\(s (172)\)$s(172)$ — open

Open. The best known published packing, Francisco Couzo’s at Evan Daniel’s exact optimum (T-098), gives \(s(172) \le 13.6189889568993984290\ldots\)$s(172)\le 13.6189889568993984290\dots$, 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

Evan 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 \(13.6189889568993984290\ldots\)$13.6189889568993984290\dots$, \(3.6 \times 10^{-11}\)$3.6\times {10}^{-11}$ below the side Francisco Couzo prints. Square for square, its pose lies within \(1.5 \times 10^{-3}\)$1.5\times {10}^{-3}$ of the binary64 pose the atlas pictures for this count. 81 squares move by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, 13 of them squares the source lists as carrying no force and the other 68 by at most \(2.1 \times 10^{-6}\)$2.1\times {10}^{-6}$; every other square moves by at most \(4.7 \times 10^{-11}\)$4.7\times {10}^{-11}$. 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. Evan 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 \(s(172) \le 13.6189889568993984290\ldots\)$s(172)\le 13.6189889568993984290\dots$, the verified upper bound; 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

Francisco 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 \(13.6189889569357306684\ldots\)$13.6189889569357306684\dots$, 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 Evan 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 Károly Hajba in 2024, via an unrecorded method. Improved by David Ellsworth in November 2024. Improved by David 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, \(s(N) \ge \min(\lceil\sqrt{N}\rceil, \sqrt{N - 2\lfloor\sqrt{N}\rfloor + 1} + 1)\)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, 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 \(6 \sqrt{26} / 85 + 2 \sqrt{2} + 851 / 85\)$6\sqrt{26}/85+2\sqrt{2}+851/85$ (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.