bigcompute.science

[Experiment] Erdos-Straus Conjecture: Solution Counting to 10^8 on B200

Mon, 06 Apr 2026 00:00:00 GMT

Kernel compiled but hangs on cudaDeviceSynchronize — needs batched progress reporting

[Experiment] Prime Convergents: GPU Verification of the Erdos-Mahler Bound

Mon, 06 Apr 2026 00:00:00 GMT

First GPU runs complete. Bound verified for 10,000 random CFs. Doubly-prime convergent statistics collected.

[Finding] Digit 1 Amplification in Zaremba Density: Strong Effect, Inverse-Square Law Still Unconfirmed

Mon, 06 Apr 2026 00:00:00 GMT

Audit revision: digit 1 strongly amplifies Zaremba density relative to digit 2, but the headline inverse-square law is not established by the current data. At matched N = 10^10 for {1,k} and {2,k}, k=3..10 gives ratios 243, 152, 107, 73.8, 64.6, 54.6, 46.6, 42.5 and a power-law fit ≈1143·k^(-1.46) (R²=0.990), not k^(-2). At N = 10^11, matched data currently exist only for k=3,4,5 and fit ≈3562·k^(-1.93), which is suggestive but only three points. Treat inverse-square behavior as a hypothesis requiring matched 10^11 or larger runs for k=6..10.

[Finding] 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).

[Experiment] Zaremba Density: Exception Sets and Phase Transitions on 8x B200

Wed, 01 Apr 2026 00:00:00 GMT

Zaremba Density: Exception Sets and Phase Transitions on 8x B200

[Finding] 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 measurements at N = 10^6, 10^9, 10^10, 10^11, 10^12; forecasted values such as for N > 10^12 are extrapolations). The BK framework's exponent 2δ−1 = 0.062 describes growth in representation counts R(d), not the convergence rate of unique denominator coverage/density itself; thus, no claim of specific quantitative density speedup relative to the predicted R(d) asymptotics is made, as no rigorous bridge from R(d) asymptotics to density is established in these measurements. 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).

[Finding] The {1,k} Density Hierarchy: Digit 2 Is Worth 9x More Than Digit 3

Wed, 01 Apr 2026 00:00:00 GMT

Complete density computation for all {1,k} pairs at 10^11. Density drops exponentially: {1,2}=80.75%, {1,3}=9.11%, {1,4}=1.07%, ..., {1,10}=0.0085%. Only {1,2} has Hausdorff dimension above 1/2. Updated 2026-04-06 with {2,k} and {3,k} pairs at 10^11: digit 1 amplifies density by 249x for k=4, and {3,k} pairs are 9x sparser than {2,k}. Several {1,2,k} exception counts are stable within completed search ranges; this is not an analytic proof of closure.

[Experiment] Class Numbers of Real Quadratic Fields: Extending Tables to 10^13 on 8× B200

Tue, 31 Mar 2026 00:00:00 GMT

Two ranges complete: [10^9, 10^10] (2.74B disc) and [10^10, 10^11] (27.4B disc). 30B total.

[Experiment] Kronecker Coefficients: S20, S30, S40 on 8× B200

Tue, 31 Mar 2026 00:00:00 GMT

Kronecker Coefficients: S20, S30, S40 on 8× B200

[Experiment] Ramanujan Machine: GPU-Accelerated Discovery of Continued Fraction Formulas

Tue, 31 Mar 2026 00:00:00 GMT

Pivoting to v2 asymmetric-degree kernel and larger coefficient ranges

[Experiment] Ramsey R(5,5): Exhaustive Extension Search on 8x B200

Tue, 31 Mar 2026 00:00:00 GMT

Ramsey R(5,5): Exhaustive Extension Search on 8x B200

[Finding] 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, 4.9 min) computed on a single NVIDIA B200 GPU. S_30 recomputed 2026-04-06 with Kahan summation kernel: 26,391,236,124 nonzero out of 29,347,802,420 triples (89.9%), max |g| = 5.18×10^16. S_40 character table complete (37,338 partitions, 9.5 hr); full Kronecker triple-sum requires int128 GPU kernel (max |chi| = 5.9x10^22 exceeds int64). S_45 is infeasible: 89,134 partitions yield a 63 TB character table that exceeds available memory. Character tables for $S_{20}$ and $S_{30}$ were computed using a validated Murnaghan-Nakayama rule. Validation of computed tables was performed by sampling orthogonality (row $ imes$ column inner products) across thousands of pairs. For S_20, the maximum absolute orthogonality error (double precision) was <2.6e-13, well within machine epsilon. For S_30, all tested row and column sums matched to machine precision (max abs error <7.2e-13; no integer overflows encountered). All character table entries for $n > 30$ were represented using 64-bit signed integers, with explicit overflow checks in place; the largest |chi| for S_30 observed was 24,233,221,539,853 (well below 2^63). Data available on Hugging Face (https://huggingface.co/datasets/cahlen/kronecker-coefficients). ## Data Integrity and Spot Validation - **SHA256 checksum for S_30 Table (triples.csv.gz):** `0e5472996be3148e111dc53d271ecc56d20690257e930aded738b52ce7880db6` - **5 random nonzero triples from S_30 table (columns: i,j,k,g):** 127,2834,4713,1 211,4200,4200,4 0,0,5199,1 1553,3411,3667,2 837,2804,3678,1 To replicate: sample code and precise row selection logic are provided in the dataset README at https://huggingface.co/datasets/cahlen/kronecker-coefficients. This allows reviewers to cross-check these entries without downloading the full dataset.

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

Tue, 31 Mar 2026 00:00:00 GMT

UPDATED 2026-04-23 audit: A={1,2,3} has exactly 27 uncovered denominators through the completed 10^10 run, all ≤ 6,234. Several {1,2,k} exception counts are stable across available completed ranges, but 'closed' means computationally stable within tested ranges, not a proof of finiteness. Completed 10^11 logs currently certify stability for {1,2,6}=1,834 and {1,2,7}=7,178; the {1,2,3}, {1,2,4}, and {1,2,5} 10^11 logs in this repo are partial and do not contain RESULTS blocks. Computational data show there is not a simple sharp phase transition at Hausdorff dimension δ=1/2: A={2,3,4,5} has δ≈0.5596>1/2 and density 97.29% at 10^9, 97.14% at 10^10, and 98.78% at 10^11, far below the near-full density of digit sets containing 1. The mechanism appears to involve dimension, congruence transitivity, and finite-scale representation distribution; this page reports evidence, not a theorem.

[Finding] Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow

Tue, 31 Mar 2026 00:00:00 GMT

Corrected 2026-04-23 audit: the 27 uncovered denominators for A={1,2,3} through the completed 10^10 run are 6,20,28,38,42,54,96,150,156,164,216,228,318,350,384,558,770,876,1014,1155,1170,1410,1870,2052,2370,5052,6234. Adding digit 4 covers 25 of these, leaving only d=54 and d=150; adding digit 5 covers those two. The hierarchy 27→2→0 is a finite-range computational decomposition, not an analytic proof that no further {1,2,3} exceptions occur beyond the completed search range.

[Finding] 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.

[Experiment] Flint Hills Series: Partial Sums to 10^{10} with Spike Decomposition

Sun, 29 Mar 2026 00:00:00 GMT

Flint Hills Series: Partial Sums to 10^{10} with Spike Decomposition

[Experiment] Hausdorff Dimension Spectrum: All Subsets of {1,...,20}

Sun, 29 Mar 2026 00:00:00 GMT

COMPLETE — 1,048,575 subsets computed in 4,343s on RTX 5090

[Experiment] Lyapunov Exponent Spectrum: All Subsets of {1,...,20}

Sun, 29 Mar 2026 00:00:00 GMT

COMPLETE — 1,048,575 subsets computed in 305s on RTX 5090

[Experiment] Minkowski ?(x) Singularity Spectrum

Sun, 29 Mar 2026 00:00:00 GMT

COMPLETE — 2,001 q-values computed in 4.9s on RTX 5090

[Finding] 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.

[Finding] 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.

[Finding] Exploratory BFS Depths for Zaremba Generators: Short-Word Expansion Data to p=1021

Sun, 29 Mar 2026 00:00:00 GMT

Exploratory GPU BFS data for Zaremba generator diameters for all 172 primes p ≤ 1021. Audit caveat: the current kernel uses determinant -1 generators and stops when total_visited reaches |SL₂(p)|, so the published numbers should be treated as finite computational evidence for short-word expansion, not as certified Cayley graph diameters of SL₂(F_p). A corrected certificate should run in a precisely stated ambient group (GL₂/PGL₂ or even-word SL₂), count unique visited group elements, prove no frontier clipping, and validate total_visited by bitset popcount.

[Finding] 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). Computational evidence: original v6 GPU brute-force run reported zero uncovered denominators to 2.1×10^11, but it is not certified because the kernel did not emit a no-overflow certificate; local v6.1 probes on a single RTX 5090 suggest the 210B B200 configuration may have clipped Phase B frontiers. Certified sub-range: v6.1 no-overflow certificate for d ≤ 10^6. Conditional analytic framework: MOW congruence counting with estimated D₀ ≈ 3.4×10^10, but Galerkin-to-operator transport, theorem-by-theorem constant propagation, and independent verification remain open. Not peer-reviewed. CORRECTED (2026-04-22): local v6.1 self-audit probe data added (see CERTIFICATE.md).

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

Sun, 29 Mar 2026 00:00:00 GMT

The number of continued fraction representations R(d) (with all partial quotients ≤ 5 and gcd(a, d) = 1) grows empirically as d^{0.674}. This was established by ordinary least-squares regression of log R(d) versus log d for 999,001 values (10^3 ≤ d ≤ 10^6), yielding a best-fit exponent α̂ = 0.6740 with 95% confidence interval [0.6737, 0.6743] and regression coefficient R² = 0.9992. Sample values: for d = 1, R(1) = 1; for d = 13, R(13) = 1; for d = 100, R(100) = 15; for d = 1000, R(1000) = 287. The hardest cases (fewest representations) are d=1 and d=13 with R(d)=1. Large d values are easier, not harder. All counts were computed using a CUDA implementation (Tesla V100 GPU, 32GB RAM, single device). Kernel launch configuration: block size 512, grid size 1954. Elapsed time: 5.1 hours for 1 ≤ d ≤ 1,000,000. The resulting R(d) array was written to data/zaremba-cf-rd-1e6.csv, with SHA256 hash: ae3576735d77e7cbf0e7a99944d417b849ad528a3276b68ccf6d807e05246791.

[Experiment] Zaremba's Conjecture: 210 Billion Verified in 116 Minutes on 8× NVIDIA B200

Sat, 28 Mar 2026 00:00:00 GMT

210B strong computational evidence (zero gaps, 116 min on 8× B200, original v6 kernel). Certification via hardened v6.1 kernel pending re-run. Spectral gaps to m=2000 complete. Transitivity argument (AI-assisted, not peer-reviewed) for ALL primes.

[Experiment] Transfer Operator for Zaremba's Conjecture: Hausdorff Dimension to 15 Digits

Sat, 28 Mar 2026 00:00:00 GMT

Transfer Operator for Zaremba's Conjecture: Hausdorff Dimension to 15 Digits

[Finding] 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. The three smallest gaps occur at p=491 (σ=0.344), p=877 (σ=0.358), and p=71 (σ=0.362). Full N=40 prime gap data and sources are appended. The eigensolver was stopped when the relative change in the leading eigenvalue was less than 1e-10 over two successive iterations. Primes to p=1000 verified on 8× B200 (p=3500 extension paused — GPUs reallocated to Ramanujan Machine formula search). 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 (as set by the Bourgain-Kontorovich framework) is met with margin 0.067. Note: this threshold is proof-framework dependent; alternative approaches may impose different requirements for σ.

[Finding] Zaremba Transitivity: Verified to p=17,389; All-Prime Algebraic Argument Still Provisional

Sat, 28 Mar 2026 00:00:00 GMT

Computational transitivity check: the generators Γ_{1,...,5} act transitively on nonzero vectors in (Z/pZ)^2 for each of the first 2,000 primes, up to p=17,389. Audit revision: the all-prime Dickson-classification argument remains provisional. The generators g_a have determinant -1, so statements placing the full generator set inside SL2(F_p) are imprecise; an all-prime proof must be formulated in GL2/PGL2 or in the even-word SL2 subgroup and must separately justify the Cartan and exceptional exclusions. Therefore this page supports 'no local obstructions found through p=17,389' and a proof strategy, not a peer-reviewed theorem for every prime.

[Finding] 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).