\(s (105)\)$s(105)$ — open

External intake, 2026-10-02. wand125’s linear-certificate source reports \(s(101) \ge 257/25 = 10.28\)$s(101)\ge 257/25=10.28$ (T-073), from a measure of points, segments and rectangles accepted there at every net angle by its linear verifier, a checker the 2 October review read with no blocking defect. A packing of 105 squares contains one of 101, so the bound holds here by monotonicity, above Nagamochi’s \(10.2736184\ldots\)$10.2736184\dots$, which this case held until Karakuş’s finding made it a reported bound (see the correction below). Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, so it is also the verified lower bound here.

Open. The best known published packing, Francisco Couzo’s (T-056), gives \(s(105) \le 10.806077865540567\)$s(105)\le 10.806077865540567$, and the verified lower bound is \(s(105) \ge 257/25 = 10.28\)$s(105)\ge 257/25=10.28$, from wand125’s n101 linear certificate by monotonicity, leaving a gap of \(0.5261\)$0.5261$.

The packing

Francisco Couzo’s square-packing reports a packing of side \(10.806077865540567\)$10.806077865540567$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side \(10.807588532111675\)$10.807588532111675$, is dated 26 September 2026. 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 \(10.8060778655405674534\ldots\)$10.8060778655405674534\dots$, \(4.534 \times 10^{-16}\)$4.534\times {10}^{-16}$ 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 \(s(105) \le 10.806077865540568\)$s(105)\le 10.806077865540568$, the verified upper bound; 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 verified upper bound (interval route).

Griffin Casson’s square-packing, dated 23 September 2026 and made with the help of Claude as its README says, reports a packing of side \(10.80758853319602153\)$10.80758853319602153$ for this count, larger than Couzo’s by \(0.00151066765545453\)$0.00151066765545453$. Casson’s is the earlier of the two by the timestamps this record’s priority notes keep: Couzo’s first packing for this count, of side \(10.807588532111675\)$10.807588532111675$, is dated 26 September 2026. The record states both dates and infers nothing about whether either packing derives from the other.

The previous best known packing

Before this intake the best known packing was the UnitSquare Project’s, of side \(10.807847913867976\)$10.807847913867976$, below the Kingbird catalogue’s \(10.80789399144854\)$10.80789399144854$.

The UnitSquare Project’s 29 July 2026 release improves the public Schadt-Ellsworth parent by \(0.000046077580564\)$0.000046077580564$. 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 \(11 \times 11\)$11\times 11$ grid construction until this intake.

The lower bound

The verified field rests on the complete replay here of wand125’s linear certificate for 101 squares (E-n101-wand125-linear-1028-source-replay, V3/C3): all 201 directions on 2 October 2026, each returning the certificate’s own record. Its mass, \(10099999/100000\)$10099999/100000$, is below 105, so a packing of 105 at a side below \(257/25\)$257/25$ would capture at least 105 from it.

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below), and from then until this replay was recorded on 3 October it was 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.

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.