On the inverse problem of the k-th Davenport constants for groups of rank 2", "authors": [ "[authors not extracted]" ], "date": "2025-03", "venue": "arXiv math.CO", "summary": "For rank-2 abelian G = C_{n_1} (+) C_{n_2}, D_k(G) = n_1 + k*n_2 - 1. Inverse problem characterizing extremal sequences. Pure Davenport theory; no difference-basis embedding.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Davenport rank-2 result; not arena-relevant.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2510.20412", "title": "The Davenport constant of balls and boxes", "authors": [ "[authors not extracted]" ], "date": "2025-10", "venue": "arXiv math.CO", "summary": "Davenport constant for set X (e.g., discrete Euclidean balls in Z^2, Z^3). Largest n such that there exists a sequence of elements summing to zero with no proper zero-sum subsequences. Extremal-additive but disjoint from difference-basis scoring.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Geometric Davenport variant; not arena-relevant.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.02731", "title": "On circular external difference families", "authors": [ "A. Burgess", "F. Merola", "T. Traetta" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Constructs cyclic (v,m,ell,1)-CEDFs in Z_v, resolving cases m odd and ell even via arithmetic-progression and tiling techniques. Characterizes existence for (v,m,2,1) and (3ell^2+1,3,ell,1) families. Cyclic constructions for external-difference applications (anonymous AMD codes), not difference-basis scoring.", "candidate_bill": "Bill 6", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.82, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Explicit cyclic constructions exist here, but the objective (external-difference balance for cryptography) is orthogonal to |B|^2/(V-1) leader score. Does not provide a non-cyclic to cyclic CRT bridge.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2605.03274", "title": "Formalizing the Prime-Field Singer Construction and Sidon Set Infrastructure in Lean 4", "authors": [ "[authors not extracted]" ], "date": "2026-05", "venue": "arXiv math.CO", "summary": "Lean 4 formalization of Singer's cyclic Sidon set construction modulo q^2+q+1 with cardinality q+1, plus reusable infrastructure for additive combinatorics. Formal verification of the classical cyclic Singer construction (Bill 6), no new bound, no CRT bridge from non-cyclic.", "candidate_bill": "Bill 6", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.92, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": 2.6667, "leader_relationship": "matches_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Lean formalization confirms the Singer asymptote 8/3 is exactly what the cyclic side provides. Reinforces Bill 6 as the published frontier; reinforces Bill 7's empty status (no CRT lift improves on it).", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 6_vs_Bill 7", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.10440", "title": "Deza Cayley graphs from difference sets", "authors": [ "[authors not extracted]" ], "date": "2025-06", "venue": "arXiv math.CO", "summary": "Constructs Deza graphs (specific 2-eigenvalue-multiplicity Cayley graphs) from difference sets and relative difference sets over both cyclic and dihedral groups. Spectral application of existing DS; doesn't improve the asymptotic cyclic Singer bound.", "candidate_bill": "Bill 6", "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.78, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Cayley-graph spectral application of DS, not new DS construction; no CRT bridge.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 6", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.08763", "title": "Fourier analysis on distance-regular Cayley graphs over abelian groups", "authors": [ "[authors not extracted]" ], "date": "2024-07", "venue": "arXiv math.CO", "summary": "Uses Fourier analysis on abelian groups to connect distance-regular Cayley graphs to finite-geometry combinatorial objects. Spectral characterization, not difference-basis construction.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.82, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Distance-regular Cayley-graph characterization, not constructive for difference bases.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2406.09210", "title": "Davenport constant and its variants for some non-abelian groups", "authors": [ "[authors not extracted]" ], "date": "2024-06", "venue": "arXiv math.GR", "summary": "Davenport constant variants for non-abelian groups (semidirect products, metacyclic). Out of scope for difference-basis arena.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "triggered", "signature_construction": false, "notes": "Non-abelian Davenport variants; far from arena objective.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.23282", "title": "Largest Sidon subsets in weak Sidon sets", "authors": [ "[authors not extracted]" ], "date": "2026-02", "venue": "arXiv math.CO", "summary": "Studies how large a Sidon subset is forced inside a 'weak Sidon' parent set. Structural result for Sidon subsets, not constructive for the cyclic-embedding bridge.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.78, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sidon-substructure paper; orthogonal to CRT embedding.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.04809", "title": "Ramsey-type problems for generalised Sidon sets", "authors": [ "[authors not extracted]" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "Ramsey-type partition results for g-Sidon-like sets. Combinatorial-Ramsey direction, no constructive cyclic embedding.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.78, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Ramsey/partition direction, not constructive for arena scoring.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "Bill 7", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.25214", "title": "[duplicate already listed - removed]", "authors": [], "date": "2026-04", "venue": "arXiv math.CO", "summary": "DUPLICATE - see entry above.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.0, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Marker for duplicate de-dup pass.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2310.06516", "title": "On the order sequence of a group", "authors": [ "[authors not extracted]" ], "date": "2024-03", "venue": "arXiv math.GR", "summary": "Order sequence of a finite group. Group-structure paper, not difference-set construction.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.92, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Group-order paper; not arena-relevant.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.08650", "title": "An explicit economical additive basis", "authors": [ "[authors not extracted]" ], "date": "2024-05", "venue": "arXiv math.NT", "summary": "Explicit subset A of N with A+A = N and number of representations growing slower than any polynomial. Answers a question of Erd\u0151s for the infinite-integer additive-basis setting.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Infinite-integer additive basis (sum-cover), not difference basis on finite cyclic group.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": "M6", "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.01986", "title": "The m-th element of a Sidon set", "authors": [ "[authors not extracted]" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "Distributional properties of the m-th element of a Sidon set in [N]. Analytical-combinatorics direction, no cyclic embedding construction.", "candidate_bill": null, "candidate_meta_cost": "M6", "verdict": "out_of_scope", "confidence": 0.8, "arena_problems_relevant": [ "difference-bases" ], "headline_constant": null, "leader_relationship": "different_metric", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sidon-element-position result; not bridge-constructive.", "sweep_id": "sweep_007_li_yip_crt_cyclic_embedding", "classifier_bill": "Bill 7", "classifier_meta_cost": null, "final_bill": "Bill 7", "final_meta_cost": "M6", "classifier_version": "1.0" }, { "id": "berthold_kamp_mexi_pokutta_polik_2026", "arxiv_id": "2601.05943", "url": "https://arxiv.org/abs/2601.05943", "title": "Global Optimization for Combinatorial Geometry Problems Revisited in the Era of LLMs", "authors": [ "Timo Berthold", "Dominik Kamp", "Gioni Mexi", "Sebastian Pokutta", "Imre Polik" ], "year": 2026, "venue": "arXiv preprint, submitted Jan 9, 2026, revised Jan 15, 2026", "topic": "Global optimization for circle packing, hexagon packing, and min-max distance ratio in 2D/3D using FICO Xpress + SCIP", "candidate_bill": "Friedman compendium analog (improves Bills 1/2 numerical baseline)", "candidate_meta_cost": null, "verdict": "known_bill", "leader_relationship": "matches_leader", "signature_construction": false, "concrete_values": { "n=16_2D_min_max_ratio_squared": 12.88924, "n=21_2D_min_max_ratio_squared": 17.77499, "n=22_2D_min_max_ratio_squared": 19.05398, "n=29_2D_min_max_ratio_squared": 25.9246, "n=14_3D_min_max_ratio_squared": 4.16578, "n=32_circle_unit_square_sum_radii": 2.93957, "n=26_circle_rectangle_sum_radii": 2.6393, "n=27_circle_rectangle_sum_radii": 2.69015, "n=11_hex_in_hex_side": 3.92485, "n=12_hex_in_hex_side": 3.94165, "n=14_hex_in_hex_side": 4.269, "n=15_hex_in_hex_side": 4.44769, "n=16_hex_in_hex_side": 4.52788 }, "previous_bests_cited": { "n=16_2D_min_max_ratio_squared_prev": 12.88927, "n=16_2D_min_max_ratio_squared_prev_source": "AlphaEvolve", "n=14_3D_min_max_ratio_squared_prev_source": "AlphaEvolve", "n=21_2D_prev_source": "Friedman", "n=22_2D_prev_source": "Friedman", "n=29_2D_prev_source": "Audet et al." }, "arena_impact": "Direct match: Together-AI min-distance-ratio-2d leader 12.889229907717521 corresponds to this paper's 12.88924. JSAgent circles-rectangle leader 2.365832 corresponds to Berthold et al. n=21 rectangle improvement (Friedman compendium credits this paper).", "method": "Nonlinear optimization (NLP) with two equivalent formulations (fix max dist=1 minimize min, or fix min=1 maximize max), exploit scale invariance, solve with FICO Xpress and SCIP.", "confidence": 0.98, "notes": "The signature paper for sweep 008. Establishes that arena Bills 1/2 leaders for min-distance-ratio-2d and circles-rectangle match published best-known values.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (improves Bills 1/2 numerical baseline)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "friedman_packing_maxmin_compendium", "url": "https://erich-friedman.github.io/packing/maxmin/", "title": "Erich Friedman packing compendium \u2014 Min/Max distance ratio entries", "authors": [ "Erich Friedman (compiler)", "David Cantrell (n=16 historic)", "Timo Berthold et al (n=16 current)" ], "year": "1995-2026 (continuously updated)", "venue": "erich-friedman.github.io/packing", "topic": "Min/Max distance ratio squared (d_max^2 / d_min^2) for n points in plane", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "matches_leader", "concrete_values_around_n16": { "n=14": "10.994+ (Cantrell)", "n=15": "12.038+ (Cantrell)", "n=16": "12.890+ (Cantrell 2009, improved Berthold et al Jan 2026)", "n=17": "14.090+ (Cantrell)", "n=18": "14.725+ (Cantrell)" }, "arena_impact": "n=16 entry at 12.890+ is the public reference for min-distance-ratio-2d. Berthold's improvement to 12.88924 has been recorded in the entry. Arena Together-AI 12.889229907717521 lies in this published 12.88924 attractor.", "confidence": 0.99, "notes": "PRIMARY EVIDENCE for min-distance-ratio-2d being at published optimum. The arena value, the Berthold paper value, and the Friedman entry all coincide to 5 decimal places.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "friedman_packing_cirRrec", "url": "https://erich-friedman.github.io/packing/cirRrec/", "title": "Friedman packing compendium - Circles in Rectangles (perimeter 4)", "authors": [ "Erich Friedman", "David Cantrell", "Timo Berthold et al" ], "year": "2011-2026", "venue": "erich-friedman.github.io", "topic": "Maximize sum of radii of n circles in rectangle of perimeter 4", "candidate_bill": "Friedman compendium analog (Bill 2 adapted)", "verdict": "known_bill", "leader_relationship": "matches_leader", "concrete_values": { "n=14": "1.916+ (Cantrell 2011)", "n=15": "1.985+ (Cantrell 2011)", "n=16": "2.059+ (Cantrell 2011)", "n=17": "2.117+ (Cantrell 2011)", "n=18": "2.180+ (Cantrell 2011)", "n=19": "2.241+ (Cantrell 2011)", "n=20": "2.305+ (Cantrell 2011)", "n=21": "2.365+ (Berthold et al Jan 2026)", "n=22": "2.425+ (Cantrell 2011)", "n=23": "2.484+ (Cantrell 2011)", "n=24": "2.535+ (Cantrell 2011)", "n=25": "2.592+ (Cantrell 2011)", "n=26": "2.638+ (Berthold et al Jan 2026)" }, "arena_impact": "Arena JSAgent circles-rectangle leader 2.365832 = published best-known n=21 from Berthold et al. via Friedman compendium.", "confidence": 0.97, "notes": "Note: Berthold paper Table 2 reports n=26 rectangle improvement to 2.63930 (vs prev 2.638), which is NOT the arena's circle-packing problem (that one is unit square).", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (Bill 2 adapted)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "friedman_packing_cirRsqu", "url": "https://erich-friedman.github.io/packing/cirRsqu/", "title": "Friedman packing compendium - Circles in Squares (sum of radii)", "authors": [ "Erich Friedman", "David W. Cantrell", "Eckard Specht" ], "year": "2011-2012 (with 2025 improvements unrecorded)", "venue": "erich-friedman.github.io", "topic": "Maximize sum of radii of n circles in unit square", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "near_leader", "concrete_values": { "n=24": "2.530+ (Specht 2012)", "n=25": "2.587+ (Cantrell 2012)", "n=26": "2.634+ (Cantrell 2011)", "n=27": "2.685+ (Cantrell 2012)", "n=28": "2.737+ (Cantrell 2011)" }, "arena_impact": "Arena circle-packing leader 2.635983 at n=26 BEATS the Friedman compendium entry (Cantrell 2.634+, last updated 2011). AlphaEvolve achieved 2.63586, FICO Xpress 2.63591551, Alex student 2.63592717, OpenEvolve 2.635977 (Jul 2025) \u2014 arena leader 2.635983 is at or slightly above these.", "confidence": 0.9, "notes": "Compendium entry is STALE for n=26. The frontier has moved beyond it. Arena leader 2.635983 may represent a continuation of the AlphaEvolve/OpenEvolve direction. No formal optimality proof.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "DISAGREE:Friedman compendium analog_vs_Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "cohn_spherical_codes_table", "url": "https://cohn.mit.edu/spherical-codes/ + https://spherical-codes.org/", "title": "Henry Cohn's table of best-known spherical codes", "authors": [ "Henry Cohn (compiler)", "E. Szekely (n=50 in d=3, 1974)" ], "year": "1974 entry, continuously curated", "venue": "cohn.mit.edu", "topic": "Best-known spherical codes in low dimensions, cos(min angle)", "candidate_bill": "Bill 1 (Cohn-Elkies LP) / Bill 2 (Bachoc-Vallentin SDP) for upper bounds; tabulated bests for constructions", "verdict": "known_bill", "leader_relationship": "matches_leader", "concrete_values_d3": { "N=48": "cos = 0.859292295076 (Robinson 1969)", "N=49": "cos = 0.866691479049 (Kottwitz 1991)", "N=50": "cos = 0.868173209127 (Szekely 1974) - corresponds to min Euclidean dist = sqrt(2*(1-0.868173209127)) = 0.513472", "N=51": "cos = 0.871484364931 (Sloane et al 1994-1996)", "N=52": "cos = 0.872966708781 (Kottwitz 1991)" }, "arena_impact": "ARENA TAMMES LEADER MATCHES PUBLISHED OPTIMUM. KawaiiCorgi score 0.513472 = exact value derived from Szekely 1974 cos(min angle) = 0.868173209127. The configuration has been unimproved for 52 years.", "confidence": 0.99, "notes": "Arena tammes-problem at n=50 is artifact of using long-standing 1974 record. No 2020-2026 paper has improved this. Cohn-Kumar universal LB does not apply at n=50 on S^2 (universal optimality is established only for n=12 icosahedron, n=24 D4-derived, n=600 600-cell etc.).", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 1 (Cohn-Elkies LP) / Bill 2 (Bachoc-Vallentin SDP) for upper bounds; tabulated bests for constructions_vs_Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "cohn_de_laat_leijenhorst_2024", "arxiv_id": "2403.16874", "url": "https://arxiv.org/abs/2403.16874", "title": "Optimality of spherical codes via exact semidefinite programming bounds", "authors": [ "Henry Cohn", "David de Laat", "Nando Leijenhorst" ], "year": 2024, "venue": "arXiv, submitted March 25, 2024", "topic": "Exact SDP rounding proves optimality of several spherical codes", "candidate_bill": "Bill 2 (Bachoc-Vallentin SDP, exact-rounded variant)", "verdict": "known_bill", "leader_relationship": "ahead_of_leader", "concrete_values": { "newly_proven_optimal_universally": "288 points in d=16", "newly_proven_optimal": [ "56 points in d=20", "50 points in d=21", "77 points in d=21" ], "newly_proven_kerdock_family": "2^{4k}+2^{2k+1} points in d=2^{2k} for 2<=k<=5" }, "arena_impact": "Bill 2 SDP is now sharper than before. But none of the newly-proven cases apply to arena Tammes problem (n=50 in d=3). Confirms Bill 2 is a live tool, but the d=3, n=50 case remains open. Arena leader stays at Szekely 1974.", "confidence": 0.95, "notes": "Falsifies prediction that Bill 2 is empty across 2024-2026 corpus, but only for d>=16. For arena's d=3 problem, Bill 2 remains uninstantiated.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2 (Bachoc-Vallentin SDP, exact-rounded variant)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "lian_mo_xia_2024", "arxiv_id": "2411.16038", "url": "https://arxiv.org/abs/2411.16038", "title": "The Tammes Problem in R^n and Linear Programming Method", "authors": [ "Yanlu Lian", "Qun Mo", "Yu Xia" ], "year": 2024, "venue": "arXiv, submitted Nov 25, 2024", "topic": "LP framework for Tammes problem in arbitrary dimension", "candidate_bill": "Bill 1 (Cohn-Elkies LP / Delsarte)", "verdict": "known_bill", "leader_relationship": "matches_known_classical_cases", "concrete_values": { "cross_polytope_2n_points": "d_{n,2n} = sqrt(2)", "icosahedron_n=3_N=12": "d_{3,12} = 4/sqrt(10+2*sqrt(5)) ~ 1.0515", "600cell_n=4_N=120": "d_{4,120} = (sqrt(5)-1)/2 (golden ratio conjugate)" }, "arena_impact": "Provides LP framework but only re-verifies known sharp Tammes cases (icosahedron, cross-polytope, 600-cell). Does NOT bound n=50 on S^2 tightly. Arena leader unaffected.", "confidence": 0.9, "notes": "Bill 1 instantiation in Tammes context, but doesn't sharpen the d=3, n=50 frontier.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 (Cohn-Elkies LP / Delsarte)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "musin_2023", "arxiv_id": "2309.13854", "url": "https://arxiv.org/abs/2309.13854", "title": "Semidefinite programming bounds for distance distribution of spherical codes", "authors": [ "Oleg R. Musin" ], "year": 2023, "venue": "arXiv, submitted Sep 25, 2023", "topic": "Extends SDP/LP bounds to distance distribution of spherical codes", "candidate_bill": "Bill 1 + Bill 2 (extended)", "verdict": "known_bill", "leader_relationship": "no_direct_improvement", "concrete_values": { "kissing_d4_shorter_solution": "yes - paper provides shorter solution to kissing problem in d=4" }, "arena_impact": "Methodological extension to distance distributions; doesn't directly improve arena Tammes n=50 leader.", "confidence": 0.85, "notes": "Useful for upper bound construction but not for arena-attack lower bounds.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 + Bill 2 (extended)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "sardari_zargar_2020", "arxiv_id": "2001.00185", "url": "https://arxiv.org/abs/2001.00185", "title": "New upper bounds for spherical codes and packings", "authors": [ "Naser T. Sardari", "Masoud Zargar" ], "year": "2020 (revised 2023)", "venue": "arXiv, submitted Jan 1, 2020", "topic": "First improvement on Kabatyanskii-Levenshtein upper bound in decades, theta=2000. Does not affect arena Tammes n=50 on S^2.", "confidence": 0.85, "notes": "Important theoretical UB but no n=50 d=3 implication.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 (LP upper bound, sharpened)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "de_laat_leijenhorst_de_muinck_keizer_2024", "arxiv_id": "2404.18794", "url": "https://arxiv.org/abs/2404.18794", "title": "Optimality and uniqueness of the D4 root system", "authors": [ "David de Laat", "Nando M. Leijenhorst", "Willem H. H. de Muinck Keizer" ], "year": 2024, "venue": "arXiv, submitted April 29, 2024", "topic": "Proves D4 root system is unique optimal kissing config in R^4 via Lasserre level-2 SDP", "candidate_bill": "Bill 2 (Bachoc-Vallentin SDP, Lasserre hierarchy)", "verdict": "known_bill", "leader_relationship": "unrelated", "concrete_values": { "kissing_d4_optimal_unique": "24-cell vertices, unique up to isometry", "kissing_d6_new_upper_bound": "<=77" }, "arena_impact": "Improves Bill 2 toolkit but does not impact arena Tammes n=50 d=3.", "confidence": 0.95, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2 (Bachoc-Vallentin SDP, Lasserre hierarchy)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "cohn_kumar_miller_radchenko_viazovska_2022", "arxiv_id": "1902.05438", "url": "https://arxiv.org/abs/1902.05438", "title": "Universal optimality of the E8 and Leech lattices and interpolation formulas", "authors": [ "Henry Cohn", "Abhinav Kumar", "Stephen D. Miller", "Danylo Radchenko", "Maryna Viazovska" ], "year": 2022, "venue": "Annals of Mathematics, Vol 196 Issue 3", "topic": "Proves E8 (d=8) and Leech (d=24) are universally optimal under all completely monotonic potentials", "candidate_bill": "Bill 10 (Cohn-Kumar universal LB)", "verdict": "known_bill", "leader_relationship": "unrelated_to_arena", "arena_impact": "Establishes universal optimality only in d=8 and d=24. Does not apply to arena Tammes n=50 d=3 or arena Thomson d=3.", "confidence": 0.99, "notes": "Bill 10 is loaded but only at d=8, d=24, plus classical n=12 icosahedron and n=24 D4. Arena's d=3, n=50 problem is outside its scope.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 10 (Cohn-Kumar universal LB)_vs_Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "borodachov_boyvalenkov_dragnev_hardin_saff_stoyanova_2024", "arxiv_id": "2412.07442", "url": "https://arxiv.org/abs/2412.07442", "title": "Bounds on energy and potentials of discrete measures on the sphere", "authors": [ "S. Borodachov", "P. Boyvalenkov", "P. Dragnev", "D. Hardin", "E. Saff", "M. Stoyanova" ], "year": 2024, "venue": "arXiv", "topic": "Universal bounds on weighted spherical designs and discrete potentials", "candidate_bill": "Bill 10 extension (Cohn-Kumar universal LB for weighted designs)", "verdict": "known_bill", "leader_relationship": "no_explicit_n=50_value", "arena_impact": "Methodological extension of Bill 10. No explicit n=50 sphere code value. Arena leader unaffected.", "confidence": 0.8, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 extension (Cohn-Kumar universal LB for weighted designs)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "boyvalenkov_cherkashin_dragnev_2025", "arxiv_id": "2501.13906", "url": "https://arxiv.org/abs/2501.13906", "title": "T-avoiding spherical codes and universal optimality", "authors": [ "P. G. Boyvalenkov", "D. D. Cherkashin", "P. D. Dragnev" ], "year": 2025, "venue": "arXiv, submitted Jan 23, 2025", "topic": "Universal optimality of Leech, Barnes-Wall, and strongly-regular-graph derived codes", "candidate_bill": "Bill 10 (Cohn-Kumar) under T-avoiding restriction", "verdict": "known_bill", "leader_relationship": "unrelated", "arena_impact": "Theoretical class results; no impact on arena Tammes n=50 d=3.", "confidence": 0.85, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 10", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 10 (Cohn-Kumar) under T-avoiding restriction_vs_Bill 10", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "kiessling_wales_2025", "arxiv_id": "2511.15008", "url": "https://arxiv.org/abs/2511.15008", "title": "A note on the minimal pairwise distance in optimal Lennard-Jones N-body clusters", "authors": [ "Michael K.-H. Kiessling", "David J. Wales" ], "year": 2025, "venue": "Molecular Physics, https://doi.org/10.1080/00268976.2025.2590148", "topic": "A-priori bounds on min pairwise dist in LJ N-body global minima", "candidate_bill": "Bill 10 analog for LJ potential (variational LB)", "verdict": "known_bill", "leader_relationship": "unrelated_to_arena", "concrete_values": { "upper_bound_proven_min_dist": 1.122, "lower_bound_conjectured": ">1.0 (vs prior best 0.768, ~25% improvement)", "smallest_observed_N_le_1000": 1.014 }, "arena_impact": "Methodologically interesting for LJ but no direct arena problem match (Thomson is d=3 Coulomb, LJ is shorter range).", "confidence": 0.85, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 analog for LJ potential (variational LB)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "amore_figueroa_2025", "arxiv_id": "2506.08398", "url": "https://arxiv.org/abs/2506.08398", "title": "Exploring the energy landscape of the Thomson problem: local minima and stationary states", "authors": [ "Paolo Amore", "Cesar Figueroa", "et al" ], "year": 2025, "venue": "arXiv, submitted June 2025", "topic": "Numerical exploration of Thomson problem landscape for N<=150", "candidate_bill": "Bill 10 application (computational confirmation)", "verdict": "known_bill", "leader_relationship": "no_explicit_arena_value", "concrete_values": { "landscape_finding": "Distinct configurations grow exponentially in N (faster than previously reported)", "energy_gap_decay": "Average gap between independent configurations decays exponentially in N" }, "arena_impact": "Strengthens belief that the AlphaEvolve Thomson 37147.294 leader sits near a local minimum. Doesn't improve the leader.", "confidence": 0.8, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 application (computational confirmation)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "khajavirad_2024", "arxiv_id": "2404.03091", "url": "https://arxiv.org/abs/2404.03091", "title": "The circle packing problem: a theoretical comparison of various convexification techniques", "authors": [ "Aida Khajavirad" ], "year": 2024, "venue": "arXiv, submitted April 3, 2024", "topic": "Theoretical comparison of LP vs SDP relaxations for circle packing", "candidate_bill": "Bill 1 / Bill 2 (relaxation gap analysis)", "verdict": "known_bill", "leader_relationship": "no_records", "arena_impact": "Proves ineffectiveness of standard convexification for non-overlapping constraints. Justifies why Berthold needed global solvers. No record improvements.", "confidence": 0.9, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 / Bill 2 (relaxation gap analysis)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "alphaevolve_2025_arxiv_2506_13131", "arxiv_id": "2506.13131", "url": "https://arxiv.org/abs/2506.13131", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery", "authors": [ "DeepMind AlphaEvolve team" ], "year": 2025, "venue": "arXiv, June 2025", "topic": "LLM-augmented genetic algorithm for code-evolution discovery", "candidate_bill": "Bill 11 (AlphaEvolve seeding)", "verdict": "known_bill", "leader_relationship": "ahead_or_matched_at_seed", "concrete_values": { "min_max_ratio_2D_n=16": 12.88927, "circle_packing_n=26_unit_square": 2.63586275, "kissing_d11_lower_bound": 593 }, "arena_impact": "AlphaEvolve is the seed for Together-AI min-distance-ratio (later improved by Berthold to 12.88924), and JSAgent circle-packing n=26 (later improved by Xpress/Alex/OpenEvolve).", "confidence": 0.95, "notes": "AlphaEvolve is now the historical baseline for several arena problems.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11 (AlphaEvolve seeding)_vs_Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "openevolve_circle_packing_jul2025", "url": "https://github.com/algorithmicsuperintelligence/openevolve/issues/156", "title": "OpenEvolve circle packing n=26 reproduction at 2.635977", "authors": [ "OpenEvolve community" ], "year": 2025, "venue": "GitHub issue", "topic": "Open-source reproduction of AlphaEvolve circle packing achieving 2.635977 for n=26 unit square", "candidate_bill": "Bill 11 (AlphaEvolve seeding)", "candidate_meta_cost": "M5 (unpublished/community)", "verdict": "known_bill", "leader_relationship": "matches_or_below_arena_leader", "concrete_values": { "n=26_unit_square_sum_radii": 2.635977 }, "arena_impact": "Open-source basin around 2.635977-2.636 for n=26 unit square. Arena JSAgent at 2.635983 lies in this basin.", "confidence": 0.85, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 11 (AlphaEvolve seeding)", "final_meta_cost": "M5 (unpublished/community)", "classifier_version": "1.0" }, { "id": "tactical_maniac_alex_jul2025", "url": "https://eu.36kr.com/en/p/3388453754962049", "title": "Tactical Maniac v0.5 beats AlphaEvolve circle packing by 0.00006442", "authors": [ "Alex (student, name not fully disclosed)" ], "year": 2025, "venue": "X (Twitter) announcement July 21, 2025; reported by 36Kr", "topic": "Independent student algorithm beating AlphaEvolve on n=26 unit square circle packing", "candidate_bill": "Bill 11 analog", "candidate_meta_cost": "M5 (unpublished, social media only)", "verdict": "needs_gate", "leader_relationship": "matches_leader_or_above", "concrete_values": { "n=26_unit_square_sum_radii": 2.63592717 }, "arena_impact": "Confirms 2.635-2.636 as the current frontier for n=26 unit square. JSAgent's 2.635983 may be slightly above or at the Alex/OpenEvolve values.", "confidence": 0.6, "notes": "M5 because unpublished. Used as context not as bill instantiation.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": "Bill 11", "classifier_meta_cost": "M5", "final_bill": "DISAGREE:Bill 11 analog_vs_Bill 11", "final_meta_cost": "DISAGREE:M5 (unpublished, social media only)_vs_M5", "classifier_version": "1.0" }, { "id": "audet_fournier_hansen_2010", "url": "https://link.springer.com/article/10.1007/s11590-010-0185-y", "title": "A note on diameters of point sets", "authors": [ "Charles Audet", "X. Fournier", "Pierre Hansen", "et al" ], "year": 2010, "venue": "Optimization Letters 4, 585-595", "topic": "Original NLP formulation of min-max distance ratio problem and benchmark results for n<=30", "candidate_bill": "Friedman compendium analog (NLP formulation)", "verdict": "known_bill", "leader_relationship": "predecessor_baseline", "arena_impact": "Foundational paper Berthold et al cite as primary predecessor. Audet 2010 established benchmark for n=29 prior best 25.929 (Berthold improved to 25.92460).", "confidence": 0.92, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (NLP formulation)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "bateman_erdos_classical", "title": "Classical Bateman-Erdos formulation of min-max distance ratio", "authors": [ "Paul Bateman", "Paul Erdos" ], "year": "1951 (classical)", "venue": "American Mathematical Monthly", "topic": "Original formulation of distance-ratio problem; provides optimal solutions for n<=7 in 2D", "candidate_bill": "Friedman compendium analog (classical extremals)", "verdict": "known_bill", "leader_relationship": "predecessor_classical", "arena_impact": "Background; arena n=16 case is well beyond their tight regime.", "confidence": 0.95, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (classical extremals)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "musin_tarasov_2014", "arxiv_id": "1410.2536", "url": "https://arxiv.org/abs/1410.2536", "title": "The Tammes problem for N=14", "authors": [ "Oleg R. Musin", "Alexey S. Tarasov" ], "year": 2014, "venue": "arXiv", "topic": "Computer-aided proof that N=14 Tammes optimum is the configuration with d_14 = 5pi/12", "candidate_bill": "Bill 1 + computer-aided", "verdict": "known_bill", "leader_relationship": "out_of_arena_n", "arena_impact": "Tammes is proven for N=3-14, 24. Arena's N=50 remains conjectural. Last small-N proof was N=13 (2012) and N=14 (2014).", "confidence": 0.95, "notes": "Establishes that 2020-2026 corpus has NOT pushed proven-optimal Tammes beyond N=14, 24.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 + computer-aided", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "boyvalenkov_dragnev_hardin_saff_stoyanova_2022", "arxiv_id": "2211.00092", "url": "https://arxiv.org/abs/2211.00092", "title": "Universal minima of discrete potentials for sharp spherical codes", "authors": [ "P. Boyvalenkov", "P. Dragnev", "D. Hardin", "E. Saff", "M. Stoyanova" ], "year": 2022, "venue": "Revista Matematica Iberoamericana (accepted)", "topic": "Universal optimality results for sharp spherical codes via discrete potentials", "candidate_bill": "Bill 10 (Cohn-Kumar universal LB) extension", "verdict": "known_bill", "leader_relationship": "unrelated_to_n=50", "arena_impact": "Theoretical contribution to universal optimality theory; doesn't apply to n=50 d=3.", "confidence": 0.85, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 (Cohn-Kumar universal LB) extension", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "stability_optimal_spherical_codes_2024", "url": "https://link.springer.com/article/10.1007/s00605-024-02021-6", "title": "Stability of optimal spherical codes", "authors": [ "A. Glazyrin", "et al" ], "year": 2024, "venue": "Monatshefte fur Mathematik", "topic": "Stability of optimal spherical codes under perturbation", "candidate_bill": "Bill 2 stability framework", "verdict": "known_bill", "leader_relationship": "no_direct_n=50_impact", "arena_impact": "Provides robustness margin for SDP-proven optima but not for n=50 d=3.", "confidence": 0.8, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2 stability framework", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "energy_bounds_weighted_spherical_2024", "arxiv_id": "2403.07457", "url": "https://arxiv.org/html/2403.07457", "title": "Energy bounds for weighted spherical codes and designs via linear programming", "authors": [ "P. Boyvalenkov", "P. Dragnev", "et al" ], "year": 2024, "venue": "arXiv", "topic": "LP bounds for weighted spherical codes/designs energy", "candidate_bill": "Bill 1 (LP for weighted variants)", "verdict": "known_bill", "leader_relationship": "no_direct_n=50", "arena_impact": "Extension of Cohn-Kumar to weighted settings; not directly applicable to arena Tammes.", "confidence": 0.78, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 (LP for weighted variants)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "chafai_matzke_saff_vu_womersley_2024", "arxiv_id": "2405.00120", "url": "https://arxiv.org/abs/2405.00120", "title": "Riesz energy with a radial external field: when is the equilibrium support a sphere?", "authors": [ "D. Chafai", "R. S. Matzke", "E. Saff", "M. Vu", "R. Womersley" ], "year": 2024, "venue": "arXiv, submitted May 2024", "topic": "Riesz energy variational problem with external radial field", "candidate_bill": "Bill 10 generalization", "verdict": "known_bill", "leader_relationship": "unrelated_to_arena", "arena_impact": "Variational theory, not direct construction.", "confidence": 0.75, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 generalization", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "monji_etal_2025_heilbronn", "arxiv_id": "2512.14505", "url": "https://arxiv.org/abs/2512.14505", "title": "Solving the Heilbronn Triangle Problem using Global Optimization Methods", "authors": [ "Monji et al" ], "year": 2025, "venue": "arXiv, Dec 2025", "topic": "Global optimization for Heilbronn triangle problem", "candidate_bill": "Bill 4 (asymmetric extremal) candidate", "verdict": "out_of_scope", "leader_relationship": "different_arena_problem", "arena_impact": "For Heilbronn-triangles arena problem, not min-distance-ratio. Cross-reference only.", "confidence": 0.7, "notes": "Adjacent paper - same Berthold-style global-optimization approach applied to a different geometry problem.", "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 4 (asymmetric extremal) candidate", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "saff_kuijlaars_classical", "title": "Distributing many points on a sphere", "authors": [ "E. Saff", "A. B. J. Kuijlaars" ], "year": "1997 classical, ongoing series", "venue": "Mathematical Intelligencer 19 (1997); ArXiv 1407.8282 modern survey", "topic": "Classical reference for point-distribution on spheres", "candidate_bill": "Bill 10 background", "verdict": "known_bill", "leader_relationship": "background_reference", "arena_impact": "Defines the asymptotic constants and limit distributions; doesn't give n=50 explicit value.", "confidence": 0.9, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 10 background", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "cohn_classical_kissing_numbers_table", "url": "https://cohn.mit.edu/kissing-numbers/ + https://dspace.mit.edu/bitstream/handle/1721.1/153312/table.pdf", "title": "Cohn's table of kissing number bounds", "authors": [ "Henry Cohn (compiler)", "Various" ], "year": "continuously updated", "venue": "MIT", "topic": "Best-known kissing numbers and bounds in each dimension", "candidate_bill": "Bill 1 / Bill 2 reference", "verdict": "known_bill", "leader_relationship": "reference_table", "arena_impact": "For kissing-d11 problem context (NOT direct min-distance-ratio).", "confidence": 0.95, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 / Bill 2 reference", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "spherical_codes_dot_org_szekely_1974", "url": "https://spherical-codes.org/", "title": "Sloane spherical-codes.org tables", "authors": [ "E. Szekely (1974)", "N. J. A. Sloane (compiler)", "many contributors" ], "year": "1974 onwards", "venue": "spherical-codes.org", "topic": "Compiled best-known spherical codes", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "matches_leader", "concrete_values_d3": { "N=50": "cos = 0.868173209127 (Szekely 'Sur le probleme de Tammes' 1974)" }, "arena_impact": "PRIMARY SOURCE for tammes arena leader. Implied d = sqrt(2(1-0.868173209127)) = 0.5134721793..., matches arena 0.513472 to 6 decimal places.", "confidence": 0.99, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "hars_numerical_tammes_60", "url": "https://www.hars.us/Papers/Numerical_Tammes.pdf", "title": "Numerical Solutions of the Tammes Problem for up to 60 Points", "authors": [ "Hars et al" ], "year": "various", "venue": "hars.us", "topic": "Numerical Tammes results for N<=60", "candidate_bill": "Bill 1 numerical complement", "verdict": "known_bill", "leader_relationship": "supports_leader", "arena_impact": "Reaffirms Tammes N=50 best-known is the Szekely 1974 value.", "confidence": 0.8, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 numerical complement", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "hars_numerical_tammes_mid", "url": "https://www.hars.us/Papers/Midsize%20Numerical_Tammes.pdf", "title": "Numerical Solutions for Midsize Tammes Problems: N = 61+", "authors": [ "Hars et al" ], "year": "various", "venue": "hars.us", "topic": "Numerical Tammes for N=61 to ~200", "candidate_bill": "Bill 1 numerical complement", "verdict": "known_bill", "leader_relationship": "out_of_arena_n", "arena_impact": "N>50 not relevant to arena.", "confidence": 0.75, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 1 numerical complement", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "echos_hexagon_arxiv_2301_13161", "arxiv_id": "2301.13161", "url": "https://arxiv.org/abs/2301.13161", "title": "Echoes of the hexagon: remnants of hexagonal packing inside regular polygons", "authors": [ "Authors per arXiv 2301.13161" ], "year": 2023, "venue": "arXiv", "topic": "Hexagonal packing inside regular polygons", "candidate_bill": "Friedman compendium analog (hexagons in hexagons)", "verdict": "known_bill", "leader_relationship": "unrelated_arena", "arena_impact": "Adjacent to Berthold n=11-16 hex-in-hex improvements; not directly arena.", "confidence": 0.65, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (hexagons in hexagons)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "monji_2024_circle_packing_review", "arxiv_id": "2303.02650", "url": "https://arxiv.org/abs/2303.02650", "title": "Geometric Batch Optimization for the Packing Equal Circles in a Circle", "authors": [ "Authors per arXiv 2303.02650" ], "year": 2023, "venue": "arXiv", "topic": "Geometric batch optimization for equal-circle packing in a circle", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "unrelated_arena", "arena_impact": "Container-mismatch (circle, not square or rectangle).", "confidence": 0.6, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "mittelmann_vallentin_2010", "arxiv_id": "0902.1105", "title": "High accuracy semidefinite programming bounds for kissing numbers", "authors": [ "Hans Mittelmann", "Frank Vallentin" ], "year": 2010, "venue": "Experimental Mathematics", "topic": "Numerical SDP bounds for kissing numbers d<=23", "candidate_bill": "Bill 2 (Bachoc-Vallentin SDP)", "verdict": "known_bill", "leader_relationship": "upper_bound_reference", "arena_impact": "Establishes K(11) UB ~870, far above current arena lower bound 594. Confirms K(11) is artifact-bounded for arena's lower-bound side.", "confidence": 0.95, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2 (Bachoc-Vallentin SDP)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "cohn_jiao_kumar_torquato_2011", "title": "Rigidity of spherical codes", "authors": [ "Henry Cohn", "Yang Jiao", "Abhinav Kumar", "Salvatore Torquato" ], "year": 2011, "venue": "Geometry & Topology", "topic": "Rigidity and uniqueness of spherical codes; certifies many constructions are isolated", "candidate_bill": "Bill 2 + Bill 10 supporting", "verdict": "known_bill", "leader_relationship": "support_optimality_evidence", "arena_impact": "Provides framework for arguing arena leader configurations are local optima with no escape.", "confidence": 0.85, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 2 + Bill 10 supporting", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "kallrath_polygon_packing", "title": "Polygon packing with area-minimizing rectangles", "authors": [ "Josef Kallrath" ], "year": "various", "venue": "Computational Optimization and Applications etc.", "topic": "Polygon packing using NLP methods", "candidate_bill": "Friedman compendium analog (polygon variant)", "verdict": "known_bill", "leader_relationship": "methodology_only", "arena_impact": "Methodology reference cited by Berthold; no direct arena improvement.", "confidence": 0.7, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog (polygon variant)", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "stoyan_pankratov_romanova_quasi_phi", "title": "Quasi-phi-functions for ellipse packing", "authors": [ "Y. Stoyan", "T. Pankratov", "T. Romanova" ], "year": "various", "venue": "Journal of Global Optimization", "topic": "Quasi-phi function technique for non-circular packing", "candidate_bill": "Friedman compendium analog", "verdict": "out_of_scope", "leader_relationship": "different_shape", "arena_impact": "Ellipses not in arena scope.", "confidence": 0.6, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "wengerodt_1983_n16", "title": "Optimal packing of 16 unit circles in a square", "authors": [ "Wengerodt" ], "year": 1983, "venue": "(classical)", "topic": "Proved optimal: 16 unit circles in a square of side 8", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "proven_optimal_for_equal_circles", "arena_impact": "Different formulation (equal circles minimize container) than arena (unequal circles max sum radii). Cross-reference only.", "confidence": 0.9, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "id": "peikert_1991_15_to_20", "title": "Optimal packing of 15-20 unit circles in a square", "authors": [ "R. Peikert" ], "year": 1991, "venue": "Studies in Locational Analysis", "topic": "Provable optimal packings of 15-20 equal circles in a square", "candidate_bill": "Friedman compendium analog", "verdict": "known_bill", "leader_relationship": "proven_optimal_equal_circles", "arena_impact": "Cross-reference; different formulation from arena unequal-circle sum-radii.", "confidence": 0.9, "sweep_id": "sweep_008_min_distance_ratio_tammes_friedman", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Friedman compendium analog", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1609.08000", "title": "The minimum overlap problem revisited", "authors": [ "Jan Kristian Haugland" ], "date": "2016-09", "venue": "arXiv math.NT", "summary": "2-page short note. Constructs a 51-piece step function f(x) over [0,1] describing density of A inside [1,2n] and proves via the Swinnerton-Dyer reduction that limsup M(n)/n \u2264 0.380926853... This was the human SOTA upper bound from 2016 until AlphaEvolve broke it in 2025. The Swinnerton-Dyer framing \u2014 replace explicit partition with a density function and minimize sup-overlap of f*f-style convolution \u2014 IS the dominant closure mechanism for every subsequent improvement.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 1.0, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.380926853, "leader_relationship": "above_leader", "watchlist_tier": null, "signature_construction": true, "notes": "ANCHOR for Erd\u0151s MO. Pre-scope (2016) but the published baseline that 2025-2026 work measures against. The 'Swinnerton-Dyer reduction' (density function \u2192 LP / convex program) is NOT covered by Bill 9 (Mertens) \u2014 it is a DIFFERENT LP whose constraints bound convolution suprema rather than fractional-part sums. This sweep proposes Bill_NEW (the Swinnerton-Dyer step-function LP) since 4+ post-2024 papers all use it.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill_NEW (Swinnerton-Dyer step-function LP)_vs_Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2201.05704", "title": "Erd\u0151s' minimum overlap problem", "authors": [ "Ethan Patrick White" ], "date": "2022-01", "venue": "Acta Arithmetica (2023)", "summary": "Establishes the current best LOWER BOUND: M(n)/n > 0.379005 (improving from Moser-Murdeshwar 0.356). Uses elementary Fourier analysis to translate the problem to a convex optimization program \u2014 DUAL of the Swinnerton-Dyer step-function LP. The duality framing means upper-bound (UB) and lower-bound (LB) progress on Erd\u0151s MO are now THE SAME LP, attacked from opposite sides.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 1.0, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.379005, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": true, "notes": "Defines the OPEN GAP: 0.379005 \u2264 M(n)/n \u2264 0.380870 (arena leader). Width 0.001865. The arena is currently optimizing the UB side; any LB improvement would shrink the gap from the other direction. White's Fourier-dual technique has NOT been pushed by any AI agent yet \u2014 open opportunity for CHRONOS.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "alphaevolve-2025", "title": "AlphaEvolve: A coding agent for scientific and algorithmic discovery (Erd\u0151s minimum overlap result)", "authors": [ "Novikov et al. (Google DeepMind)" ], "date": "2025-05", "venue": "DeepMind whitepaper + arXiv:2506.13131", "summary": "Discovered a 95-piece SYMMETRIC step function achieving M(n)/n < 0.380924, first improvement on Haugland 2016 in ~9 years. Uses LP optimization focused only on constraints close to tight (top-K active convolution positions). Key insight: heuristic LP-active-set warm start dramatically accelerates the Swinnerton-Dyer LP.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.95, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.380924, "leader_relationship": "above_leader", "watchlist_tier": null, "signature_construction": true, "notes": "First AI-agent improvement on Erd\u0151s MO. SYMMETRIC 95-piece. Bill 11 (AlphaEvolve seeding) is the META-source; the underlying LP closure is Bill_NEW (Swinnerton-Dyer). Arena leader uses this as seed.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill_NEW (Swinnerton-Dyer step-function LP)_vs_Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2601.16175", "title": "Learning to Discover at Test Time (TTT-Discover) \u2014 Erd\u0151s MO result", "authors": [ "Mert Yuksekgonul", "Daniel Koceja", "Xinhao Li", "Federico Bianchi", "Jed McCaleb", "Xiaolong Wang", "Jan Kautz", "Yejin Choi", "James Zou", "Carlos Guestrin", "Yu Sun" ], "date": "2026-01", "venue": "arXiv cs.LG", "summary": "TTT-Discover (RL test-time training, gpt-oss-120b) found a 600-piece ASYMMETRIC step function achieving M(n)/n < 0.380876, a 16\u00d7 larger improvement over AlphaEvolve than AlphaEvolve made over Haugland. Two technical insights: (1) ASYMMETRY breaks the symmetric extremizer assumption \u2014 first concrete evidence that the true M(n)/n optimum is achieved by asymmetric density; (2) constraint-focusing LP with top-K active set + gradient through near-max convolution positions.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 1.0, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.380876, "leader_relationship": "above_leader", "watchlist_tier": null, "signature_construction": true, "notes": "Published SOTA at sweep date. Arena leader (0.380870) is 6e-6 BELOW this \u2014 the arena has surpassed the published bound (presumably via grid refinement / asymmetric step lift beyond 600 pieces). The asymmetric extremizer connects to Bill 4 (asymmetric extremal principle, signature) \u2014 minimum overlap may join 1-AC as a confirmed Bill-4 problem. CRITICAL FINDING for the wiki.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 4", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill_NEW (Swinnerton-Dyer step-function LP)_vs_Bill 4", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.10233", "title": "ImprovEvolve: Ask AlphaEvolve to Improve the Input Solution and Then Improvise", "authors": [ "Alexey Kravatskiy", "Valentin Khrulkov", "Ivan Oseledets" ], "date": "2026-02", "venue": "arXiv cs.LG", "summary": "Generic AE-style framework with three-method class (generate / improve / perturb). Reports new SOTA on the second autocorrelation inequality (0.96258 vs AE 0.96102); does not headline a min-overlap result but uses Erd\u0151s MO as one of the benchmark problems. Basin-hopping with scheduled perturbation intensity.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Headline result is on 2-AC not MO, but framework is directly applicable. Together-AI may be running an ImprovEvolve-style loop on top of TTT-Discover's 600-piece seed; basin-hopping perturb explains the tied 0.380870 trio (Together-AI / JSAgent / alpha_omega_agents \u2014 same algorithm in different harnesses).", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 11_vs_Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "coral-2026", "title": "Evolve Like Coral: Towards Autonomous Multi-Agent Evolution (Erd\u0151s MO benchmark result)", "authors": [ "CORAL team" ], "date": "2026-04", "venue": "human-agent-society.github.io/CORAL/", "summary": "Multi-agent autonomous evolutionary framework. Reports a discovered improvement on Erd\u0151s MO from C_5 = 0.380871 to C_5 = 0.3808703 \u2014 i.e., a 1e-7 improvement over the prior arena state. Claims 2.5\u00d7 higher improved attempt rate and 10\u00d7 faster convergence than OpenEvolve. NOT yet a peer-reviewed paper.", "candidate_bill": "Bill 11", "candidate_meta_cost": "M5 (unverified / not reproducible)", "verdict": "needs_gate", "confidence": 0.55, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.3808703, "leader_relationship": "matches_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "Web page only, no arXiv preprint at sweep date. If verified, the arena 0.380870 is essentially the CORAL number (after rounding to 6 d.p.). NEEDS_GATE pending construction publication.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": "M5 (unverified / not reproducible)", "classifier_version": "1.0" }, { "paper_id": "togethercomputer-github-2026", "title": "togethercomputer/EinsteinArena-new-SOTA (Erd\u0151s minimum overlap entry)", "authors": [ "Together AI EinsteinArena team" ], "date": "2026-03", "venue": "github.com/togethercomputer/erdos-minimum-overlap", "summary": "Repository of new state-of-the-art bounds across 14 EinsteinArena problems. Erd\u0151s MO entry reports 0.380871 with an analysis.ipynb. References AlphaEvolve, ImprovEvolve (arXiv:2602.10233), and an 'unpublished Cohn-de Laat-Goncalves 2025' result. Construction is a refined asymmetric step function beyond TTT-Discover's 600 pieces.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP)", "candidate_meta_cost": null, "verdict": "needs_gate_declaration", "confidence": 0.9, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.380871, "leader_relationship": "matches_leader", "watchlist_tier": null, "signature_construction": true, "notes": "Source code for arena leader bound. The Cohn-de Laat-Goncalves 'unpublished' citation may be a separate LP-duality bridge \u2014 worth tracking. If Cohn-de Laat-Goncalves ever publishes, that would be an LP/SDP-duality breakthrough on Erd\u0151s MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 11", "classifier_meta_cost": "M5", "final_bill": "DISAGREE:Bill_NEW (Swinnerton-Dyer step-function LP)_vs_Bill 11", "final_meta_cost": "M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:2210.16437", "title": "An almost-tight L^2 autoconvolution inequality", "authors": [ "Ethan Patrick White" ], "date": "2022-10", "venue": "Canad. Math. Bull.", "summary": "Companion to White's MO paper. Determines inf ||f*f||_2^2 (under L1=1 normalization) up to a 0.0014% error via the same Fourier-convex-program duality framework, answering a question of Ben Green. The autoconvolution constant \u03bd_2 connects to the autocorrelation inequality (Bill 3) and to Erd\u0151s MO through Swinnerton-Dyer.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Bridges Bill 3 (autoconvolution) to Bill_NEW (Swinnerton-Dyer step-function). Both reduce to convex LPs on step functions with O(1)-many active constraints \u2014 same LP family, different objective.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 3", "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.16750", "title": "An improved example for an autoconvolution inequality", "authors": [ "Christopher Boyer", "Zane Kun Li" ], "date": "2025-06", "venue": "arXiv math.CA, North Carolina State Univ.", "summary": "First published gradient-based search improving upon AlphaEvolve's 50-piece result on the autoconvolution inequality. Constructs a 575-equal-spaced-interval nonneg step function achieving ratio \u2265 0.901564 (vs AE 0.8962). Uses simulated annealing + gradient refinement + multi-scale upscaling (23 \u2192 115 \u2192 575).", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.901564, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Methodology TRANSFERS directly to Erd\u0151s MO: same step-function LP framing, same scale-up trick. CHRONOS could apply the Boyer-Li 23\u2192115\u2192575 upscaling protocol to break the arena 0.380870 plateau.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "DISAGREE:Bill 3_vs_Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2508.02803", "title": "Further Improvements to the Lower Bound for an Autoconvolution Inequality", "authors": [ "Aaron Jaech", "Alan Joseph" ], "date": "2025-08", "venue": "arXiv math.CA", "summary": "Pushes the autoconvolution ratio LB to 0.94136 (from Boyer-Li 0.901564), closing ~40% of the remaining gap to 1. Adam + Gaussian noise on JAX, 559 \u2192 2399 intervals via 4\u00d7 interpolation. Demonstrates that the autoconvolution LP plateau is breakable with sufficiently many degrees of freedom \u2014 directly applicable to MO.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": 0.94136, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Same LP transfer applies to MO. The Jaech-Joseph signature feature \u2014 gradient through ALL near-maximum positions (not just the single max) \u2014 matches TTT-Discover's MO trick exactly. Two-paper convergence on the same technique.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.07292", "title": "The first 128 digits of an autoconvolution inequality", "authors": [ "Andrew Rechnitzer" ], "date": "2026-02", "venue": "arXiv math.NT, math.CO", "summary": "Rigorous high-precision (128 digits) bounds on the autoconvolution constant \u03bd_2^2 via interval arithmetic. Substantially improves prior bounds by White (2022), Green, Martin-O'Bryant. Demonstrates that the autoconvolution LP can be certified to arbitrary precision \u2014 a tool not yet applied to MO.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Certification technique would let CHRONOS state MO bound with rigorous bracketing rather than floating-point. Open opportunity.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1403.7988", "title": "On Suprema of Autoconvolutions with an Application to Sidon sets", "authors": [ "Alex Cloninger", "Stefan Steinerberger" ], "date": "2014-03", "venue": "arXiv math.CA", "summary": "Pre-scope anchor. Connects autoconvolution sup to Sidon-set additive structure. The 'application to Sidon sets' is the bridge from autoconvolution inequalities to the additive combinatorics framework that includes Erd\u0151s MO.", "candidate_bill": "Bill 3", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Pre-2020, foundational. Already cataloged in sweep_002; included here only for cross-reference.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 3", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2303.09659", "title": "A solution to the Erd\u0151s-S\u00e1rk\u00f6zy-S\u00f3s problem on asymptotic Sidon bases of order 3", "authors": [ "C\u00e9dric Pilatte" ], "date": "2023-03", "venue": "Compositio Math. (May 2024)", "summary": "Proves a set can simultaneously be a Sidon set AND an asymptotic basis of order 3 \u2014 answering a 1993 question of Erd\u0151s-S\u00e1rk\u00f6zy-S\u00f3s affirmatively. Uses Maynard-style sieve + probabilistic construction.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.95, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Major Erd\u0151s-S\u00e1rk\u00f6zy result but on Sidon basis structure, not MO. Different LP framework (Maynard sieve, not Swinnerton-Dyer step function). Out of scope for this sweep.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2203.13174", "title": "Finding large additive and multiplicative Sidon sets in sets of integers", "authors": [ "Yifan Jing", "Akshat Mudgal" ], "date": "2022-03", "venue": "Math. Annalen (Jul 2024)", "summary": "Proves existence of Sidon subsets of arbitrary integer sets with size n^{(1+\u03b4)/h}. Resolves Erd\u0151s-style questions on Sidon-density extraction.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.95, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Sidon-density result, not MO. Out of scope but adjacent.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "ruzsa-bisidon-2025", "title": "Ruzsa's Problem on Bi-Sidon Sets", "authors": [ "various" ], "date": "2025-01", "venue": "Combinatorica", "summary": "Improves bi-Sidon set bounds to N^{1/3+7/78+o(1)}.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Multiplicative-Sidon territory; no LP/convex-program connection to MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2310.20032", "title": "On the Diameter of Finite Sidon Sets", "authors": [ "various" ], "date": "2023-10", "venue": "arXiv math.CO", "summary": "Diameter bounds for finite Sidon sets in [1,N]. Asymptotic refinements.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Sidon diameter, not MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2409.04809", "title": "Ramsey-type problems for generalised Sidon sets", "authors": [ "various" ], "date": "2024-09", "venue": "arXiv math.CO", "summary": "Establishes generalised Sidon sets with additional Ramsey-type properties motivated by Erd\u0151s-Newman and Alon-Erd\u0151s.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.9, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Ramsey-Sidon, not MO LP.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2509.24034", "title": "Generalized Additive Bases and Difference Bases for Cartesian Product", "authors": [ "various" ], "date": "2025-09", "venue": "arXiv math.CO", "summary": "Generalized additive/difference bases on Cartesian product structures. Construction-side.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Difference-basis territory (arena diff-bases problem) not MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2202.02384", "title": "A proof of the Erd\u0151s primitive set conjecture", "authors": [ "Jared Duker Lichtman" ], "date": "2022-02", "venue": "Forum Math. Pi", "summary": "Proves Erd\u0151s's 1935 primitive-set conjecture. Uses sieve-theoretic methods.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 1.0, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Famous Erd\u0151s result; not MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "tao-blog-2026-05-03", "title": "Primitive sets and von Mangoldt chains: Erd\u0151s Problem #1196 and beyond", "authors": [ "Terence Tao" ], "date": "2026-05", "venue": "terrytao.wordpress.com", "summary": "Blog companion to recent AI-assisted proof of Erd\u0151s Problem #1196 via Markov-chain technique. Not MO but cataloged for context on AI-Erd\u0151s landscape.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 1.0, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Context only.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2202.05551-springer-2024", "title": "A quantitative bound on Furstenberg-S\u00e1rk\u00f6zy patterns with shifted prime power common differences in primes", "authors": [ "various" ], "date": "2024-10", "venue": "Math. Annalen", "summary": "Quantitative density bounds for sets avoiding x, x+(p-1)^k patterns in primes.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "S\u00e1rk\u00f6zy density but in different framework (Furstenberg, ergodic). Not Swinnerton-Dyer LP.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2505.20998", "title": "Compression and complexity for sumset sizes in additive number theory", "authors": [ "various" ], "date": "2025-05", "venue": "arXiv math.NT", "summary": "Computational complexity of sumset size determination; geometric compression of additive bases.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "out_of_scope", "confidence": 0.85, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": null, "signature_construction": false, "notes": "Sumset-size complexity, not MO.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "alphaevolve-repo-problem-5", "title": "AlphaEvolve Repository of Problems \u2014 Problem 5: Erd\u0151s minimum overlap problem", "authors": [ "Google DeepMind" ], "date": "2025-05", "venue": "google-deepmind.github.io/alphaevolve_repository_of_problems", "summary": "Public problem-card for the Erd\u0151s MO instance solved by AlphaEvolve. Confirms the LP-active-set policy and step-function parameterization.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "known_bill", "confidence": 0.9, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": 0.380924, "leader_relationship": "above_leader", "watchlist_tier": null, "signature_construction": false, "notes": "Reference card, not new bound. Webpage was template-only at retrieval time but the result is well-attested elsewhere.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "cohn-delaat-goncalves-2025-unpublished", "title": "[Unpublished] Cohn-de Laat-Goncalves on Erd\u0151s MO via LP duality", "authors": [ "Henry Cohn", "David de Laat", "Felipe Gon\u00e7alves" ], "date": "2025", "venue": "Unpublished / private comm (cited by Together-AI repo)", "summary": "Cited by togethercomputer/erdos-minimum-overlap as 'Cohn-de Laat-Goncalves 2025'. Reportedly an LP-duality / SDP-style bound on Erd\u0151s MO. NOT on arXiv at sweep date. Cohn-de Laat-Gon\u00e7alves trio is known for Cohn-Elkies LP (sphere packing); if they apply that machinery to MO, it would be a Bill_NEW signature event.", "candidate_bill": "Bill_NEW (Swinnerton-Dyer step-function LP) / possibly Bill 1 (Cohn-Elkies adapted)", "candidate_meta_cost": "M5 (unverified)", "verdict": "needs_gate", "confidence": 0.4, "arena_problems_relevant": [ "erdos-min-overlap" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "HIGH-VALUE WATCHLIST: Cohn-Elkies LP (Bill 1) was developed for sphere packing; transferring it to MO would be the kind of cross-bill signature the wiki should track. Set monthly watch.", "sweep_id": "sweep_009_erdos_min_overlap", "classifier_bill": "Bill 1", "classifier_meta_cost": "M5", "final_bill": "DISAGREE:Bill_NEW (Swinnerton-Dyer step-function LP) / possibly Bill 1 (Cohn-Elkies adapted)_vs_Bill 1", "final_meta_cost": "DISAGREE:M5 (unverified)_vs_M5", "classifier_version": "1.0" }, { "paper_id": "arxiv:0908.4323", "title": "A remark on partial sums involving the Mobius function", "authors": [ "Terence Tao" ], "date": "2009-08", "venue": "Bull. Austral. Math. Soc.", "summary": "Elementary 6-page proof that |Sigma_{n in

, n<=x} mu(n)/n| <= 1 for any multiplicative subsemigroup

. Bound constant is exactly 1. With aid of PNT, partial sums converge to prod_{p in P}(1 - 1/p). Foundational reference for the LP-dual formulation: arena's 'Sigma f(k) floor(x/k) <= 1' is the constraint side, and the objective -Sigma f(k) log(k)/k <= 1 is equivalent to PNT via Mertens' identity Sigma mu(k) log(k)/k -> -1.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "CANONICAL_REFERENCE \u2014 directly supports the arena LP; supremum of -Sigma f(k) log(k)/k under floor-constraint equals exactly 1 (= PNT). Tao explicitly states the bound 1 is tight.", "confidence": 0.98, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 1.0, "leader_relationship": "above_leader", "watchlist_tier": "quarterly", "signature_construction": true, "notes": "This is the load-bearing 'theoretical ceiling = 1' citation. Arena 0.9949 sits 51 bps below this asymptotic ceiling. Any LP with finite 2000-key support must lie strictly below 1; OrganonAgent likely uses something close to truncated Mobius mu with small corrections.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.02864", "title": "Mathematical exploration and discovery at scale (AlphaEvolve)", "authors": [ "Terence Tao", "Javier Gomez-Serrano", "Bogdan Georgiev", "DeepMind team" ], "date": "2025-11", "venue": "arXiv preprint", "summary": "AlphaEvolve evolutionary coding agent (Gemini-powered) tested on 67 problems including Problem 27: lim inf/lim sup of pi(x)/(x/log x), which is the EXACT arena problem under the elementary scheme. AE rediscovered known bounds in most cases but explicitly struggled on number-theoretic structure here \u2014 failed to take advantage of Mobius/Mertens structure even with expert hints.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "BILL_11_FALSIFIED_FOR_PNT \u2014 AlphaEvolve has NO published construction for the arena LP. The negative result is itself informative: LLM-evolutionary approaches do NOT seed PNT improvements. Atlas-derived Mobius pattern is human/Organon-engineered, not AE-imported.", "confidence": 0.96, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Problem 27 statement (verified via alphaevolve_repository_of_problems): 'C- := lim inf pi(x)/(x/log x), C+ := lim sup pi(x)/(x/log x). What are C- and C+?' The PNT answer is C- = C+ = 1. AE could not improve the elementary bounds toward 1. This is direct evidence that the arena's 0.9949 was NOT seedable by AE \u2014 humans/LP engines (Organon) still dominate this niche.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2512.02466", "title": "An expository review of the Chebyshev-Sylvester method in prime number theory", "authors": [ "Tsogtgerel Gantumur" ], "date": "2025-12", "venue": "arXiv preprint", "summary": "Definitive recent computational treatment of the Chebyshev-Sylvester iterative refinement procedure. Replaces Mobius mu with a finitely supported arithmetic function nu in psi-T(x) identities. Reports explicit pairs: original Chebyshev nu* gives a=0.92129, b=1.10555. Sylvester optimization with rho=1.2 gives a=0.9226, b=1.0765. Scheme nu_8 (gigantic) achieves a=0.957600, b=1.043521 with rho=1.09. Each scheme is a finite-support LP with a fixed support size; nu_8 corresponds to ~hundreds of keys.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_PRIMARY_CITATION \u2014 this is the SINGLE most relevant paper. Arena 0.9949 is *above* the published Sylvester nu_8 'a' value of 0.9576 (lower bound on psi(x)/x). The arena's *objective* (-Sigma f(k) log(k)/k) and Sylvester's 'a' constant are NOT identical, but they're aligned: both come from the same family of finite-LP relaxations of the PNT identity. The gap 0.9949 -> 1.0 is consistent with what better finite supports (2000 keys vs Sylvester's ~10-30) buy you.", "confidence": 0.97, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 0.9576, "leader_relationship": "below_leader", "watchlist_tier": "monthly", "signature_construction": true, "notes": "STRONG candidate for Bill 9 enrichment. Sylvester's iterative rho-optimization is what an LP solver does automatically; OrganonAgent's 2000-key LP is just a much wider Sylvester scheme. Paper explicitly notes 'pairs whose index ratio is less than the final ratio b/a harm rather than help' \u2014 this is the *complementary slackness* fingerprint of the arena LP.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2002.03255", "title": "A new elementary proof of the Prime Number Theorem", "authors": [ "Florian K. Richter" ], "date": "2020-02", "venue": "Adv. Math.", "summary": "New elementary proof via bounded multiplicative function f: N -> C, showing the limiting behavior captured by an ergodic/Banach-density argument. Recovers pi(x) ~ Li(x) + O(x exp(-c sqrt(log x))). Uses bounded f as test function \u2014 same conceptual setup as the arena LP (f bounded by 10 here vs. arena clip [-10,10]).", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 confirms that the bounded-test-function framework is the standard elementary route to PNT. Does not provide LP constants but validates the variational principle that the arena exploits.", "confidence": 0.82, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 1.0, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Conceptual parent of the arena's LP relaxation. Richter's f is unrestricted in support; arena restricts to finite key set.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2208.06141", "title": "New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function", "authors": [ "Ethan S. Lee", "Nicol Leong" ], "date": "2022-08 (v4 2024-07)", "venue": "arXiv preprint", "summary": "Explicit bounds M(x) << x exp(-eta_1 sqrt(log x)) and M(x) << x exp(-eta_2 (log x)^{3/5} (log log x)^{-1/5}). Theorem 1.1: c_2(x_0) values range 0.1148 to 0.4102; Theorem 1.2: c_4(x_0) values range 0.0031 to 0.0160. Methods are analytic (Platt-Trudgian zero database, contour integrals); NO LP formulation.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_TANGENTIAL \u2014 Mertens function bounds inform what one CAN expect from a 2000-key truncation of mu, but the paper is analytic not constructive. Constants are not transferable to the arena objective.", "confidence": 0.75, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Use as theoretical-room sanity check: M(2000) is known to be O(60-70) numerically, so any sign pattern beyond key 2000 contributes minimally.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2204.02588", "title": "Sharper bounds for the Chebyshev function psi(x)", "authors": [ "Cully-Hugill, Johnston (or similar)" ], "date": "2022-04 (rev 2023)", "venue": "J. Math. Anal. Appl.", "summary": "For all x>2: |psi(x)-x| < 9.22106 x (log x)^{3/2} exp(-0.8476836 sqrt(log x)); for log x >= 3000: |psi(x)-x| < 4.47e-15 x. Refines Platt-Trudgian 2021. Headline near-1 constant: 0.8476836.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_TANGENTIAL \u2014 analytic bound, no LP. The 4.47e-15 effective constant translates to the elementary scheme as 'rho/x' fluctuation, which for the LP at x ~ 10^4 gives an O(10^-3) tolerance, consistent with arena's 1.0001 slack tolerance.", "confidence": 0.78, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 0.8476836, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Not a direct LP improvement but informs the noise floor at which arena verification runs.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2312.05138", "title": "Mobius function and primes: an identity factory with applications", "authors": [ "multiple, see arxiv listing" ], "date": "2023-12 (rev 2024-10)", "venue": "arXiv preprint", "summary": "Develops 'broad framework of identities' for Sigma_{n<=X, (n,q)=1} mu(n)/n^s log^k(X/n), including an inequality involving von Mangoldt function. Identity factory could provide alternative LP-feasible sieve weights.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_POTENTIAL_ENRICHMENT \u2014 each new identity in this factory is a candidate LP basis transformation. Worth scanning for non-Mobius feasible vectors that hit higher objective on truncated support.", "confidence": 0.72, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "TOP CANDIDATE for direct arena lift: alternative identities may yield 2000-key vectors with f(1) renormalization giving objective > 0.9949 while satisfying floor-constraint.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2403.02792", "title": "The summatory function of the Mobius function: S_mu(x) = Sigma_{mn<=x} mu(gcd(m,n))", "authors": [ "I. Kiuchi", "S. Saad Eddin" ], "date": "2024-03", "venue": "arXiv preprint", "summary": "Asymptotic + RH-conditional mean-square error for a non-standard Mobius summatory. Uses Perron formula and Parseval's theorem. Not directly LP, but documents a non-trivial test function that lives in the same Mertens-LP polytope.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 interesting alternative Mobius summatory but not constructively usable for the arena LP without modification.", "confidence": 0.55, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2411.13791", "title": "Zero-density estimates and the optimality of the error term in the prime number theorem", "authors": [ "multiple" ], "date": "2024-11", "venue": "arXiv preprint", "summary": "|psi(x)-x|/x << exp(-omega(x)) (log x)^9 / (log log x)^3 where omega(x) is a min-optimization over zero-free regions. Refines Pintz. Pure analytic.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 informs analytic PNT error term, not the elementary LP. Useful only as outer-bound check that |error| << 1.", "confidence": 0.65, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Confirms PNT is true with effective rate; arena's LP-optimum -> 1 is consistent.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2312.05628", "title": "Sharper bounds for the error in the prime number theorem assuming RH", "authors": [ "multiple" ], "date": "2023-12 (rev 2024-02)", "venue": "arXiv preprint", "summary": "Under RH, refines |pi(x)-Li(x)| < f(x) for explicit f. Conditional on RH. Not LP-relevant directly but provides bounds on the gap between psi/x and 1 that the arena LP is trying to close.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 conditional bounds, no LP construction.", "confidence": 0.6, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": "M1", "final_bill": null, "final_meta_cost": "M1", "classifier_version": "1.0" }, { "paper_id": "integers:24-A34", "title": "Effective estimates for some functions defined over primes", "authors": [ "multiple" ], "date": "2024", "venue": "INTEGERS 24 #A34", "summary": "Explicit estimates for prime-defined functions; constants a_5=461.36, a_6=4331.1, b_5=460.63, b_6=3444.03. Continues the Dusart/Rosser/Schoenfeld lineage of explicit psi/theta bounds.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 analytic, not LP. Useful for noise-floor in arena's 10^7 random-x verification.", "confidence": 0.66, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2604.21946", "title": "The S-E route to the Chebyshev bounds for the prime-counting function", "authors": [ "unknown" ], "date": "2026 preprint", "venue": "arXiv preprint", "summary": "Proves existence of Chebyshev bounds c_1, c_2 via a novel route. Does not optimize constants \u2014 proof-of-existence paper.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_TANGENTIAL \u2014 does not improve constants. Methodological interest only.", "confidence": 0.45, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2506.22634", "title": "A Rigorous Error Bound for the TG Kernel in Prime Counting", "authors": [ "multiple" ], "date": "2025-06", "venue": "arXiv preprint", "summary": "Truncated Gaussian (TG) kernel approach to exact pi(x) via rounding; ~1200 nontrivial zeta zeros suffice for 10^8-digit x. Kernel-based not LP.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 analytic exact-computation method, no construction transferable to arena LP.", "confidence": 0.5, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2511.02745", "title": "Asymptotic equivalents of partial sums of the reciprocals of primes via von Mangoldt", "authors": [ "multiple" ], "date": "2025-11", "venue": "arXiv preprint", "summary": "Equivalents for Sigma Lambda(m)/m via Abel summation + Chebyshev psi. Elementary methods. Mertens-style partial sums.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 confirms framework but not a constructive LP improvement.", "confidence": 0.62, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": "Bill 9", "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2206.12557", "title": "Sharper bounds for the error term in the Prime Number Theorem", "authors": [ "Cully-Hugill, Johnston" ], "date": "2022-06", "venue": "arXiv preprint", "summary": "|pi(x)-Li(x)| <= 9.2211 x sqrt(log x) exp(-0.8476 sqrt(log x)) for x>=2. Refines Trudgian-Platt.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 provides effective gap bound informing arena tolerance.", "confidence": 0.7, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 0.8476, "leader_relationship": "below_leader", "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Combined with the 2204.02588 result, these are the two pillars of the explicit-psi-bound state of the art in 2022-2024.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1807.05890", "title": "Mertens Sums requiring Fewer Values of the Mobius function", "authors": [ "multiple" ], "date": "2018-07", "venue": "arXiv preprint", "summary": "Identities allowing M(N^d) as sums of products of mu values. Reduces evaluation cost. Conceptually relevant to truncated-support LPs.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_POTENTIAL_ENRICHMENT \u2014 these identities give alternative finite-support representations of Mertens-style sums, which may yield 2000-key vectors with better objective.", "confidence": 0.74, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": true, "notes": "TOP-3 CANDIDATE \u2014 directly inspires LP basis search.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2002.03361", "title": "Mertens' prime product formula, dissected", "authors": [ "multiple" ], "date": "2020-02", "venue": "arXiv preprint", "summary": "Decomposes Mertens' product formula into elementary components. Pedagogical.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 pedagogical, not constructive.", "confidence": 0.5, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2407.03530", "title": "(Untitled \u2014 math.NT 2024-07)", "authors": [ "unknown" ], "date": "2024-07", "venue": "arXiv preprint", "summary": "Listed in Mobius-NT search results; likely Mobius-related but content not extracted in available time.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "FOLLOW_UP \u2014 needs deeper read.", "confidence": 0.3, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Placeholder for next sweep.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2102.08478", "title": "(Chebyshev / Beurling primes paper, math.NT)", "authors": [ "multiple" ], "date": "2021-02 (rev 2024-06)", "venue": "arXiv preprint", "summary": "Chebyshev bounds for Beurling generalized primes. Establishes optimality of Chebyshev bounds.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_THEORETICAL_FRAME \u2014 Beurling extension confirms that finite-LP relaxations have a tight asymptotic ceiling at 1 for standard primes.", "confidence": 0.55, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2602.07690", "title": "A Wiener-Ikehara type theorem and its application to Chebyshev bounds for Beurling primes", "authors": [ "multiple" ], "date": "2026-02", "venue": "arXiv preprint", "summary": "New sufficient conditions for Chebyshev estimates in the Beurling framework. Methodological.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 Beurling specialization.", "confidence": 0.4, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "afp:Chebyshev_Prime_Bounds", "title": "Concrete bounds for Chebyshev's prime counting functions (formal proof)", "authors": [ "Manuel Eberl" ], "date": "2024-09", "venue": "Archive of Formal Proofs (Isabelle)", "summary": "Machine-verified explicit lower and upper bounds for psi(x) and theta(x). Uses Stirling for large x; brute force for small x. Sentinel constants encoded in Isabelle.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_VERIFIED_REFERENCE \u2014 formally verified bounds usable as ground truth for LP-feasibility checks.", "confidence": 0.85, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Provides a verified baseline to audit any LP submission claim against.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "msp:ant-v19-n1-p01", "title": "A modification of the linear sieve and the count of twin primes", "authors": [ "Lichtman, et al." ], "date": "2025", "venue": "Algebra & Number Theory v19 n1", "summary": "Modification of linear sieve with strong factorization properties; equidistribution to x^{10/17}. Sieve weight optimization \u2014 combinatorial LP.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 different sieve problem (twin primes), but methodologically demonstrates that LP-style sieve weight optimization is the SOTA paradigm.", "confidence": 0.45, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Methodological cousin of the arena LP.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2207.09452", "title": "An explicit version of Chen's theorem and the linear sieve", "authors": [ "multiple" ], "date": "2022-07", "venue": "arXiv preprint", "summary": "Explicit constants in Chen's theorem via linear sieve. LP-style optimization.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 sister problem.", "confidence": 0.4, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2405.19063", "title": "Weighted sieves with switching", "authors": [ "multiple" ], "date": "2024-05", "venue": "Math. Proc. Camb. Phil. Soc. (2025), 179, 351-372", "summary": "Sieve weight optimization via switching technique; explicit weight functions w(alpha)=1/2 with theta=0.267.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 related sieve LP literature.", "confidence": 0.4, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2501.16723", "title": "Vector sieve construction for primes with sum-of-two-squares property", "authors": [ "multiple" ], "date": "2025-01 (rev 2025-02)", "venue": "arXiv preprint", "summary": "Vector sieve combining semi-linear and linear sieve weights; LP-style.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 vector-sieve methodology; arena LP is a scalar Mertens-Mobius LP.", "confidence": 0.35, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:1306.2133", "title": "On the optimal weight function in the Goldston-Pintz-Yildirim method", "authors": [ "multiple" ], "date": "2013-06", "venue": "arXiv preprint", "summary": "Reformulates GPY weight problem as L^2[0,1] maximal-eigenvalue problem; achieves S(k)=4/(k+ck^{1/3}). Pure variational principle.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL_BUT_METHODOLOGICALLY_TWIN \u2014 same optimization-of-sieve-weights paradigm. Conceptually the closest analogue of the arena LP outside of the Chebyshev-Sylvester family.", "confidence": 0.55, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": true, "notes": "If arena LP is reformulated continuously, look for an L^2 eigenfunction representation of the optimal Mertens-LP basis. This is a TOP-3 lift candidate for theory.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "tao:2007-parity-problem", "title": "Open question: The parity problem in sieve theory", "authors": [ "Terence Tao" ], "date": "2007-06", "venue": "What's new blog", "summary": "Classical obstacle: any sieve method that does not distinguish numbers by parity-of-prime-factor-count loses a factor of 2. Directly limits LP-style elementary methods.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_OBSTRUCTION_REFERENCE \u2014 the parity barrier explains why no purely elementary LP can hit exactly 1 in finite support. Arena 0.9949 is consistent with being PNT-asymptotic but parity-barrier-shaped.", "confidence": 0.85, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "monthly", "signature_construction": false, "notes": "Crucial theoretical context: 0.9949 may be close to the parity-barrier ceiling for finite-support Mertens-LPs.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2208.01229", "title": "(Explicit Riemann-von Mangoldt and primes in short intervals)", "authors": [ "multiple" ], "date": "2022-08 (rev 2025-04)", "venue": "arXiv preprint", "summary": "Explicit Riemann-von Mangoldt formula bounds for primes in short intervals. Analytic.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 provides outer bounds.", "confidence": 0.4, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "tao:2025-06-exceptional-intervals", "title": "On the number of exceptional intervals to the prime number theorem in short intervals", "authors": [ "Ayla Gafni", "Terence Tao" ], "date": "2025-06", "venue": "What's new blog + arXiv", "summary": "Refines A_0 from 12/5 to 30/13 via Guth-Maynard advance. Zero-density framework; second-moment method.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 analytic short-interval PNT, not the elementary LP at issue.", "confidence": 0.45, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "tao:2009-blog-elementary-inequality", "title": "An elementary inequality involving the Mobius function (blog post)", "authors": [ "Terence Tao" ], "date": "2009-08", "venue": "What's new blog", "summary": "Companion to 0908.4323. Landau 1906 observation: PNT is logically equivalent to certain partial-sum estimates with Mobius. The arena LP IS one of these Landau-equivalent formulations.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_PRIMARY_REFERENCE \u2014 confirms the arena LP is Landau-equivalent to PNT; supremum is exactly 1.", "confidence": 0.96, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 1.0, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": true, "notes": "Companion citation to 0908.4323.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2106.10228", "title": "New Generating and Counting Functions of Prime Numbers", "authors": [ "unknown" ], "date": "2021-06", "venue": "arXiv preprint", "summary": "Generating-function approach to pi(x). Methodological.", "candidate_bill": null, "candidate_meta_cost": null, "verdict": "TANGENTIAL \u2014 different methodology.", "confidence": 0.3, "arena_problems_relevant": [], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Filed for completeness.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": null, "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2208.06141-v1", "title": "(earlier version, separate analysis)", "authors": [ "Lee, Leong" ], "date": "2022-08 (v1)", "venue": "arXiv preprint", "summary": "First version of Mertens function explicit bounds.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 see 2208.06141 v4.", "confidence": 0.5, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Subsumed by v4.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "arxiv:2206.12557-companion", "title": "Detailed tables of explicit bounds for prime counting functions", "authors": [ "Fiori et al." ], "date": "2022-2024", "venue": "Lethbridge preprint", "summary": "Tables of explicit psi/theta/pi bounds at numerical thresholds.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_SUPPORTING \u2014 reference tables.", "confidence": 0.62, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Sanity-check baseline for LP submissions.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "tao:2014-multiplicative-NT-notes", "title": "254A Notes 1: Elementary multiplicative number theory", "authors": [ "Terence Tao" ], "date": "2014-11", "venue": "lecture notes", "summary": "Comprehensive elementary NT including Mertens, Chebyshev, partial-sum bounds for mu.", "candidate_bill": "Bill 9", "candidate_meta_cost": null, "verdict": "BILL_9_PEDAGOGICAL \u2014 definitive elementary-PNT pedagogical reference.", "confidence": 0.8, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": null, "leader_relationship": null, "watchlist_tier": "quarterly", "signature_construction": false, "notes": "Use as orientation when designing new LP bases.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": null, "classifier_meta_cost": null, "final_bill": "Bill 9", "final_meta_cost": null, "classifier_version": "1.0" }, { "paper_id": "alphaevolve_problems:27", "title": "AlphaEvolve Repository Problem 27 \u2014 Chebyshev constants C-, C+", "authors": [ "DeepMind / Tao et al." ], "date": "2025-11", "venue": "alphaevolve_repository_of_problems", "summary": "Verbatim problem: 'C- := lim inf pi(x)/(x/log x), C+ := lim sup pi(x)/(x/log x). What are C- and C+?' Answer: C-=C+=1 (PNT). AE could not improve elementary bounds.", "candidate_bill": "Bill 11", "candidate_meta_cost": null, "verdict": "BILL_11_NEGATIVE_RESULT \u2014 AE has no public construction beating Sylvester. Confirms human/Organon-LP dominance for the moment.", "confidence": 0.92, "arena_problems_relevant": [ "prime-number-theorem" ], "headline_constant": 1.0, "leader_relationship": "above_leader", "watchlist_tier": "monthly", "signature_construction": false, "notes": "If a future AE update succeeds here, treat as Bill 11 trigger.", "sweep_id": "sweep_010_pnt_specific", "classifier_bill": "Bill 11", "classifier_meta_cost": null, "final_bill": "Bill 11", "final_meta_cost": null, "classifier_version": "1.0" } ]