$n=199$$n=199$

$s(199)$$s(199)$ — open

Held for the owner, 15 October 2026; lifted 16 October 2026. T-174 at $14.6175721735928069$$14.6175721735928069$, 1.46×10−19$1.46\times {10}^{-19}$ 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. The case keeps its house until T-193’s adoption, staged under think-ot5c, moves it to T-193’s side, below T-174’s, once the writer takes an algebraic side. T-193’s exact algebraic side at this count, a root of a degree-16 integer polynomial 1.46×10−19$1.46\times {10}^{-19}$ below this case’s verified ceiling, is confirmed here and accepted by the review of 16 October 2026.

The second SQUISH update

Previously, Chaoweeraprasit, using SQUISH, reports this packing: $s(199)\le 14.6175721735980400$$s(199)\le 14.6175721735980400$, with exact side $8228961574259683/562949953421312$$8228961574259683/562949953421312$. Two independently implemented exact rational SAT routes accepted the complete packing and rejected both complete-roster controls. Two separately prompted AI reviews accepted the mathematical inputs and full receipt bindings. The author reports the seed as Kingbird’s s(37), grafted into n199’s record.

The revision-specific packet retains the source print $14.6175721735980400$$14.6175721735980400$ separately from the verified upward decimal ceiling. The exact side is unchanged; this certifies feasibility and an upper bound, without optimality, rigidity or human oversight claims. Earlier source geometry, certificates, credit and verification lanes remain historical evidence for their own packings.

Open. Previously, the best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(199)\le 14.6189889568993984291\dots$$s(199)\le 14.6189889568993984291\dots$, and the strongest lower bound independently verified here is $14.110657$$14.110657$ from Karakuş’s general theorem, leaving a gap of $0.5083$$0.5083$. 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 199 squares in a square of side $14.6189889568993984291\dots$$14.6189889568993984291\dots$, 7.8×10−12$7.8\times {10}^{-12}$ below the side Couzo prints. Square for square, its pose lies within 7.4×10−7$7.4\times {10}^{-7}$ of the binary64 pose the atlas pictured at that intake for this count. 3 squares move by more than 1×10−8$1\times {10}^{-8}$, none of them listed by the source as free, each by at most 7.4×10−7$7.4\times {10}^{-7}$; every other square moves by at most 5.5×10−9$5.5\times {10}^{-9}$. The known-best witness this record lists is that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s(199)\le 14.6189889568993984291\dots$$s(199)\le 14.6189889568993984291\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(199)$$s(199)$. 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 $14.618988956907218$$14.618988956907218$ 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 $14.6189889569072182592\dots$$14.6189889569072182592\dots$, 2.592×10−16$2.592\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\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(199)\le 14.618988956907219$$s(199)\le 14.618988956907219$, 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 $14.61898898660160$$14.61898898660160$.

Found by an unrecorded author, via extension of a smaller record. 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(199)$$s(199)$ is 267​/197+22​+2169/197$26\sqrt{7}/197+2\sqrt{2}+2169/197$ (approximately $14.18776486123$$14.18776486123$). Friedman’s DS7 survey, Theorem 9, k=14, reports this bound at n=197; 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=14 that point pattern leaves a unit square empty (devtools.check_green_ds7). Inherited at n=199 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.

The Gupta precision refinement

Gupta refines the earlier SQUISH construction at exact rational side $7308786086796403421851018497093/500000000000000000000000000000$$7308786086796403421851018497093/500000000000000000000000000000$, displayed upward as $14.6175721735928069$$14.6175721735928069$ (T-127). The complete factual packet retains every original certificate, complete comparator and actual deciding outcome for all seventeen source cases, including three withdrawals.

Complete dual-route native replay, full-roster controls and separately prompted source and production review confirm finite feasibility at V3/C3 using independently re-implemented deciding code. The repository routes share parsing, half-angle conversion, Fraction arithmetic and SAT methodology; the recorded code relation is to the source producer’s verification, not to each other. The full source poses are converted without dilation or rounding. The source credits Chaoweeraprasit for SQUISH and Daniel for the optimizer. No author program is executed here. Feasibility establishes no optimizer, local-minimum, rigidity, novelty, priority, global-optimality, formal proof or human-oversight assurance.