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Épi is Bake AI’s infrastructure for autonomous research. This collection contains 120 record-setting results across science and engineering: 20 numerical bound records and 100 construction records, plus one theorem extension.

20 Numerical bound records

100 Construction records

1 Theorem extension

The results span 21 problem families: 12 numerical-bound families, 8 construction families and the Tuza theorem extension.

Selected results

Problem	Previous result	Épi	Explore
Polyomino growth	≤ 4.5238	≤ 4.29569	Exact recurrence certificate
n-queens constant	Interval width 3.30 × 10⁻⁷	3.9367 × 10⁻⁹, about 84× narrower	Both endpoint witnesses
Water-network design	Global lower bound 655.5441707	≥ 662.80702839	Proof bundle
Cohn–Elkies method	Sharpness unresolved in dimensions 10 and 11	Non-sharp in both dimensions	Exact dual certificates

A theorem extension: Tuza’s inequality

Épi proved τ△(G) ≤ 2ν△(G) for a larger class of split graphs. The clique part has eight vertices; vertices in the independent part with at least two clique neighbors may now have three distinct neighborhoods, up from two. Each type can occur arbitrarily many times.

The proof combines a reduction with exact triangle covers and packings. The release includes the argument, its certificate and an independent checker.

Read the proof   ·   Inspect the certificate   ·   Run the checker

Explore all mathematical results

Area	Results
Growth and entropy	Klarner’s constant · Dimer constant · n-queens asymptotics
Combinatorics	Wellens’ bound · Spencer discrepancy · Ramsey c₄,₅ · Bₕ[g] coefficients · Tuza’s inequality
Geometry and dynamics	Cohn–Elkies bounds · Quartic Feigenbaum dimension
Optimization and information	MAX-4-CUT hardness · Water-network lower bound · Quantum-capacity thresholds

Full table of values and sources →

Constructions

34 pentagons Triangle	87 circles Quadrant	74 circles Semicircle

The figures are drawn from the released coordinates. Open a construction to see its value, comparison source and verification details.

Browse 100 construction records across 8 categories

Category	Records	Objective
Circles in a quadrant	46	Larger common radius
Balls in four dimensions	36	Larger common radius
Pentagons in a triangle	9	Smaller enclosing triangle
Circles in a semicircle	5	Larger common radius
Circles in an octagon	1	Larger common radius
Vehicle routing	1	Shorter feasible routes
Third autocorrelation inequality	1	Smaller signed-convolution objective
Zhang–Zagier essential minimum	1	Smaller certified upper bound

Downloads

Collection	Files
120 records + 1 theorem extension	Counts and result membership
100 construction records	CSV index · JSON index
Mathematical results	Result pages · JSON index
Large certificates	Release v1.1.0
File verification	Download sizes and SHA-256 · Repository checksums

Small witnesses live alongside their result pages. The larger certificates are Release downloads, including the n-queens upper witness and water-network proof bundle, keeping the repository lightweight.

Citation

@misc{epiteam2026results,
  author       = {{Epi Team}},
  title        = {{\'E}pi Results},
  year         = {2026},
  howpublished = {GitHub},
  url          = {https://github.com/BakeLab/Epi-results}
}

_{Comparisons last reviewed on September 11, 2026. If you know a stronger result, please contact us.}

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Bake AI · Épi
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