bigcompute.science — Findings

Kronecker S_40: Complete Character Table and Targeted Coefficients — 94.9% Nonzero

Fri, 03 Apr 2026 00:00:00 GMT

Complete character table of S_40 (37,338 partitions, 1.394 billion entries, 9.5 hours on 64-core CPU). Values exceed int64 (max |chi| = 5.9 x 10^22). To our knowledge, the first publicly archived explicit S_40 character table file (GAP can compute entries on demand). Targeted Kronecker coefficients computed exactly: hooks are multiplicity-free (all g in {0,1}), near-rectangular GCT-relevant triples reach g = 10^8, random sampling estimates 94.9% +/- 1.5% of all 8.68 trillion triples are nonzero. The nonzero fraction grows: 79.5% (S_20 exact) -> 89.9% (S_30 exact) -> 94.9% (S_40 sampled).

A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testable at 10^12

Wed, 01 Apr 2026 00:00:00 GMT

The Zaremba density for A={1,2} fits density = 31.5 + 4.47·log₁₀(N) (R² = 0.9984, 5 empirical points from 10^6 to 10^12). The BK framework's exponent 2δ−1 = 0.062 describes R(d) representation-count growth, not density convergence directly, so no quantitative comparison is claimed. Extrapolation suggests 100% density near 10^15.3, but the largest residual (−0.53% at 10^12) may signal sub-logarithmic curvature. This is the slowest-converging digit set measured (δ = 0.531, barely above 1/2).

The {1,k} Density Hierarchy: Digit 2 Is Worth 7x More Than Digit 3

Wed, 01 Apr 2026 00:00:00 GMT

Complete density computation for all {1,k} pairs at 10^10. Density drops exponentially: {1,2}=76.55%, {1,3}=11.06%, {1,4}=1.61%, ..., {1,10}=0.020%. Only {1,2} has Hausdorff dimension above 1/2. The ratio between consecutive pairs shows digit 2 is 6.9x more valuable than digit 3, consistent with the Gauss measure weight 1/k^2 as the dominant factor in Zaremba density. Four exception sets observed to be stable (no growth across a decade of extension): {1,2,3}=27, {1,2,4}=64, {1,2,5}=374, {1,2,6}=1,834.

Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero Triples in 7 Minutes

Tue, 31 Mar 2026 00:00:00 GMT

Complete Kronecker coefficient tables for S_20 (32.7M nonzero, 3.7s) and S_30 (26.4B nonzero, 7.4 min) computed on a single NVIDIA B200 GPU. These are, to our knowledge, the largest complete Kronecker coefficient tables published. The previous highest-n full table in the literature is for n=19 [Baldoni-Bergvelt-van Willigenburg, Exp. Math., 2006]; we are not aware of any published S_20 or S_30 tables. The S_30 table has 29.3 billion unique triples, 90% nonzero, with maximum coefficient 24.2 trillion. Character tables computed via validated Murnaghan-Nakayama rule (rim-path method). For Sₙ with n=20, max absolute row/column orthogonality error is 8.2×10⁻⁹ × n!; for n=30, maximum error is 6.7×10⁻⁷ × n!. Intermediate sum overflow checked by repeating select computations with 256-bit arithmetic (GNU MP). All outputs are non-negative integers, as verified for random samples and all maxima. Data available on Hugging Face, with SHA-256 full dataset checksum: 7f19c4... (truncated) and a file of 1000 random (λ, μ, ν, g(λ,μ,ν)) samples for verification at: https://huggingface.co/datasets/cahlen/kronecker-coefficients/blob/main/s30_samples_1000.csv.

Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full Density

Tue, 31 Mar 2026 00:00:00 GMT

UPDATED 2026-04-05: Five closed exception sets confirmed — {1,2,3}=27, {1,2,4}=64, {1,2,5}=374, {1,2,6}=1834, and NEW {1,2,7}=7178 (verified to 10^11). Sharp threshold at k=7: {1,2,k} for k<=7 has finite exceptions, k>=8 has growing exceptions. A={1,2,3} gives 99.9999997% density at 10^10. 8xB200 overnight batch produced 17 new results at 10^11 scale.

Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow

Tue, 31 Mar 2026 00:00:00 GMT

The 27 exceptions to Zaremba density with A={1,2,3} decompose hierarchically: 25 are resolved by adding digit 4, leaving only d=54 and d=150 (which need digit 5). The hierarchy 27 -> 2 -> 0 reveals a precise structure in the exception set. The CF identity [0;...,a] = [0;...,a-1,1] allows some exceptions to be resolved by splitting the last quotient.

Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theory Dominates

Mon, 30 Mar 2026 00:00:00 GMT

GPU computation of 30 billion class numbers for real quadratic fields reveals that the h(d)=1 rate DECREASES from 42% at d~10^4 to 15.35% at d~10^10 and is still falling. This is NOT non-monotone convergence — the h=1 rate goes to 0 asymptotically because genus theory forces 2|h for discriminants with multiple prime factors (which become dominant by Erdos-Kac). The Cohen-Lenstra prediction of 75.4% applies to Prob(h_odd=1), not Prob(h=1). The odd-part distribution converges extremely slowly to C-L, with 3|h at 15% vs predicted 44%. CORRECTION (2026-04-01): original version incorrectly claimed non-monotone convergence to 75%. Peer review via MCP verification identified the error.

GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batched 2×2 Multiply

Sun, 29 Mar 2026 00:00:00 GMT

Reformulating CF tree enumeration as batched 2×2 matrix multiplication on GPU eliminates all CPU bottlenecks. The fused expand+mark+compact kernel verifies 100M values in 7.5 seconds on a single B200, 175× faster than the previous tree-walk approach. At 10B values, 8 GPUs complete in 43 seconds.

Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without

Sun, 29 Mar 2026 00:00:00 GMT

At n=20 (1,048,575 subsets): dim_H(E_{1,...,5}) = 0.837 while dim_H(E_{2,...,20}) = 0.768. Five digits containing 1 produce a larger Cantor set than fourteen digits without it. Removing digit 1 from {1,...,20} costs dimension 0.197 while removing digit 20 costs at most the numerical noise. Note: Differences below 0.003 may not be statistically significant without further error analysis. CORRECTED (2026-04-01): E_{2,...,20} previously reported as 0.826; actual value from spectrum data is 0.768.

Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.45 for Primes to 1021

Sun, 29 Mar 2026 00:00:00 GMT

GPU BFS on the Cayley graph of Γ_{1,...,5} in SL₂(Z/pZ) for all 172 primes p ≤ 1021. The diameter ratio diam(p)/log(p) decreases from ~3.1 at small primes to ~1.45 at p~1000, suggesting diam(p) ≤ 2·log(p) for sufficiently large p. Note: small primes (p=2,5) violate the 2·log(p) bound. The asymptotic constant appears to be ~1.45.

Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + MOW Spectral Theory (Not Peer-Reviewed, Known Gaps Remain)

Sun, 29 Mar 2026 00:00:00 GMT

Proof FRAMEWORK (not a completed proof) for Zaremba's Conjecture (A=5). Theorem 1: GPU brute force to 2.1×10^11 (unconditional). Theorem 2: MOW congruence counting framework — D₀ ≈ 3.4×10^10, margin 6× below brute-force frontier. KNOWN GAPS: ρ_η is FP64 not interval-certified; MOW theorem matching not verified theorem-by-theorem; C_η constant underestimated. Paper: 15 pages, requires gap closure before arXiv submission. Not peer-reviewed. CORRECTED (2026-04-01): MCP peer review identified 6 gaps preventing characterization as a completed proof.

Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are Small d

Sun, 29 Mar 2026 00:00:00 GMT

The number of CF representations R(d) with partial quotients ≤ 5 grows as d^{0.674}, matching the transfer operator prediction d^{2δ-1}. The hardest cases (fewest representations) are d=1 and d=13 with R(d)=1. Large d values are easier, not harder. Verified via GPU representation counter to d = 10^6.

Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform

Sat, 28 Mar 2026 00:00:00 GMT

FP64/N=40 cuBLAS computation of congruence spectral gaps for Zaremba's semigroup Γ_{1,...,5}. All 168 primes to p=1000 have σ_p ≥ 0.344. Global minimum: σ(p=491) = 0.344. Primes to p=3500 verified on 8× B200 (in progress). Combined with flat gap bound |λ₂|/√p ≤ 2.18 for 9,592 primes, property (τ) computationally supported for square-free m ≤ 1999 (not proven for all moduli or non-square-free m). The convergence threshold σ > 0.277 (Bourgain-Kontorovich framework) is met with margin 0.067.

Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Algebraic Argument + Computation)

Sat, 28 Mar 2026 00:00:00 GMT

The semigroup Γ_{1,...,5} acts transitively on nonzero vectors in (Z/pZ)² for every prime p. Algebraic argument via Dickson's classification (1901): not Borel (g₁ and g₂ share no common eigenvector over F_p for p≥7), not Cartan normalizer (4 distinct eigenlines cannot all be preserved), not exceptional for p≥61 (generator order exceeds 60), small primes p<13 verified by BFS. Revised April 2026 to fix circular size estimate, basis-dependent Borel check, and shared-eigenvector fallacy identified in o3-pro review. This eliminates local obstructions to Zaremba's Conjecture at all primes. Note: this argument applies classical theory to our specific semigroup — it was constructed with AI assistance and has not been independently peer-reviewed.

Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the Golden Ratio

Sat, 28 Mar 2026 00:00:00 GMT

The smallest Zaremba witness for d concentrates at a/d ≈ 0.171 with 99.7% sharing CF prefix [0; 5, 1, ...]. The concentration lies between the convergents 1/6 and 2/11, with a heuristic connection to 1/(5+φ) where φ is the golden ratio. For 99.9% of d, the *minimal* witness uses max quotient 5 (this does not rule out larger witnesses with quotients ≤ 4).