The values
| n | Continuous optimum | Grid maximum |
|---|---|---|
| 1 | √3/6 | 0.288675134 |
| 2 | 2√3/9 | 0.384900179 |
| 3 | 3(√3−1)/4 | 0.549038104 |
| 4 | 0.617524884780866… | 0.617524883 |
| 5 | 0.693904526840437… | 0.693904525 |
| 6 | 6−3√3 | 0.803847575 |
| 7 | 0.855687593470088… | 0.855687591 |
| 8 | 0.912454563520543… | 0.912454561 |
| 9 | 0.976596443373822… | 0.976596440 |
| 10 | 5(3−√3)/6 | 1.056624324 |
| 11 | 1.098475709724551… | 1.098475706 |
| 15 | (60−15√3)/26 | 1.308432222 |
1. Continuous optimum
Some optimal packing has every gap closed: among the maximisers, the one farthest from the origin cannot move along any direction that keeps its contacts, so it has 3n independent contacts, and Euler's formula makes the contacts (with the three walls as vertices) a triangulation. Each triangulation has at most one packing, fixed by the angle sums around each circle and by the side length. So the continuous optimum is the best of finitely many packings.
For n ≤ 3 this is settled by hand. For n = 4–11 every triangulation is enumerated (16, 78, 457, 2 938, 20 118, 144 113, 1 065 328 and 8 068 332 of them). For n = 15 a power-cell argument shows that any arrangement other than the equal-circle triangular array falls short.
2. The nine-decimal grid
Every packing that would beat the record lies within a few grid units of the optimal packing: an explicit inequality forbids it from leaving a small ball, and a connected set cannot jump across. Inside those boxes, every integer point is excluded by a branch tree whose leaves carry exact rational certificates.
Credit and verification
zzzcy #308 submitted the proofs by email on 4 October 2026, stating that they were written with AI assistance. MinMax Arena checked every claim with its own code, reading only the submitted data and never running the submitted scripts: our own enumeration of the triangulations, and our own replay of every certificate. This adopted contribution earns one permanent +2 proof award.
The grid maxima equal the current records, so these rows close with their holders unchanged. The same package also proves the continuous optima for n = 21, 28, 36 and 45; those rows stay open for now, since their grid scores can still rise.