T-156: $s(290)\le 17.5355339059327376\dots$$s(290)\le 17.5355339059327376\dots$, by monotonicity from T-155, with its own exact certificate

S1 V3 C3 Upper bound confirmed

2026-10-13 established · Levy after Daniel · $n=290$$n=290$

$s(290)\le 17.53553390593273762200422181052425$$s(290)\le 17.53553390593273762200422181052425$, which is 70142135623730950488016887242097/4000000000000000000000000000000 exactly, by monotonicity from T-155 (a packing of 291 unit squares contains a packing of 290): T-155 certifies $n=291$$n=291$ at exactly this side. It is also the exact side of a rational packing that this project made from the register's known-best witness at $n=290$$n=290$ with its grid-slack regularizer, its port of Daniel's coincidence placement, and decided here over Q, a certificate of the catalogue's own packing of 290 squares at that side. Measured here, it is the witness's packing.

The bound is not new. It has followed from T-155's certificate at $n=291$$n=291$ since 12 October 2026, when T-155 registered that count at this side; what this result adds is $n=290$$n=290$'s verified upper bound moved onto the side, and the second certificate, which is of $n=290$$n=290$'s own packing.

The side is the witness's own, the 36-digit rational promotion of the decimal witness at centre dilation 1. It lies 2.4e-15 below the verified upper bound the record held, 17.53553390593274 exactly, the side of Daniel's exact certificate of the same packing rounded up at the fourteen decimals the catalogue prints (T-101), and 1.8e-19 below that certificate's own side. It is now the exact form of the case's verified upper bound, which the case record still prints as 17.53553390593274.

This is not a new packing: it is the same construction certified at a tighter rational side. The catalogue gives the packing's side as 14 + (5/2)sqrt(2), an irrational number 4.8e-33 below the certificate's side, which no rational certificate reaches, so the ceiling still trails the report there.

The certificate was decided here on 13 October 2026, every pair and every wall over Q, by sqpack.witness.exact_verify and devtools.check_rational_witness_independent, which share no geometry or verification code with each other: valid under both, 41905 pair decisions each. These are the checkers the regularization run decided the view with, so the replay reproduces that decision with the producer's code.

Both refuse the certificate with its tightest pair pushed into overlap and with its box shrunk by one unit of its side's denominator, 2.5e-31: it touches a wall and a neighbour exactly, so that shrink is also the shrink past its least top or right clearance, and the three controls are two distinct mutants.

Measured against the witness's exact frame under the box's eight symmetries, the certificate pairs one to one with the witness by nearest centres, the largest move is 6.0e-31 and none turns by more than 1e-6 as a sine. It draws the same picture as the witness, so the atlas drawing does not change.

The packing is the one the Kingbird catalogue lists, as T-101 records it, and the regularization below names that result. The regularizer is this project's port of Daniel's coincidence placement (https://github.com/evand/square-packing, MIT), devtools.regularize_grid_slack; the run, the exact re-solve, the exact decisions and the registration are this project's. No optimality, local minimum or rigidity is claimed.