$n=105$$n=105$

$s(105)$$s(105)$ — open

Hosono’s closed form, 14 October 2026. Hosono’s closed form (wand125, 13 October 2026, T-180) states $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, from a measure of line segments on 34 intervals of the fractional part. At $n=105$$n=105$, $k=10$$k=10$ and interval 6 give $a(105)=10.354760059211635897156128812338\dots$$a(105)=10.354760059211635897156128812338\dots$, so the reported lower bound here is now $s(105)\ge 10.354760059211$$s(105)\ge 10.354760059211$, rounded down, above a smaller count’s bound by monotonicity, $10.28$$10.28$. The source replayed its certificate with its own checker; no record of it has been replayed here, so the verified lower bound is unchanged. The source states that its search, certificates, checkers and text were produced with AI assistance (Claude and Codex), under the direction of and reviewed by Hosono.

House adoption, 17 October 2026. PingYou Ltd’s tighter certificate of Couzo’s refinement of Xu’s packing (T-174) is now this case’s house: its exact certificate proves $s(105)\le 10.789303783748158831$$s(105)\le 10.789303783748158831$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.08×10−79$1.08\times {10}^{-79}$. The earlier ceiling, displayed outward as $10.78930378374815884$$10.78930378374815884$ from Xu’s packing, improved by Couzo and PingYou Ltd (T-170), applied until this adoption and stays retained as history (the certificate).

House adoption, 15 October 2026. PingYou Ltd’s tighter certificate of Couzo’s refinement of Xu’s packing (T-170) was this case’s house until 17 October 2026: its exact certificate proves $s(105)\le 10.78930378374815884$$s(105)\le 10.78930378374815884$, the reported and the verified upper bound, 1.08×10−19$1.08\times {10}^{-19}$ below the ceiling the case held until then, $10.7893037837481589$$10.7893037837481589$ from Xu’s packing, improved by Couzo (T-130), which stays retained as history (the certificate).

House adoption, 11 October 2026. Couzo’s refinement of Xu’s packing (T-130) was this case’s house until 15 October 2026: its exact certificate proves $s(105)\le 10.7893037837481589$$s(105)\le 10.7893037837481589$, the reported and the verified upper bound, 1.37×10−3$1.37\times {10}^{-3}$ below the ceiling the case held until then, $10.7906765754107907$$10.7906765754107907$ from Xu’s packing (T-125), which stays retained as history (the certificate).

External intake, 2026-10-02. Hosono’s linear-certificate source reports $s(101)\ge 257/25=10.28$$s(101)\ge 257/25=10.28$ (T-073), from a measure of points, segments and rectangles accepted there at every net angle by its linear verifier, a checker the 2 October review read with no blocking defect. A packing of 105 squares contains one of 101, so the bound holds here by monotonicity, above Nagamochi’s $10.2736184\dots$$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.

Before the 7 October refinement, the best known published packing, Couzo’s (T-056), reported the displayed side $10.806077865540567$$10.806077865540567$, and the verified lower bound is $s(105)\ge 257/25=10.28$$s(105)\ge 257/25=10.28$, from Hosono’s n101 linear certificate by monotonicity, leaving a gap of $0.5261$$0.5261$.

The packing

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\dots$$10.8060778655405674534\dots$, 4.534×10−16$4.534\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That replay established the earlier verified ceiling $10.806077865540568$$10.806077865540568$; 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).

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 Hosono’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)≥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.

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-117) at exact side $540303893275985231628745267522385760199372637/50000000000000000000000000000000000000000000$$540303893275985231628745267522385760199372637/50000000000000000000000000000000000000000000$, whose complete terminating decimal is $10.80607786551970463257490535044771520398745274$$10.80607786551970463257490535044771520398745274$. 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 Xu construction

Xu reports (T-125) a construction at exact side $53953382877053953382877/5000000000000000000000$$53953382877053953382877/5000000000000000000000$, displayed upward as $10.7906765754107907$$10.7906765754107907$. The pinned factual packet retains the entire source input and complete actual deciding outcomes.

The complete rational source certificate already includes the source’s stated dilation. No additional dilation or coordinate rounding is applied here.

Complete native replay, required full-roster controls and scoped independent review confirm finite feasibility at V3/C3. The source discloses LLM assistance. No global optimum, local-minimum theorem, rigidity or novelty is established. The source’s 191-touching-pair assertion for n51 is unconfirmed and is not admitted.

Francisco Couzo’s certificate (T-130)

Couzo’s refinement of Xu’s packing, reported in a comment on pull request #460, at exact side $10789303783748158831034729697921/1000000000000000000000000000000$$10789303783748158831034729697921/1000000000000000000000000000000$, displayed upward as $10.7893037837481589$$10.7893037837481589$ (T-130). The certificate gives a rational side and, for each of the 105 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. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The review of 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(105)\le 10.7893037837481589$$s(105)\le 10.7893037837481589$, the verified upper bound until 15 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The house it replaces, $10.7906765754107907$$10.7906765754107907$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $10.789303783751$$10.789303783751$, T-128 at $10.7906182681071446$$10.7906182681071446$ and T-117 at $10.8060778655197047$$10.8060778655197047$, stay verified and are not selected.

PingYou Ltd’s certificate (T-170)

PingYou Ltd’s tighter certificate of Couzo’s refinement of Xu’s packing, reported on jlevy/squares#568, at an exact rational side, displayed upward as $10.78930378374815884$$10.78930378374815884$ (T-170). The certificate gives a rational side and, for each of the 105 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(105)\le 10.78930378374815884$$s(105)\le 10.78930378374815884$, the verified upper bound until 17 October 2026: 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-130 records, measured here pose by pose as that packing (T-170’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The issue states that computation, checking and submission preparation were AI-assisted; the pinned source makes no statement about AI assistance.

The house it replaces, $10.7893037837481589$$10.7893037837481589$ from Xu’s packing, improved by Couzo (T-130), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $10.789303783751$$10.789303783751$, T-128 at $10.7906182681071446$$10.7906182681071446$, T-125 at $10.7906765754107907$$10.7906765754107907$ and T-117 at $10.8060778655197047$$10.8060778655197047$, stay verified and are not selected. T-174 at $10.7893037837481589$$10.7893037837481589$, 1.08×10−79$1.08\times {10}^{-79}$ below this house, 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.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s tighter certificate of Couzo’s refinement of Xu’s packing, reported in pull request #574, at an exact rational side, displayed upward as $10.789303783748158831$$10.789303783748158831$ (T-174). The certificate gives a rational side and, for each of the 105 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(105)\le 10.789303783748158831$$s(105)\le 10.789303783748158831$, 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-130 records, measured here pose by pose as that packing (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, $10.78930378374815884$$10.78930378374815884$ from Xu’s packing, improved by Couzo and PingYou Ltd (T-170), stays retained with its own replay as history. The other certificates confirmed here at this count, T-130 at $10.7893037837481589$$10.7893037837481589$, T-138 at $10.789303783751$$10.789303783751$, T-128 at $10.7906182681071446$$10.7906182681071446$, T-125 at $10.7906765754107907$$10.7906765754107907$ and T-117 at $10.8060778655197047$$10.8060778655197047$, stay verified and are not selected.