T-247: Kim's exact degree-21 side at 297, as reported, of the packing the record credits

S2 V3 C3 Upper bound confirmed

2026-10-25 published · Kim after Couzo, Daniel, PingYou Ltd, Levy · $n=297$$n=297$

Kim's comment of 25 October 2026 on issue 804 reports that the stationary side of the best-known packing at 297 is the only real root of an irreducible degree-21 integer polynomial, which the comment prints, so $s(297)\le 17.7404172875438018509758015948842895936406549234461952734252335613502680498177981816058180396096276933\dots$$s(297)\le 17.7404172875438018509758015948842895936406549234461952734252335613502680498177981816058180396096276933\dots$, and attaches an exact configuration at that side under CC0 1.0. The attachment could not be downloaded here; the author's public copy of 31 October serves the same bytes, which the packet keeps since 1 November 2026.

The author's configuration was decided here on 1 November 2026 on two exact routes (route B), so the bound is shown. sqpack's exact witness verifier, over the configuration's field of degree 21, decided every unit-square identity, wall and pair and refused a duplicated square and a square moved outside the box. Daniel's verify_exact.py, the checker the author reports running, run unchanged, printed VALID, with the 1,750 corner incidences it counts as touching (a square's corner on a wall or on a separating side) proved as identities, and refused four mutated controls. Before deciding, the file was held to the digests the comment states and its side to this entry's: the same polynomial, a root of it in the field, and the same real root.

This repository also built its own exact configuration at the same side from T-174's retained certificate (route A, devtools.build_field_configuration): the contacts and the side's stationarity solved to 2,600 digits, the field K of degree 21 generated by the band's half-angle tangent found by integer relation, and every centre and rotation solved exactly in K, where all 1,208 contact equations are identities and the comment's polynomial vanishes at the side. Its feasibility, every other pair and wall separated, is not yet decided exactly: sqpack.witness.exact_verify over K did not finish within the item's CPU cap.

The side lies below T-174's rational certificate, the case's house, by 3.359e-90, and below issue 639's degree-12 side by 6.624e-63, decided exactly. Credit Kim with the exact side, a refinement of the packing it solves, after that packing's finder and improvers as the case record credits them: Couzo's, as Daniel and PingYou Ltd improved it (T-174). The comment says Claude (Anthropic) did the computation, the construction and the checks at the author's direction, and that he reviewed and reran them.