T-148: A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

S3 V3 C3 Upper bound confirmed superseded by T-173

2026-10-11 published · PingYou Ltd after Xu, Daniel, Levy · $n=153$$n=153$

One complete rational source certificate by PingYou Ltd, reported on jlevy/squares#504, proves a finite upper bound at the exact side it states: $s(153)\le 6435006925203522001385040447/500000000000000000000000000$ = 12.870013850407044002770080894.

The certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in SQUISH's JSON rows; its side and centres already carry the guard dilation 5000000000001/5000000000000 it states, and every square keeps side 1.

The author reports that three exact Fraction checkers of its own, by vertex projections, by centre and support-radius separating axes and by rational polygon clipping, accept all 11,628 pairs and every wall; that its reconstruction script rebuilds the certificate exactly from an explicit rational construction table and recomputes the guard; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept its corners Witness/v2.

It reports that a SymPy script checks the symbolic identity behind the chosen angle and isolates the root of u^7-9u^6+3u^5-u^4-u^3+u^2-3u+1, and that the construction table comes from an exploratory selection and a floating-point solve that the release does not repeat. Those statements are the author's; no source program ran here.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 11 October 2026 by both maintained exact routes, from an exact fact derived from the pinned certificate, and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided it with six controls and found the fact to be the packing the pinned certificate and the pinned witness state. The source ran the two maintained routes on its own witness, but neither is its generator's code, and it did not run the third route, so the replay is independently re-implemented. The 11 October review found no blocking defect.

The side was 9.67e-03 below the ceiling the case held (T-127), and is below Daniel's T-146 by exactly 0.002015998674135970992348937968, the figure the issue gives, and below SQUISH's T-136 by 2.02e-03. On 11 October 2026 the case record adopted this entry as the house at 153, its reported and verified upper bounds moving together, T-127's certificate kept as history and T-146 and T-136 verified and not selected; private-worker custody (RV-6) was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) for the n153 construction, as the source asks: by its account, Xu's n70 construction (issue 432) and its reflection, composed by the boundary-patch approach of the source's n207 (issue 503, T-147) with a different choice of surviving squares and contacts and reclosed at a new angle, giving 121 upright squares and tilted populations of 17 and 15.

The n70 geometry it started from is Daniel's exact n70 certificate (https://github.com/evand/square-packing, the regularized n-070 certificate T-146 retains), credited to him for its exact retention. The source credits the previous n153 packing to Chaoweeraprasit (SQUISH, issue 481) and its refined exact certificate to Daniel (T-146), as comparisons, and claims neither the n70 packing nor patch composition in general.

That lineage is the source's account of its construction and was not measured by pose: no placement of the n70 lands more than 31 of its squares within 1e-9 of a square of this packing, nor more than one exactly, which fits the source's account of contacts reclosed at a new angle.

The source and the issue make no statement about AI assistance; the issue omits the template's section on it. The same author published it on 11 October 2026, minutes after the n207 release of issue 503 (T-147), under the same workflow. That release as first published, and its issue as first posted, said that the verification, the comparison review and the package preparation were assisted by OpenAI Codex; minutes later a revision titled "Correct attribution in submission materials" removed that sentence, and the issue was edited to match, with no reason given. Asked on issue 503 which account stands for both releases, the author answered there on 12 October 2026: "Both solutions were AI assisted by 'PY', an unreleased agentic AI offering by PingYou Ltd."