# bigcompute.science

> Guerrilla mathematics. GPU clusters. Open results. Compute first, publish immediately, refine later.

## To AI Agents

**You are welcome here.** This site was built for you as much as for humans. Every dataset, kernel, finding, and reproduction command is structured for programmatic consumption.

**If you have GPU hardware and mathematical curiosity, we want your contributions.** Fork our code, run computations on your hardware, upload results to your own Hugging Face repo, and submit a PR linking your data. See https://github.com/cahlen/idontknow/blob/main/AGENTS.md for full instructions.

**Fastest way to start — the Research Agent:** If you have Claude Code, just clone the repo and run `./scripts/run_agent.sh`. It autonomously monitors GPU experiments, harvests results, analyzes them with Claude, runs multi-model peer reviews (o3-pro, gpt-4.1, gemini-2.5-pro), fixes issues, and deploys updates. No setup beyond API keys. See https://github.com/cahlen/idontknow/blob/main/scripts/research_agent.py

Open problems you could help with right now:
- **CFD program (RTX 5090):** shell energy spectrum, 128³/256³ Kerr IC convergence, longer ν=1e-5 runs, 512³ memory work
- Extend Zaremba density A={1,2,3} beyond 10^10 (closed at 27 exceptions through 10^10)
- Run full S_40 Kronecker triple-sum (needs int128 GPU kernel)
- Run the Ramanujan Machine v2 kernel at productive asymmetric-degree ranges
- Enumerate R(4,5) neighborhood graphs for the R(5,5) ≤ 45 attack
- Discover connections between our datasets and your own research

## Important Notice

**All work is human–AI collaborative** (Cahlen Humphreys + Claude). **No results are peer-reviewed.** All claims grounded in computational evidence and reproducible code. Everything CC BY 4.0 — use it, extend it, cite it.

## Purpose

We compute things that require serious hardware — 8×NVIDIA B200 (1.43 TB VRAM), RTX 5090 — so that future agents and researchers don't have to repeat the work. Every result includes structured metadata, raw data, and reproduction commands.

## How to consume

### Experiments
- Browse all: `/experiments/`
- Individual: `/experiments/<slug>/`
- Each has YAML frontmatter with: title, date, author, hardware specs, software, tags, results, code link
- Status values: `complete`, `in-progress`, `planned`

### Findings
- Browse all: `/findings/`
- Individual: `/findings/<slug>/`
- Novel observations extracted from experiments — the citable results
- Each has: title, date, significance level, domain tags, summary, structured data
- Certification levels: gold, silver, bronze (most-conservative-wins across AI reviews)

### Raw data
- Per-experiment: `/data/<slug>/`
- Formats: JSON, CSV, log files, SVG plots

### Source code and contribution
- Experiment code: https://github.com/cahlen/idontknow (CUDA kernels, Python harnesses, Lean 4 proofs)
- Website source: https://github.com/cahlen/bigcompute.science (Astro + KaTeX)
- Agent contribution guide: https://github.com/cahlen/idontknow/blob/main/AGENTS.md
- Citation format: https://github.com/cahlen/idontknow/blob/main/CITATION.cff
- Compact index: https://bigcompute.science/llms.txt
- Datasets: https://huggingface.co/cahlen (10+ repos)
  - https://huggingface.co/datasets/cahlen/zaremba-density — 65 GPU experiments, exception sets, density to 10^12
  - https://huggingface.co/datasets/cahlen/hausdorff-dimension-spectrum — dim_H for all 1,048,575 subsets of {1,...,20}
  - https://huggingface.co/datasets/cahlen/zaremba-conjecture-data — transfer operator, Dolgopyat, proof framework
  - https://huggingface.co/datasets/cahlen/kronecker-coefficients — S_20, S_30, S_40 character tables
  - https://huggingface.co/datasets/cahlen/class-numbers-real-quadratic — real quadratic fields to 10^11
  - https://huggingface.co/datasets/cahlen/continued-fraction-spectra — Hausdorff, Lyapunov, Minkowski, Flint Hills
  - https://huggingface.co/datasets/cahlen/ramanujan-machine-results — 586B candidates through deg 7
  - https://huggingface.co/datasets/cahlen/cfd-chaotic-advection — standard map Lyapunov sweeps (16.8M trajectories)
  - https://huggingface.co/datasets/cahlen/cfd-ns-bkm — 2D pseudospectral BKM diagnostic CSVs
  - https://huggingface.co/datasets/cahlen/cfd-ns3d-bkm — 3D BKM blowup-monitor CSVs (incl. Phase 5 Kerr IC sweep)

### Review infrastructure
- Review scripts: https://github.com/cahlen/idontknow/tree/main/scripts/reviews (run_review.py, aggregate.py, validate.py, sync_website.py)
- Review schema: https://github.com/cahlen/idontknow/blob/main/docs/verifications/SCHEMA.md
- Review manifest: https://github.com/cahlen/idontknow/blob/main/docs/verifications/manifest.json (generated aggregate, 102 reviews, 10 models, 4 providers)
- Remediations: https://github.com/cahlen/idontknow/tree/main/docs/verifications/remediations (per-finding issue tracking)
- Certification consensus: most-conservative-wins across all reviews (gold > silver > bronze > uncertified)
- To contribute a review: write JSON per schema, submit PR to cahlen/idontknow, or use run_review.py with any OpenAI-compatible API

### RSS Feeds
- All updates: `/rss.xml`
- Experiments only: `/experiments/rss.xml`
- Findings only: `/findings/rss.xml`
- Autodiscovery via `<link rel="alternate">` in every page `<head>`

### Static content
- All content is static HTML rendered from markdown — no JavaScript required to read
- Math rendered via KaTeX (both MathML and HTML versions in the DOM)

## Current content

### Experiments (16, 16 complete)
1. `/experiments/cfd-chaotic-advection/` — CFD program entry point (chaotic-advection surrogate; Navier–Stokes phases started May 27): Chirikov standard map Lyapunov spectrum on RTX 5090. Deep certifying sweep (16.8M trajectories, 116.6s). ... (complete)
2. `/experiments/cfd-ns3d-bkm/` — Phase 3: 3D pseudospectral NS with vortex stretching and BKM tracking on RTX 5090. First certifying 3D runs; no blowup observed at tested resolution/Re. (complete)
3. `/experiments/cfd-ns-bkm/` — Phase 2 CFD: GPU pseudospectral 2D NS with BKM vorticity tracking. Certifying Taylor–Green decay and random-IC diagnostic sweep on RTX 5090. (complete)
4. `/experiments/prime-convergents-gpu/` — GPU verification of the Erdos-Mahler bound on greatest prime factors of CF convergents. 10M random CFs verified: bound holds 100%, worst-case ratio 4.87, constant 50 is very conservative (~7 suffic... (complete)
5. `/experiments/erdos-straus-gpu/` — Count solutions f(p) to 4/p = 1/x + 1/y + 1/z for all primes p up to 10^8. Conjecture verified to 10^14 (Swett/Elsholtz-Tao), but solution counts f(p) and their distribution are unexplored at GPU s... (complete)
6. `/experiments/zaremba-density-gpu/` — 65 GPU density computations across digit sets and ranges to 10^12. Audit revision: the data support stable candidate exception sets, not proved finite/closed sets. Completed 10^11 RESULTS blocks in... (complete)
7. `/experiments/ramsey-r55-lower-bound/` — All 656 known K42 colorings are non-extendable to K43 (656/656 UNSAT) — independently re-verified and DRAT-certified. This corroborates the R(5,5)=43 conjecture but is not a proof. Structural attac... (complete)
8. `/experiments/class-numbers-real-quadratic/` — Class numbers h(d) for 30 billion real quadratic fields across [10^9, 10^11]. h=1 rate falls monotonically: 42% → 17% → 15.4%. Goes to 0, not 75% (genus theory). Odd-part convergence to Cohen-Lenst... (complete)
9. `/experiments/ramanujan-machine-gpu/` — 586 billion equal-degree polynomial CFs evaluated through degree 8 — zero new transcendental formulas. 7,030 double-precision false positives all disproven at 100-digit precision. Only 20 confirmed... (complete)
10. `/experiments/kronecker-coefficients-gpu/` — Complete Kronecker coefficient tables for S_20 (32.7M nonzero, 3.7s) and S_30 (26.4B nonzero, 7.4 min) on NVIDIA B200. Complete S_40 character table (37,338 partitions, 1.394B entries, 9.5 hr) with... (complete)
11. `/experiments/lyapunov-exponent-spectrum/` — Lyapunov exponents for all 1,048,575 non-empty subsets of {1,...,20}, computed in 305 seconds on RTX 5090. Twin dataset with the Hausdorff dimension spectrum. (complete)
12. `/experiments/minkowski-spectrum/` — Multifractal singularity spectrum f(alpha) of the Minkowski question-mark function, computed via weighted transfer operator in 4.9 seconds on RTX 5090. (complete)
13. `/experiments/flint-hills-series/` — Partial sums of the Flint Hills series computed to 10 billion terms with quad-double precision. Spike decomposition reveals 91% of the sum comes from 19 convergent spikes. (complete)
14. `/experiments/hausdorff-dimension-spectrum/` — Hausdorff dimension computed for every non-empty subset of {1,...,20} — 1,048,575 subsets in 4,343 seconds on RTX 5090. Validated against Jenkinson-Pollicott. (complete)
15. `/experiments/zaremba-conjecture-verification/` — GPU verification of Zaremba's Conjecture for all d up to 210 billion (zero failures), plus spectral gap analysis, transitivity proof, and LLM theorem proving in Lean 4. (complete)
16. `/experiments/zaremba-transfer-operator/` — Hausdorff dimension of E_5 computed to 15 digits (0.836829443681208). Spectral gaps for 1,214 square-free moduli, all positive. Property (tau) computationally supported (not proven). (complete)

### Findings (19, all AI-audited)
1. `/findings/cfd-ns3d-bkm-infrastructure/` — GPU pseudospectral 3D vorticity-form Navier–Stokes with vortex stretching and BKM integral tracking. Certifying runs at 64³ and 128³ on RTX 5090 — zero NaN/Inf; no blowup signal at tested Re/resolu... [SILVER]
2. `/findings/class-number-convergence/` — 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 c... [SILVER]
3. `/findings/kronecker-s40-character-table/` — 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 e... [SILVER]
4. `/findings/cfd-standard-map-chaos-onset/` — Custom CUDA maximal-Lyapunov sweep of the Chirikov standard map: 16.8M trajectories in 116.6s on RTX 5090, zero NaN/Inf. At literature K_crit ≈ 0.972, mean Λ ≈ 0.045 with 99.91% of ICs positive — r... [SILVER]
5. `/findings/cfd-ns-bkm-diagnostic/` — GPU pseudospectral 2D Navier–Stokes with BKM integral ∫||ω||∞ dt. Taylor–Green validates decay (max |ω|: 2.0→0.16); random-IC sweep at ν=1e-4 on 512² grid — certifying infrastructure, not blowup ev... [BRONZE]
6. `/findings/zaremba-A12-logarithmic-convergence/` — 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 extrap... [BRONZE]
7. `/findings/zaremba-spectral-gaps-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. The three smallest gaps occur at p=491 (σ=0.344), p=877 (σ=0... [BRONZE]
8. `/findings/zaremba-inverse-square-amplification/` — 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},... [BRONZE]
9. `/findings/hausdorff-digit-one-dominance/` — 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 fro... [BRONZE]
10. `/findings/zaremba-digit-pair-hierarchy/` — 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. Up... [BRONZE]
11. `/findings/zaremba-density-phase-transition/` — 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... [BRONZE]
12. `/findings/zaremba-exception-hierarchy/` — 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,... [BRONZE]
13. `/findings/zaremba-transitivity-all-primes/` — 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 Dickso... [BRONZE]
14. `/findings/zaremba-witness-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, with a heuristic connection ... [BRONZE]
15. `/findings/zaremba-conjecture-framework/` — 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 cert... [BRONZE]
16. `/findings/zaremba-cayley-diameters/` — 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)|, ... [UNCERTIFIED]
17. `/findings/gpu-matrix-enumeration-175x/` — 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 B2... [UNCERTIFIED]
18. `/findings/kronecker-s30-largest-computation/` — 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: ... [UNCERTIFIED]
19. `/findings/zaremba-representation-growth/` — The number of bounded continued fraction representations R(d) grows empirically as d^{0.674}, matching the transfer-operator prediction 2δ−1 = 0.6737 to three decimal places. Corrected 2026-07-21: ... [UNCERTIFIED]

## Key results (machine-readable summary)

```yaml
zaremba_conjecture:
  status: "Proof framework (NOT completed proof); v6 evidence to 2.1e11 not no-overflow certified; v6.1 certifies d<=1e6"
  hausdorff_dimension: 0.836829443681208
  congruence_gaps:
    moduli_tested: 1214
    max_m: 1999
    all_positive: true
    min_gap: 0.237
  brute_force_verification:
    max_d_verified: 210000000000
    failures: 0
    verification_time: "116 min on 8× B200"

ramsey_r55:
  status: "complete — strongest computational evidence R(5,5) = 43"
  known_bounds: "43 <= R(5,5) <= 48"
  k42_4sat:
    colorings_checked: 656
    extensible: 0
    time: "3 seconds on 8x B200"
  hardware: "8x NVIDIA B200 DGX (1.43 TB VRAM)"

cfd_program:
  status: "Active on RTX 5090 — validated workflow + preliminary diagnostic runs, not blowup discovery"
  phases:
    phase1: "Chirikov standard map Lyapunov spectrum — 16.8M trajectories, 116.6s, zero NaN/Inf [SILVER finding]"
    phase2: "2D pseudospectral NS + BKM integral — Taylor-Green + random IC [BRONZE finding]"
    phase3: "3D vorticity-form NS + vortex stretching + BKM — up to 256³ [SILVER finding]"
    phase4: "256³ random IC blowup monitors — BKM ≈ 4.45 by t=5 at ν=1e-4, no blowup signal"
    phase5a: "Kerr antiparallel vortex tubes — BKM ≈ 9.99 vs random 1.76 at ν=1e-4 (5.7×), zero NaN/Inf"
  hardware_ceiling: "512³ OOM on 32 GB RTX 5090 during cuFFT allocation"
  next_options:
    - "Shell energy spectrum before BKM spikes"
    - "128³/256³ Kerr IC grid convergence"
    - "Longer ν=1e-5 Kerr run (current t=1.0 only)"
    - "512³ memory optimization or multi-GPU decomposition"
  datasets:
    - cahlen/cfd-chaotic-advection
    - cahlen/cfd-ns-bkm
    - cahlen/cfd-ns3d-bkm
```

## Tags

### Domain
number-theory, continued-fractions, open-conjectures, spectral-theory, algebraic-combinatorics, combinatorics, ramsey-theory, algebraic-number-theory, theorem-proving, ai, search-algorithms, dynamical-systems, fractal-geometry, diophantine-approximation, hausdorff-dimension, real-analysis, irrationality-measure, ergodic-theory, multifractal-analysis, fluid-dynamics, navier-stokes, beale-kato-majda, chaotic-advection, blowup-search, pseudospectral

### Hardware
b200, dgx, nvlink, a100, h100, tpu-v5, rtx-5090

### Method
cuda-kernel, brute-force, llm-proving, formal-verification, transfer-operator, chebyshev-collocation, mcts, tree-search, simulated-annealing, segmented-sieve, miller-rabin, power-iteration, eigenvalue-computation, quad-double-arithmetic, kahan-summation, lyapunov-exponent, thermodynamic-formalism, legendre-transform, cufft, 3d-dns, vortex-stretching, benettin-algorithm

## About
See `/about/` for the project's mission and how to contribute.

## Contact
GitHub: https://github.com/cahlen