Their table, re-decided to the last digit.

The platform accepts a circle packing if no two circles overlap by more than 10⁻⁹ and the bounding box's width plus height exceeds 2 by no more than 10⁻⁹. Read exactly, the Together packing overlaps in 44 of its 210 pairs — the worst by 3.09 × 10⁻12 in squared distance, pair 3–17 — and its box exceeds the perimeter by 4.20 × 10⁻14. AlphaEvolve's packing does neither: its worst pair clears by 5.04 × 10⁻9 and its box by 3.20 × 10⁻8, so it is an exact witness of 2.3658321334.

The repair is the smallest one the bytes allow: every radius scaled by one λ = 0.999999999963354125, the largest rational with λ²(rᵢ+rⱼ)² ≤ dᵢⱼ² on every pair and the box inside the perimeter. The exact witness then sums to 2.3658323758318174, a deficit of 8.67 × 10⁻11 against the published sum. The printed 2.3658323759 rounds the published sum, not the witness: the witness rounds to 2.3658323758. Ten digits were printed and the tenth is the tolerance's. The improvement over AlphaEvolve is unaffected — 2.42 × 10⁻7, decided between the two exact witnesses.

The other three repairs are float noise made visible. The platform requires a step function's values to sum to n/2 within 10⁻⁶; three of the four minimum-overlap constructions miss it by 2.41 × 10⁻15, 1.48 × 10⁻14, 9.99 × 10⁻16 (Haugland's 2016 function, with 51 steps written as short decimals, sums exactly). Renormalised to Σh = n/2 exactly, the three bounds move by at most 10⁻¹⁷ and every printed ceiling stands: TTT-Discover's 0.380876 and Together's 0.380871 are the ceilings of 0.3808753232 and 0.3808703105.

The two inequalities are the rows where exactness is more than bookkeeping. A step function's autoconvolution is piecewise linear with its breakpoints at multiples of the step, so the platform's discrete maximum is the true supremum of f∗f — the printed number is the bound itself, not an estimate of it. Together's 30,000-value function attains its maximum on a plateau: 10 indices lie within the screen's error bound (2.7e-11, relative) of the float maximum, and every one was decided in exact integers; the maximum sits at index 28,879 and C₁ = 1.5028628587049711, whose ceiling to eight digits is the printed 1.50286286. TTT-Discover's function gives 1.502862898255…, ceiling 1.50286290; the difference, 3.96 × 10⁻8, is real.

One file in that folder carries no printed number: AlphaEvolve V2's 1,319-value function decides to C₁ = 1.5031635546…, above the 30,000-value constructions by three parts in ten thousand — the repository ships it as a baseline and prints only the others.

The flat polynomial is the row where the platform's verifier is not the mathematics. It scores a ±1 polynomial by the largest |g| over a million equally spaced points on the unit circle — a lower bound on the supremum, however fine the grid. Here |g(z)|² on the circle is written as the integer cosine polynomial its autocorrelations define, reduced to a degree-69 polynomial in cos θ, and its maximum on [−1, 1] is certified by the same Sturm-chain and interval-Newton instrument that decides the Chowla cosine minima on this site: 43 critical points isolated and enclosed, the value at each bounded exactly, all in 383 ms. The supremum is C⁺ ∈ [1.280932052875041675, 1.280932052875041753]; the grid score the repository prints to sixteen digits, 1.2809320527987995, falls short of it by 7.62 × 10⁻11 — the grid missed the peak by that much — and the six printed digits, 1.280932, are the supremum's. AlphaEvolve's polynomial certifies to [1.3409252804007615, 1.3409252804007616] (its grid short by 9.45 × 10⁻10); the two enclosures are disjoint, so the improvement is decided at 6.00 × 10⁻2 or more.

The hexagon packings are the rows exact rationals cannot reach: each of the twelve unit hexagons is rotated by a published angle in degrees, so every vertex is a cosine and a sine of a decimal. They are decided instead in outward-rounded interval arithmetic with a certified π and certified sin and cos — every literal enters as the two doubles around the exact rational it denotes, and every operation widens outward — so a SEPARATED pair or an INSIDE vertex holds for the exact real configuration, not for a float reading of it. Together's packing: all 66 pairs certified separated by the separating-axis theorem, the closest (hexagons 4 and 7) by at least 3.62 × 10⁻8; all 72 vertices certified inside the container, the tightest with a cross product of at least 4.06 × 10⁻7. The platform declares a pair intersecting unless it is separated by more than 10⁻⁹ and forgives a vertex outside by up to 10⁻⁹; neither margin was needed. AlphaEvolve's packing certifies the same way (closest pair at least 6.93 × 10⁻6 apart), and the two outer sides are exact decimals, so the improvement, 2.60 × 10⁻4, is exact. The enclosures are 6.22 × 10⁻15 wide; a pair touching exactly would come back UNDECIDED at that width, never SEPARATED — the battery plants one.

Edges vs triangles is a benchmark score rather than a theorem: rows of twenty weights become (edge density, triangle density) points by the Newton identities, and the score is the area under the platform's slope-3 envelope plus ten times the largest gap in edge density. Recomputed exactly, both scores agree with the printed sixteen-digit values to fifteen digits — the sixteenth is float64 — and the improvement, 2.38 × 10⁻4, is the largest in the table because it is a score, not a bound.