Findings
Each finding is peer-reviewed claim-by-claim against live academic literature. Badges show the certification level.
A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testable at 10^12
The Zaremba density for A={1,2} fits density = 30.1 + 4.65·log₁₀(N) with residuals < 0.1%, predicting 100% density at N ~ 10^15. This is logarithmic convergence, NOT the power-law N^(2δ-1) = N^0.062 p...
Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero Triples in 7 Minutes
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 Kronecker coefficie...
Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full Density
CONFIRMED TO 10^10: A={1,2,3} has exactly 27 exceptions (all ≤ 6234), giving 99.9999997% density at d ≤ 10^{10}. Zero new exceptions between d=6234 and d=10^10. The exception set appears finite and cl...
Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow
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 ...
Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theory Dominates
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...
GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batched 2×2 Multiply
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,...
Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without
CONFIRMED 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 nineteen digits without it. Removing digi...
Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.45 for Primes to 1021
GPU BFS on the Cayley graph of Γ_{1,...,5} in SL₂(Z/pZ) for all 669 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...
Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + MOW Spectral Theory (Not Peer-Reviewed)
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...
Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are Small d
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...
Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform
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 ...
Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Algebraic Argument + Computation)
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 (nonzero (2,1) entry), not Cartan normali...
Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the Golden Ratio
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, connected to 1/(5+φ) where φ is...