T-244: Kim's exact algebraic sides at 172 and 263, as reported, of packings the record credits

S2 V3 C3 Upper bound confirmed

2026-10-25 published · Kim after Couzo, Chaoweeraprasit, Daniel, PingYou Ltd, Levy · $n=172,263$$n=172,263$

Kim's issue 804 reports an exact side for the best-known packing at 172 and at 263: each the packing's side at a stationary (KKT) point of minimizing the side, and a root of an integer polynomial irreducible over Q. At 172 it is the least of the four real roots of a degree-24 polynomial, and at 263 the only real root of a degree-17 polynomial, so $s(172)\le 13.6189889568993984288635016293504898995040150521961793814779549035630128682876859236484103034563728798\dots$$s(172)\le 13.6189889568993984288635016293504898995040150521961793814779549035630128682876859236484103034563728798\dots$ and $s(263)\le 16.7331660078993789705122814839596734462471472483871051710853314027161147491596783918785611086390994182\dots$$s(263)\le 16.7331660078993789705122814839596734462471472483871051710853314027161147491596783918785611086390994182\dots$.

The source determined each side numerically: it solved the KKT system on the active contacts to 3,000 digits from the binary64 packings of Daniel's atlas, and found each polynomial as an LLL relation at 1,500 digits, and says the nonnegativity of its multipliers is not certified. In a later comment the same day the author attached an exact configuration at each side: every centre and half-angle tangent in a number field K of degree 24 at 172 and 17 at 263 that holds the side.

Both configurations were decided here on 31 October 2026 on two exact routes, so both bounds are shown. sqpack's exact witness verifier, over each configuration's field, 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 here, printed VALID and refused four mutated controls at each count.

Before deciding, each configuration was held to the digest the comment states, and its side to this entry's: the same polynomial, a root of it in K, and the same real root.

Here SymPy and sqpack.field found each polynomial primitive and irreducible, and each side is the root the source's 100-digit print names, decided exactly. Each lies below T-174's rational certificate at its count, the case's house when registered, by 3.898e-89 at 172 and 2.126e-94 at 263. It also lies below the degree-12 side of Daniel's issue 639 there, by 1.185e-62 and 9.234e-63, which T-174 is already below. The atlas packings the source read are T-174's witnesses at both counts, to binary64 at every square. The ledger of 25 October for issue 804 records both in report order.

At 172 the house moved on 31 October 2026 to Daniel's issue 845 certificate (T-257), 1.053e-3 below this side, so the configuration there realizes T-174's retired packing; at 263 T-174's certificate is still the house.

Credit Kim with each exact side, a refinement of the packing it solves, after that packing's finders as the case records credit them: Couzo's at 172, as Daniel and PingYou Ltd improved it, and Chaoweeraprasit's (itsnaka) at 263, as Daniel and PingYou Ltd improved it (T-174). The issue says Claude (Anthropic) did the search for open cases, the pipeline, the computations and the verification script at the author's direction, and that he reviewed the outputs.