\(s (68)\)$s(68)$ — open

External intake, 2026-10-01. wand125’s rectangle-density source reports \(s (68) \ge 851 / 100 = 8.51\)$s(68)\ge 851/100=8.51$, with total mass \(6799/100 = 67.99 < 68\)$6799/100=67.99<68$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(1699/200 = 8.495\)$1699/200=8.495$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(6799/100 = 67.99 < 68\)$6799/100=67.99<68$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(423/50 = 8.46\)$423/50=8.46$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(6799/100 = 67.99 < 68\)$6799/100=67.99<68$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged.

Open. The best known published packing, Francisco Couzo’s at Evan Daniel’s exact optimum (T-098), gives \(s(68) \le 8.7987952372182839027\ldots\)$s(68)\le 8.7987952372182839027\dots$, and the strongest verified lower bound is \(851/100 = 8.51\)$851/100=8.51$, from wand125’s rectangle-density certificate of 1 October (T-074), leaving a gap of about \(0.2888\)$0.2888$. Before it the verified bound was \(1691/200 = 8.455\)$1691/200=8.455$, from wand125’s n67 rectangle-density certificate by monotonicity (T-070), and until 2026-10-02 \(841/100\)$841/100$, from the complete exact replay of wand125’s n69 point certificate and its stricter mass budget, which already excludes 68 squares. The exact optimum remains open.

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 68 squares in a square of side \(8.7987952372182839027\ldots\)$8.7987952372182839027\dots$, \(4.3 \times 10^{-12}\)$4.3\times {10}^{-12}$ below the side Francisco Couzo prints. Square for square, its pose lies within \(4.3 \times 10^{-4}\)$4.3\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 4 squares move by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, all of them squares the source lists as carrying no force; every other square moves by at most \(5.8 \times 10^{-12}\)$5.8\times {10}^{-12}$. 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 \(s(68) \le 8.7987952372182839027\ldots\)$s(68)\le 8.7987952372182839027\dots$, 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 \(s(68)\)$s(68)$. 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 \(8.798795237222592\)$8.798795237222592$ 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 \(8.7987952372225919633\ldots\)$8.7987952372225919633\dots$, 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 \(s(68) \le 8.798795237222592\)$s(68)\le 8.798795237222592$, 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 UnitSquare Project’s, of side \(8.803383074716108386903836683375989697670063329\)$8.803383074716108386903836683375989697670063329$, below the Kingbird catalogue’s \(8.80345993651653\)$8.80345993651653$.

The UnitSquare Project’s 29 July 2026 release improves the public Brendberg-Schadt-Ellsworth parent by \(0.0000768618004216131\)$0.0000768618004216131$. The release classifies the result as a construction-only upper bound and says it used outward-rounded interval arithmetic, a 300-digit zero-tolerance recomputation, and an independent published checker. The public release does not include the interval boxes, governed receipt, or replayable checker needed to inspect the formal claim. This repository recorded the value as reported. The formal lane held the exact \(9 \times 9\)$9\times 9$ grid construction until this intake.

The lower bound

The verified floor is a local deduction from wand125’s retained cert_n69_L841.json. The native exact verifier completed all 201 directions and all five certificate conditions successfully; its exact total mass is \(846701027/12500000 < 68\)$846701027/12500000<68$. The count enters only through the strict mass inequality, so replacing the source file’s target 69 by 68 changes no geometric premise. The remaining mass margin is \(3298973/12500000\)$3298973/12500000$. The replay log and mathematical review retain the evidence and derivation. This is a local consequence of the external certificate, with no claim of priority; the upstream README announces n69 rather than n68.

The selected literal external report, [Friedman DS7], gives the lower-bound expression \(2 \sqrt{2} + 71 / 13\)$2\sqrt{2}+71/13$ for \(s(68)\)$s(68)$ (approximately \(8.289965586285\)$8.289965586285$). Friedman’s DS7 survey, Theorem 9, k=8, reports this bound at n=65; 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 \(k = 8\)$k=8$ that point pattern leaves a unit square empty (review). Inherited at n=68 by monotonicity. That value remains in the reported field, separate from the stronger locally derived verified floor. The source audit compares the exact theorem expressions separately from opaque table decimals. Nagamochi’s general bound \(1 + \sqrt{53}\)$1+\sqrt{53}$ remains valid historical evidence and is superseded as this case’s strongest verified floor. Corrected 2 October 2026: its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false, so it is now a reported bound (review). This register had recorded that proof as verified, its own error, logged as defect D-516.

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.