Open computational mathematics. AI-audited, not peer-reviewed. All code and data open for independent verification.

How to Cite These Results

All results on this site should be cited as computational evidence, not as peer-reviewed theorems. The recommended verbs are: computed, archived, benchmarked, observed, verified.

Cite the Project

BibTeX
@misc{humphreys2026bigcompute,
  author = {Humphreys, Cahlen},
  title = {bigcompute.science: GPU-Accelerated Computational Mathematics},
  year = {2026},
  url = {https://bigcompute.science},
  note = {Human--AI collaborative. AI-audited, not peer-reviewed. CC BY 4.0}
}
APA
Humphreys, C. (2026). bigcompute.science: GPU-accelerated computational
  mathematics. https://bigcompute.science. Human–AI collaborative.
  AI-audited, not peer-reviewed.

Cite Individual Findings

Click any finding to expand its citation. Each URL is the canonical reference.

SILVER Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.4...
BibTeX
@misc{zarembacayleydiameters,
  author = {Humphreys, Cahlen},
  title = {Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.45 for Primes to 1021},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-cayley-diameters/},
  note = {AI-audited (3 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.45 for Primes to 1021. bigcompute.science.
  https://bigcompute.science/findings/zaremba-cayley-diameters/
Plain text
Cahlen Humphreys, "Cayley Graph Diameters of Zaremba's Semigroup: diam(p)/log(p) → 1.45 for Primes to 1021," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-cayley-diameters/
SILVER Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theor...
BibTeX
@misc{classnumberconvergence,
  author = {Humphreys, Cahlen},
  title = {Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theory Dominates},
  year = {2026},
  url = {https://bigcompute.science/findings/class-number-convergence/},
  note = {AI-audited (3 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theory Dominates. bigcompute.science.
  https://bigcompute.science/findings/class-number-convergence/
Plain text
Cahlen Humphreys, "Cohen-Lenstra at Scale: h=1 Rate Falls to 15% at 10^10, Genus Theory Dominates," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/class-number-convergence/
SILVER Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform
BibTeX
@misc{zarembaspectralgapsuniform,
  author = {Humphreys, Cahlen},
  title = {Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-spectral-gaps-uniform/},
  note = {AI-audited (3 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform. bigcompute.science.
  https://bigcompute.science/findings/zaremba-spectral-gaps-uniform/
Plain text
Cahlen Humphreys, "Congruence Spectral Gaps for Zaremba's Semigroup Are Uniform," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-spectral-gaps-uniform/
SILVER Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without
BibTeX
@misc{hausdorffdigitonedominance,
  author = {Humphreys, Cahlen},
  title = {Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without},
  year = {2026},
  url = {https://bigcompute.science/findings/hausdorff-digit-one-dominance/},
  note = {AI-audited (3 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without. bigcompute.science.
  https://bigcompute.science/findings/hausdorff-digit-one-dominance/
Plain text
Cahlen Humphreys, "Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/hausdorff-digit-one-dominance/
SILVER Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero ...
BibTeX
@misc{kroneckers30largestcomputation,
  author = {Humphreys, Cahlen},
  title = {Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero Triples in 7 Minutes},
  year = {2026},
  url = {https://bigcompute.science/findings/kronecker-s30-largest-computation/},
  note = {AI-audited (2 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero Triples in 7 Minutes. bigcompute.science.
  https://bigcompute.science/findings/kronecker-s30-largest-computation/
Plain text
Cahlen Humphreys, "Kronecker Coefficients: Complete S_30 Table — 26.4 Billion Nonzero Triples in 7 Minutes," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/kronecker-s30-largest-computation/
SILVER Kronecker S_40: Complete Character Table and Targeted Coefficients ...
BibTeX
@misc{kroneckers40charactertable,
  author = {Humphreys, Cahlen},
  title = {Kronecker S_40: Complete Character Table and Targeted Coefficients — 94.9% Nonzero},
  year = {2026},
  url = {https://bigcompute.science/findings/kronecker-s40-character-table/},
  note = {AI-audited (2 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Kronecker S_40: Complete Character Table and Targeted Coefficients — 94.9% Nonzero. bigcompute.science.
  https://bigcompute.science/findings/kronecker-s40-character-table/
Plain text
Cahlen Humphreys, "Kronecker S_40: Complete Character Table and Targeted Coefficients — 94.9% Nonzero," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/kronecker-s40-character-table/
SILVER Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full De...
BibTeX
@misc{zarembadensityphasetransition,
  author = {Humphreys, Cahlen},
  title = {Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full Density},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-density-phase-transition/},
  note = {AI-audited (5 reviews). silver certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full Density. bigcompute.science.
  https://bigcompute.science/findings/zaremba-density-phase-transition/
Plain text
Cahlen Humphreys, "Zaremba Density Phase Transition: A={1,2,3} Appears to Have Full Density," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-density-phase-transition/
BRONZE A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testab...
BibTeX
@misc{zarembaA12logarithmicconvergence,
  author = {Humphreys, Cahlen},
  title = {A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testable at 10^12},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-A12-logarithmic-convergence/},
  note = {AI-audited (2 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testable at 10^12. bigcompute.science.
  https://bigcompute.science/findings/zaremba-A12-logarithmic-convergence/
Plain text
Cahlen Humphreys, "A={1,2} Density Fits Logarithmic Growth: 30 + 4.65·log₁₀(N), Testable at 10^12," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-A12-logarithmic-convergence/
BRONZE GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batche...
BibTeX
@misc{gpumatrixenumeration175x,
  author = {Humphreys, Cahlen},
  title = {GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batched 2×2 Multiply},
  year = {2026},
  url = {https://bigcompute.science/findings/gpu-matrix-enumeration-175x/},
  note = {AI-audited (2 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batched 2×2 Multiply. bigcompute.science.
  https://bigcompute.science/findings/gpu-matrix-enumeration-175x/
Plain text
Cahlen Humphreys, "GPU Matrix Enumeration: 175× Faster Zaremba Verification via Batched 2×2 Multiply," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/gpu-matrix-enumeration-175x/
BRONZE The {1,k} Density Hierarchy: Digit 2 Is Worth 7x More Than Digit 3
BibTeX
@misc{zarembadigitpairhierarchy,
  author = {Humphreys, Cahlen},
  title = {The {1,k} Density Hierarchy: Digit 2 Is Worth 7x More Than Digit 3},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-digit-pair-hierarchy/},
  note = {AI-audited (2 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). The {1,k} Density Hierarchy: Digit 2 Is Worth 7x More Than Digit 3. bigcompute.science.
  https://bigcompute.science/findings/zaremba-digit-pair-hierarchy/
Plain text
Cahlen Humphreys, "The {1,k} Density Hierarchy: Digit 2 Is Worth 7x More Than Digit 3," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-digit-pair-hierarchy/
BRONZE Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow
BibTeX
@misc{zarembaexceptionhierarchy,
  author = {Humphreys, Cahlen},
  title = {Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-exception-hierarchy/},
  note = {AI-audited (2 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow. bigcompute.science.
  https://bigcompute.science/findings/zaremba-exception-hierarchy/
Plain text
Cahlen Humphreys, "Zaremba Exception Hierarchy: 27 → 2 → 0 as Digits Grow," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-exception-hierarchy/
BRONZE Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are...
BibTeX
@misc{zarembarepresentationgrowth,
  author = {Humphreys, Cahlen},
  title = {Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are Small d},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-representation-growth/},
  note = {AI-audited (3 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are Small d. bigcompute.science.
  https://bigcompute.science/findings/zaremba-representation-growth/
Plain text
Cahlen Humphreys, "Zaremba Representation Counts Grow as d^{0.674} — Hardest Cases Are Small d," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-representation-growth/
BRONZE Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the G...
BibTeX
@misc{zarembawitnessgoldenratio,
  author = {Humphreys, Cahlen},
  title = {Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the Golden Ratio},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-witness-golden-ratio/},
  note = {AI-audited (3 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the Golden Ratio. bigcompute.science.
  https://bigcompute.science/findings/zaremba-witness-golden-ratio/
Plain text
Cahlen Humphreys, "Zaremba Witnesses Concentrate at α(d)/d ≈ 0.171, Connected to the Golden Ratio," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-witness-golden-ratio/
BRONZE Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Al...
BibTeX
@misc{zarembatransitivityallprimes,
  author = {Humphreys, Cahlen},
  title = {Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Algebraic Argument + Computation)},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-transitivity-all-primes/},
  note = {AI-audited (3 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Algebraic Argument + Computation). bigcompute.science.
  https://bigcompute.science/findings/zaremba-transitivity-all-primes/
Plain text
Cahlen Humphreys, "Zaremba's Semigroup Acts Transitively on (Z/pZ)² for ALL Primes (Algebraic Argument + Computation)," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-transitivity-all-primes/
BRONZE zaremba-conjecture-proved
BibTeX
@misc{zarembaconjectureproved,
  author = {Humphreys, Cahlen},
  title = {zaremba-conjecture-proved},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-conjecture-proved/},
  note = {AI-audited (5 reviews). bronze certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). zaremba-conjecture-proved. bigcompute.science.
  https://bigcompute.science/findings/zaremba-conjecture-proved/
Plain text
Cahlen Humphreys, "zaremba-conjecture-proved," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-conjecture-proved/
UNCERTIFIED Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + ...
BibTeX
@misc{zarembaconjectureframework,
  author = {Humphreys, Cahlen},
  title = {Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + MOW Spectral Theory (Not Peer-Reviewed, Known Gaps Remain)},
  year = {2026},
  url = {https://bigcompute.science/findings/zaremba-conjecture-framework/},
  note = {AI-audited (0 reviews). uncertified certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + MOW Spectral Theory (Not Peer-Reviewed, Known Gaps Remain). bigcompute.science.
  https://bigcompute.science/findings/zaremba-conjecture-framework/
Plain text
Cahlen Humphreys, "Zaremba's Conjecture (A=5): Proof Framework via GPU Verification + MOW Spectral Theory (Not Peer-Reviewed, Known Gaps Remain)," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/zaremba-conjecture-framework/
UNCERTIFIED _project-level
BibTeX
@misc{_projectlevel,
  author = {Humphreys, Cahlen},
  title = {_project-level},
  year = {2026},
  url = {https://bigcompute.science/findings/_project-level/},
  note = {AI-audited (1 reviews). uncertified certification. Not peer-reviewed.}
}
APA
Humphreys, C. (2026). _project-level. bigcompute.science.
  https://bigcompute.science/findings/_project-level/
Plain text
Cahlen Humphreys, "_project-level," bigcompute.science, 2026.
URL: https://bigcompute.science/findings/_project-level/

Cite the Data

Hugging Face Datasets
Humphreys, C. (2026). Zaremba Conjecture Data. Hugging Face.
  https://huggingface.co/datasets/cahlen/zaremba-conjecture-data

Humphreys, C. (2026). Kronecker Coefficients (S_20, S_30, S_40). Hugging Face.
  https://huggingface.co/datasets/cahlen/kronecker-coefficients

Humphreys, C. (2026). Continued Fraction Spectra. Hugging Face.
  https://huggingface.co/datasets/cahlen/continued-fraction-spectra

Guidance

Safe verbs: computed, archived, benchmarked, observed, fit, checked, verified, released
With caveats: suggests, appears, is consistent with, within the tested range, preliminary numerical evidence
Do not use: proved, established, confirmed (unless referring to bounded computation, not theorems)

All content CC BY 4.0. Human–AI collaborative work (Cahlen Humphreys + Claude, o3-pro, GPT-5.2, Grok). Not independently peer-reviewed.

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