$n=108$$n=108$

s(108) — Square Packing Case

Fine-net lower bound, 18 October 2026. The certificate mixed_n108_L1062, published by Hosono on 16 October, raises both lower lanes to $s(108)\ge 531/50=10.62$$s(108)\ge 531/50=10.62$. Its exact nonnegative rectangle measure has total mass $10799999/100000<108$$10799999/100000<108$. The retained maintained sqverify-fast census verifies capture at every centre in all 201 declared-net directions, and its whole-net control receipt refuses the scaled mutants. The two accepted mathematical reviews check the certificate’s implication and premises; the separate exact audits check saved points. No native verifier was rerun during those reviews or this adoption. The producer’s checker and the maintained replay share sqverify_fast components, so this is shared component confirmation. It establishes neither independent full confirmation nor optimality. The upper lanes remain unchanged.

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=108$$n=108$, $k=10$$k=10$ and interval 23 give $a(108)=10.484093855851092636476980027521\dots$$a(108)=10.484093855851092636476980027521\dots$, so the reported lower bound here was $s(108)\ge 10.484093855851$$s(108)\ge 10.484093855851$, rounded down, above Nagamochi’s closed form, $10.43398113205$$10.43398113205$. The source replayed its certificate with its own checker; no record of it has been replayed here; it remains a historical reported bound. 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 refinement of Couzo’s refinement of Xu’s packing (T-174) is now this case’s house: its exact certificate proves $s(108)\le 10.90482101232035643$$s(108)\le 10.90482101232035643$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.09×10−19$1.09\times {10}^{-19}$. The earlier ceiling, displayed outward as $10.9048210123203565$$10.9048210123203565$ from Xu’s packing, improved by Couzo (T-128), applied until this adoption and stays retained as history (the certificate).

House adoption, 11 October 2026. Couzo’s refinement of Xu’s packing (T-128) was this case’s house until 17 October 2026: its exact certificate proves $s(108)\le 10.9048210123203565$$s(108)\le 10.9048210123203565$, the reported and the verified upper bound, 3.77×10−6$3.77\times {10}^{-6}$ below the ceiling the case held until then, $10.9048247851109049$$10.9048247851109049$ from Xu’s packing (T-125), which stays retained as history (the certificate).

The second SQUISH update

Previously, nate Chaoweeraprasit, using SQUISH, reports this packing: $s(108)\le 10.9099400734448775$$s(108)\le 10.9099400734448775$, with exact side $6141750256175099/562949953421312$$6141750256175099/562949953421312$. 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 SQUISH’s own s(110), two squares removed, then nearby search.

The revision-specific packet retains the source print $10.9099400734448775$$10.9099400734448775$ 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.

The Upper Bound

Previously, nate Chaoweeraprasit (itsnaka), using SQUISH, reports s(108)≤Sn​$s(108)\le S_n$, with Sn​=5629499534213126147784441267127​$S_n=\frac{6147784441267127}{562949953421312}$ in SQUISH’s ten-packing release. The certificate gives rational centres and half-angle parameters for 108 unit squares. The source’s original finite decimal display is $10.9206589394033085$$10.9206589394033085$; the rational fraction states the claimed bound exactly.

The new packing improves the earlier reported side $10.92591939016138$$10.92591939016138$ from [Kingbird]. This repository confirmed the upper bound by independently re-implemented exact-rational checks: native exact_verify and devtools.check_rational_witness_independent accept every pair and wall of the new packing and reject both geometric controls. The verified display is $10.9206589394033085$$10.9206589394033085$, the exact source side rounded upward at sixteen decimals. Both accepted reviews cover the shared half-angle conversion and the complete roster. The submitted certificates establish feasible upper bounds and make no optimality or rigidity claim. Separately, this repository’s numerical translation screen finds a nontrivial feasible motion in the retained configuration. That numerical assessment concerns this configuration alone; it does not settle the optimum or all packings at this side.

The author credits Ellsworth’s records and tools and states that SQUISH’s solver, verification and issue were written with Claude Opus 5.5 under his direction, management, review and steering. The source packet retains derived packing facts and source metadata.

Earlier Packing

Hajba found the earlier packing; Ellsworth improved it. Its reported side was $10.92591939016138$$10.92591939016138$. The earlier certified ceiling $10.92591939016139$$10.92591939016139$ was the verified upper bound before the SQUISH certificate (T-101), supported by Daniel’s certificate packet. This repository replayed Daniel’s two source checkers and independently decided the rational certificate with its own exact-rational routes; the packet records the shared premises and receipts. Any optimality or rigidity observations about that packing concern its own pose.

The lower bound

The current verified lower bound is $s(108)\ge 531/50=10.62$$s(108)\ge 531/50=10.62$, from Hosono’s fine-net measure and the maintained replay described above. That confirmation shares the producer’s verifier components.

Karakuş’s general bound supplied the earlier verified lower bound. It 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 stood above that earlier Karakuş bound and was 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 Xu construction

Xu reports (T-125) a construction at exact side $109048247851109048247851/10000000000000000000000$$109048247851109048247851/10000000000000000000000$, displayed upward as $10.9048247851109049$$10.9048247851109049$. 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-128)

Couzo’s refinement of Xu’s packing, reported on jlevy/squares#451, at exact side $5452410506160178214527814352111/500000000000000000000000000000$$5452410506160178214527814352111/500000000000000000000000000000$, displayed upward as $10.9048210123203565$$10.9048210123203565$ (T-128). The certificate gives a rational side and, for each of the 108 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(108)\le 10.9048210123203565$$s(108)\le 10.9048210123203565$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The house it replaces, $10.9048247851109049$$10.9048247851109049$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $10.904821012324$$10.904821012324$, T-127 at $10.9099400734438859$$10.9099400734438859$, T-116 at $10.9099400734448775$$10.9099400734448775$ and T-113 at $10.9206589394033085$$10.9206589394033085$, stay verified and are not selected. T-174 at $10.9048210123203565$$10.9048210123203565$, 1.09×10−19$1.09\times {10}^{-19}$ 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 refinement of Couzo’s refinement of Xu’s packing, reported in pull request #574, at an exact rational side, displayed upward as $10.90482101232035643$$10.90482101232035643$ (T-174). The certificate gives a rational side and, for each of the 108 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(108)\le 10.90482101232035643$$s(108)\le 10.90482101232035643$, 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-128 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, $10.9048210123203565$$10.9048210123203565$ from Xu’s packing, improved by Couzo (T-128), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $10.904821012324$$10.904821012324$, T-125 at $10.9048247851109049$$10.9048247851109049$, T-127 at $10.9099400734438859$$10.9099400734438859$, T-116 at $10.9099400734448775$$10.9099400734448775$ and T-113 at $10.9206589394033085$$10.9206589394033085$, stay verified and are not selected.