Open computational mathematics. AI-audited, not peer-reviewed. All code and data open for independent verification.
17 computational results from GPU-accelerated mathematics research. Each finding is AI-audited claim-by-claim against published literature (44 reviews from 6 models). Not a substitute for formal peer review. How to cite →
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 expl...
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...
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 fin...
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 conv...
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 {...
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·lo...
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 ...
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 g...
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 r...
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 ...
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,...
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...
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 wit...
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 ov...
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 ...