$n=172$$n=172$

$s(172)$$s(172)$ — open

House adoption, 17 October 2026. PingYou Ltd’s refinement of Daniel’s exact optimum of Couzo’s packing (T-174) is now this case’s house: its exact certificate proves $s(172)\le 13.6189889568993984289$$s(172)\le 13.6189889568993984289$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.36×10−19$1.36\times {10}^{-19}$. The earlier ceiling, displayed outward as $13.61898895689939842899969151892$$13.61898895689939842899969151892$ from Couzo’s packing, improved by Daniel (T-098), applied until this adoption and stays retained as history (the certificate).

Held for the owner, 15 October 2026; lifted 16 October 2026. T-174 at $13.6189889568993985$$13.6189889568993985$, 1.36×10−19$1.36\times {10}^{-19}$ below this case’s verified ceiling, is confirmed here and accepted by the review of 14 October 2026. From 15 October 2026 until the owner’s house rule of 16 October 2026 lifted the hold, PR #574’s houses were held for the owner, since their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker’s snapshot.

Open. Previously, the best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(172)\le 13.6189889568993984290\dots$$s(172)\le 13.6189889568993984290\dots$, and the strongest lower bound independently verified here is $13.119429$$13.119429$ from Karakuş’s general theorem, leaving a gap of $0.4996$$0.4996$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The exact optimum

Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 172 squares in a square of side $13.6189889568993984290\dots$$13.6189889568993984290\dots$, 3.6×10−11$3.6\times {10}^{-11}$ below the side Couzo prints. Square for square, its pose lies within 1.5×10−3$1.5\times {10}^{-3}$ of the binary64 pose the atlas pictures for this count. 81 squares move by more than 1×10−8$1\times {10}^{-8}$, 13 of them squares the source lists as carrying no force and the other 68 by at most 2.1×10−6$2.1\times {10}^{-6}$; every other square moves by at most 4.7×10−11$4.7\times {10}^{-11}$. The known-best witness this record lists is that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(172)\le 13.6189889568993984290\dots$$s(172)\le 13.6189889568993984290\dots$, the verified upper bound until 17 October 2026; it says nothing about optimality. The source reports the exact point as a KKT local minimum: multipliers that keep it in equilibrium, a reduced Hessian positive definite once its exact flat motions are set aside (its per-count report), and no first-order descent across corner-to-corner contacts, each computed numerically at that point; none of that is verified here, and it bears on this packing alone, not on $s(172)$$s(172)$. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The packing

Couzo’s square-packing reports a packing of side $13.618988956935731$$13.618988956935731$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). The repository names no method and no tolerance, and itself states no AI assistance; its author said on issue #227 that he found the 102 and 103 packings “with the help of Claude”.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side $13.6189889569357306684\dots$$13.6189889569357306684\dots$, no larger than the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That proves $s(172)\le 13.618988956935731$$s(172)\le 13.618988956935731$, which was the verified upper bound until Daniel’s exact optimum above replaced it; it says nothing about optimality. Interval arithmetic on the printed pose itself, with each angle’s true cosine and sine and no rational rounding, decides every pair and wall again and gives the same bound (interval route).

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side $13.61898898660160$$13.61898898660160$.

Found by Hajba in 2024, via an unrecorded method. Improved by Ellsworth in November 2024. Improved by Ellsworth in December 2024. Its side length is algebraic of degree 8 over $\mathbb{Q}$$\mathbb{Q}$.

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which applies to every nonsquare $N\ge 8$$N\ge 8$:

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, s(N)≥min(⌈N​⌉,N−2⌊N​⌋+1​+1)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, which is stronger at this $n$$n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

Source-reported bounds are recorded separately from this independently verified theorem.

The reported lower-bound expression for $s(172)$$s(172)$ is 626​/85+22​+851/85$6\sqrt{26}/85+2\sqrt{2}+851/85$ (approximately $13.20012261982$$13.20012261982$). Friedman’s DS7 survey, Theorem 9, k=13, reports this bound at n=170; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at k=13 that point pattern leaves a unit square empty (devtools.check_green_ds7). Inherited at n=172 by monotonicity. This is a source-reported lower bound; it does not replace the verified bound above. The exact specialization and comparison are retained by audit_ds7_lower_bounds.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Daniel’s exact optimum of Couzo’s packing, reported in pull request #574, at an exact rational side, displayed upward as $13.6189889568993984289$$13.6189889568993984289$ (T-174). The certificate gives a rational side and, for each of the 172 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. The source ran both maintained routes itself on its corners witnesses; those runs are its own, and the replay here is independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(172)\le 13.6189889568993984289$$s(172)\le 13.6189889568993984289$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is PingYou Ltd’s tightening of the packing T-098 records, that packing by the source’s statement, not decided here (T-174’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by ‘PY’, an unreleased agentic AI offering by PingYou Ltd.

The house it replaces, $13.61898895689939842899969151892$$13.61898895689939842899969151892$ from Couzo’s packing, improved by Daniel (T-098), stays retained with its own replay as history.