$n=68$$n=68$

$s(68)$$s(68)$ — open

Fine-net validation, 19 October 2026 (T-205). The retained candidate mixed_n68_L853 raises both lower lanes from $851/100$$851/100$ to $853/100$$853/100$. The complete 201-direction maintained native replay and mathematical review confirm it by a re-implementation sharing components with the producer’s checker, Hosono’s adaptation of this repository’s sqverify_fast crate, apart from the declared-net readers. The exact audits check premises and saved poses only. The original 18 source reports and historical native receipts remain retained.

House adoption, 17 October 2026. PingYou Ltd’s refinement of Rehwaldt’s refinement of Couzo’s packing (T-174) is now this case’s house: its exact certificate proves $s(68)\le 8.7987952372182839025766809670213940835210190302278792632403696$$s(68)\le 8.7987952372182839025766809670213940835210190302278792632403696$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 4.58×10−61$4.58\times {10}^{-61}$. The earlier ceiling, displayed outward as $8.79879523721828390257668096702139408352101903022787926324037$$8.79879523721828390257668096702139408352101903022787926324037$ from Couzo’s packing, improved by Rehwaldt (T-118), 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 $8.7987952372182840$$8.7987952372182840$, 4.58×10−61$4.58\times {10}^{-61}$ 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.

External intake, 2026-10-01. Hosono’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. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. Hosono’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. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. Hosono’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.

Before the 7 October refinement, the best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(68)\le 8.7987952372182839027\dots$$s(68)\le 8.7987952372182839027\dots$, and the strongest verified lower bound then was $851/100=8.51$$851/100=8.51$, from Hosono’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 Hosono’s n67 rectangle-density certificate by monotonicity (T-070), and until 2026-10-02 $841/100$$841/100$, from the complete exact replay of Hosono’s n69 point certificate and its stricter mass budget, which already excludes 68 squares. The exact optimum remains open.

The earlier exact packing

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\dots$$8.7987952372182839027\dots$, 4.3×10−12$4.3\times {10}^{-12}$ below the side Couzo prints. Square for square, Daniel’s pose lay within 4.3×10−4$4.3\times {10}^{-4}$ of the binary64 pose the atlas pictured at that time. In that source comparison, 4 squares moved by more than 1×10−8$1\times {10}^{-8}$, all of them squares the source listed as carrying no force; every other square moved by at most 5.8×10−12$5.8\times {10}^{-12}$. The known-best witness listed then was that binary64 pose at its finder’s larger side; Daniel’s certificate witnessed the earlier ceiling above. The current known-best witness is the 7 October exact rational refinement. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That replay established the earlier verified ceiling $8.7987952372182839027\dots$$8.7987952372182839027\dots$; 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

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\dots$$8.7987952372225919633\dots$, no larger than the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\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 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 Hosono’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 22​+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+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.

The 7 October rational refinement

Rehwaldt reports the finite rational witness (T-118) at exact side $879879523721828390257668096702139408352101903022787926324037/100000000000000000000000000000000000000000000000000000000000$$879879523721828390257668096702139408352101903022787926324037/100000000000000000000000000000000000000000000000000000000000$, whose complete terminating decimal is $8.79879523721828390257668096702139408352101903022787926324037$$8.79879523721828390257668096702139408352101903022787926324037$. The pinned source refines Couzo’s existing construction with disclosed OpenAI Codex assistance. The complete finite witness is now exact-replayed here at V3/C3 after scoped review and public recovery of all original scientific evidence. Earlier ceilings and verification remain historical evidence for their own geometries. No unrestricted lower bound, optimality or rigidity claim is added. The earlier rigidity screen belongs to the earlier geometry and is preserved in the packet’s prior-state record; the current numerical translation screen is separate from the exact feasibility evidence.

The exact side is below Daniel’s retained rational ceiling even though an 18-place upward-rounded display is larger. The full rational and terminating decimal, rather than that rounded display, determine this comparison.

The v1.2 Exact-Root and Restricted-Family Report

Rehwaldt’s separate v1.2 packet attributes exact-root feasibility and attainment inside a specified contact family. Independent mathematical review and replay remain pending. The root packing is distinct from the finite rational witness above. Neither v1.2 statement changes the standing upper or lower lanes, and restricted-family attainment supplies no unrestricted optimum.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Rehwaldt’s refinement of Couzo’s packing, reported in pull request #574, at an exact rational side, displayed upward as $8.7987952372182839025766809670213940835210190302278792632403696$$8.7987952372182839025766809670213940835210190302278792632403696$ (T-174). The certificate gives a rational side and, for each of the 68 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(68)\le 8.7987952372182839025766809670213940835210190302278792632403696$$s(68)\le 8.7987952372182839025766809670213940835210190302278792632403696$, 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-118 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, $8.79879523721828390257668096702139408352101903022787926324037$$8.79879523721828390257668096702139408352101903022787926324037$ from Couzo’s packing, improved by Rehwaldt (T-118), stays retained with its own replay as history.