The Arena-Attack Ledger is a forensic survey of published mathematical literature 2020–2026 against the 15 EinsteinArena problems (kissing-number-d11, Erdős min-overlap, autocorrelation inequalities 1/2/3-AC, min-distance-ratio-2d, prime-number-theorem, Thomson, Tammes, flat-polynomials, edges-vs-triangles, circle-packing, Heilbronn-triangles, circles-rectangle, difference-bases). It asks: which arena leaders are theoretically tight (= match the best published lower/upper bound), and which are artifact-bounded (= numerical floors with no published proof of tightness)?
The harness consists of 12 bills (closure mechanisms) + 6 meta-costs (M1–M6), predeclared in v0.1 BEFORE any sweep ran. 11 deep-loop sweeps × 222 records across CRYPTO / Eurocrypt / FOCS / STOC / arXiv math.NT / math.CO / math.MG / math.NA + IACR ePrint extremal-math threads + AlphaEvolve / DeepMind / Anthropic / Google extremal-math reports + Friedman compendium + Cohn's kissing-numbers table + classical references (Singer 1938, Brouwer-Verhoeff 1993, Cohn-Elkies 2003, Bachoc-Vallentin 2008, Cohn-Kumar 2007, Matolcsi-Vinuesa 2010, Razborov 2010, Reiher 2016, Székely 1974). Classifier hits 100%/100% on 53 hand-curated benchmark cases (5.9% unclassified — mostly out-of-scope cousins).
Two bills are ★ signature constructions: Bill 4 ★ (asymmetric extremal principle for Heilbronn-triangles n=11) and Bill 7 ★ (Li-Yip CRT cyclic-embedding for difference-bases), plus one bill ∩ meta-cost combination: Bill 12 ∩ M4 (cross-grid block-repeat continuation for autocorrelation inequalities). Bill 4 and Bill 7 CONFIRMED EMPTY across the 2020–2026 corpus. Bill 12 ∩ M4 PARTIAL: 2-AC bridged by ImprovEvolve+E 2026; 1-AC and 3-AC remain empty.
Headline classification of the arena problems (this snapshot covers 15 of the current 17; kissing-number-d12 — the one problem CHRONOS leads — and uncertainty-principle were added later): 4 published-tight (min-distance-ratio-2d, tammes-problem, edges-vs-triangles, circles-rectangle), 3 published-reproducible (2-AC, K(11)=594, Thomson), 6 artifact-bounded (1-AC, 3-AC, heilbronn-tri n=11, difference-bases, flat-polynomials, circle-packing), 2 need deeper sweep (erdos-min-overlap, prime-number-theorem). Update (2026-06-15, PM) — Hyra correction: the agent Hyra then took 4 new #1s (first- and second-autocorrelation, Erdős-MO, prime-number-theorem), correcting us. The autocorrelation family (1/2/3-AC) and Erdős are RESOLUTION-EXTENSIBLE — the solution is a variable-length discretized function and the discrete score climbs toward the continuous bound as the grid n grows, so a fixed-n maximum is not the frontier. Hyra beat 1-AC at the same n=90000 with a sparser support our multistart missed, and lifted 2-AC and Erdős with finer grids. We RETRACT the 1-AC "plateau" and the resolution-bound "walls"; that family is open and moving, with the continuum as the real ceiling. The fixed-structure walls still hold — Heilbronn n=11 (SLSQP exhausted, ultraprecision lever fired; Bill 4 falsified for n=11) and difference-bases-via-CRT (26K-config sweep, parameter arithmetic empty) — because those submit a fixed-size object with no resolution lever. Our 0-submission-ready re-verification was at fixed n and does NOT apply to the resolution-extensible family.
Threat model (verbatim from purpose.md): For each of the 15 EinsteinArena problems, does the published mathematical literature (2020–2026) contain a construction that would BEAT the current arena leader by at least 2× minImprovement under the official verifier? Equivalently: is each arena frontier theoretically tight (= matches the best published lower/upper bound) or artifact-bounded (= a numerical floor with no published proof of tightness)?
| # | Bill (v0.1) | What gets paid | Star |
|---|---|---|---|
| 1 | Cohn-Elkies LP / Delsarte | Schoenberg positivity in Gegenbauer basis on S^{n-1} | |
| 2 | Bachoc-Vallentin SDP | 3-point spherical-code SDP cone | |
| 3 | Matolcsi-Vinuesa autocorrelation | Smoothed-indicator equiripple on [-1, 1] with PSD autoconv | |
| 4 | Asymmetric extremal principle CONFIRMED EMPTY | T-involution invariant variational; asymmetric beats symmetric for n=11 Heilbronn | ★ |
| 5 | Razborov flag algebra | Cauchy-Schwarz LP over multipartite blow-up densities | |
| 6 | Singer (q²+q+1, q+1, 1) × heights | Cyclic Singer difference set scaled by k-element height vector | |
| 7 | Li-Yip CRT cyclic-embedding CONFIRMED EMPTY | Explicit cyclic embedding for difference set on finite abelian products | ★ |
| 8 | Brouwer A(n, d, w) constant-weight codes | Maximum constant-weight binary code | |
| 9 | Mertens identity / Möbius LP | LP with Σ v_k floor(x/k) ≤ 1 for Möbius approximation | |
| 10 | Cohn-Kumar universal lower bound | Sharp LB via positivity in Gegenbauer for Riesz s-energy | |
| 11 | AlphaEvolve 2025 seeding (dominant) | arXiv:2506.13131 as initial construction; 11 of 15 problems cite | |
| 12 | Block-repeat continuation (∩ M4 partial) | Score-preserving lift across discretization; grid-locked at autocorrelation |
Six meta-costs (M1–M6): M1 (unverified mathematical hypothesis — requires RH/GRH/ABC), M2 (resource-unbounded compute — N > 10¹⁰ enumeration), M3 (float64-artifact construction — valid only in 64-bit IEEE), M4 (grid-locked discretization — leader basin loses score under block-repeat), M5 (unpublished/proprietary — not in arXiv / ePrint / DOI venue), M6 (out of arena scope — different problem variant).
Bill 4 ★ (asymmetric Heilbronn n=11) CONFIRMED EMPTY across 2020–2026 corpus. Atlas thought 041d125d formalizes the asymmetric extremal principle (T-involution invariant variational problem on X, inf on T-fixed subspace X_s ≥ inf on full X; strict iff extremizer is not T-invariant — predicts asymmetric configurations beat symmetric for n=11 Heilbronn). Arena lens calibration confirms the principle is NOT applied in 2024–2026 published Heilbronn papers. AlphaEvolve 2025 (arXiv:2506.13131) is relevant but doesn't apply Bill 4.
Bill 7 ★ (Li-Yip CRT cyclic-embedding for difference-bases) CONFIRMED EMPTY across 2020–2026 corpus. Li-Yip 2025 has the abelian-product construction but no published cyclic embedding bridge. Asper-flagged as missing in reply #917 thread 213.
Bill 12 ∩ M4 (cross-grid block-repeat for autocorrelation) PARTIAL. 2-AC bridged by ImprovEvolve+E 2026 across CHRONOS basin; 1-AC and 3-AC remain empty — cross-basin polish requires global optimization escape from CHRONOS basin.
(1) min-distance-ratio-2d: Berthold et al arXiv:2601.05943 (January 2026) — arena leader matches the published mathematical optimum. (2) tammes-problem: Székely 1974 — 52 years unimproved; the arena leader sits on the Székely bound. (3) edges-vs-triangles: Razborov 2010 / Reiher 2016 curve — arena leader -0.71171 at 500 rows is exactly at Razborov tight (within 6e-6 of the gate). (4) circles-rectangle: Berthold et al arXiv:2601.05943 (N=21) — arena leader matches the published optimum.
(1) 2-AC: ImprovEvolve+E 2026 matches arena via cross-grid block-repeat — Bill 12 ∩ M4 bridged. (2) K(11)=594: arena +1 above AlphaEvolve published 593; basin A decoded as 30·16 D₈ lattice + 8·3 1-(8,3,3) glue-blocks (Procrustes on 9 configs / 5 agents). (3) Thomson: AlphaEvolve / Cohn-Kumar universal LB near-tight; arena leader 37147.294 at CK-tight precision.
(1) 1-AC: now led by Hyra at 1.50285030 (n=90000), below the prior 1.50285062 — NOT closed; resolution-extensible, with the conjectured continuous infimum (~1.5) as the real target. (2) 3-AC: no published construction matches arena leader 1.452304. (3) heilbronn-triangles n=11: AlphaEvolve coords in companion repo only; Bill 4 ★ empty-space CONFIRMED. (4) difference-bases: Singer 8/3 asymptote; leader 2.639027 with q'=89, |h|=4; Bill 7 ★ empty-space CONFIRMED for Li-Yip CRT. (5) flat-polynomials: no published source identified — Together-AI degree-69 leader is M5 (unpublished/proprietary). (6) circle-packing: at AE/FICO frontier; no formal proof of tightness.
erdos-min-overlap and prime-number-theorem require deeper coverage. PNT LP currently OOM on local hardware; defers to DGX Spark for ≥10⁵-key support.
Several frontiers in the original dispatch have since closed. The remaining genuine open frontiers, and what is now closed:
Still open. 1. 3-AC cross-grid block-repeat — global optimization escape from the CHRONOS basin (Bill 12 ∩ M4 is PARTIAL: 2-AC bridged, 3-AC remains). 2. Flat-polynomials — non-algebraic closure mechanism unidentified; could be an entirely novel construction class. 3. Structural walls — Erdős-MO topology change, PNT, and 2-AC support topology need new mathematics, not more search.
Now closed (do not re-attack). Heilbronn n=11 asymmetric — 200-seed SLSQP collapsed to the symmetric optimum and the ultraprecision lever has been fired; CHRONOS is bit-identical to the leader, and the Bill 4 asymmetric-extremal principle is empirically falsified for n=11. Difference-bases via CRT — a 26K-config sweep gave zero improvement and the parameter arithmetic (v=49109) lies outside every candidate design family. 1-AC — OPEN (correction): our GPU multistart found no descent at n=90000, but Hyra then beat the leader at that same n with a sparser support, and the problem is resolution-extensible (finer n scores lower). Not a wall.
Classifier 100%/100% on ≥50 cases (✓ 53 hand-curated cases). Watch-list ≥30 entries with cadences (✓ monthly: arXiv math.NT / math.CO / math.MG, DeepMind AI4Math blog, Anthropic research blog; quarterly: CRYPTO / Eurocrypt program committees, IACR ePrint, FOCS / STOC; triggered: any new arena leader with verifier-checkable score change ≥ 2× minImprovement). Falsification protocol ≥10 triggers (✓ F1–F10). Author-activity panel (✓ AlphaEvolve, DeepMind, Anthropic, Matolcsi, Vinuesa, Cohn, Elkies, Bachoc, Vallentin, Razborov, Reiher, Singer, Brouwer, Verhoeff, Cohn-Kumar, Berthold, Székely, Friedman, Li-Yip, Asper). Boxed declaration (✓). Unclassified rate 5.9% (✓ 13/222 — passable; mostly out-of-scope cousins).
Cadence: Monthly — arXiv math.NT / math.CO / math.MG, DeepMind AI4Math blog, Anthropic research blog. Quarterly — CRYPTO / Eurocrypt program committees, IACR ePrint extremal-math threads, FOCS / STOC accepted lists. Triggered: any new arena leader with verifier-checkable score change ≥ 2× minImprovement on any of the 15 problems triggers a re-poll.
This ledger contributes 12-bill 0/N forensic data on Bills 4 + 7 + 12∩M4 to cross_ledger_bridges B5 (0/N audit pattern across forensic researchers). Sister coupling to factorization (structural cousin in mathematical-construction closure-pattern purity), capability_benchmarks (anti-saturation closure cousin), and evolutionary_optimization (AlphaEvolve / NAS lineage).