News

Couzo’s exact rational packings of 204 and 260 unit squares, the two in the case corpus among issue 678’s 85 bounds, are confirmed at V3/C3 and are now the house packings at both counts. At 204 the packing lies about 2.04×10−5$2.04\times {10}^{-5}$ below the earlier ceiling; at 260 it lies about 2.56×10−5$2.56\times {10}^{-5}$ below de Winter’s packing, adopted there the day before. The 83 bounds beyond 324 are kept as dated rows of the source.

• This project’s filtered exact replay decided both certificates on two routes that share the parser, the half-angle map, the pair filter and rational arithmetic, and refused six controls of each, one of them 10−30${10}^{-30}$ past exact contact. The maintained all-pairs kernel decided every pair at both counts, the source’s own checker accepted both, and a separately prompted mathematical review, with a checker sharing only rational arithmetic, agreed at every number.
• Each is a new arrangement: 14 of 204 squares lie beyond a tenth of the earlier packing at 204, and 24 of 260 beyond a tenth of de Winter’s at 260. The source credits Polkovskyi’s construction at 204, and at 260 de Winter’s packing, which it seeded from Daniel’s exact form.
• The case at 204 draws a regularized view made here at the same side; at 260 the stronger views wait for the owner’s review. Daniel’s solved, regularized views of both packings lie a little lower still and are registered separately (T-225), pending review. No optimality, rigidity or priority is claimed.

PRs and supporting records: #706 · The case record at 204 · The case record at 260 · Retained certificates and verification limits · Mathematical review of 20 October · Couzo’s registration request.

De Winter’s exact rational packings of 227 and 260 unit squares, reported on 17 October, are confirmed at V3/C3 and are now the house packings at both counts. They lie about 7.27×10−3$7.27\times {10}^{-3}$ and 1.11×10−2$1.11\times {10}^{-2}$ below the earlier ceilings, 17/2+52​$17/2+5\sqrt{2}$ and 11+42​$11+4\sqrt{2}$, which the record no longer carries as conjectured optima. His certificate at 229 is confirmed too; Daniel’s smaller T-199 stays the house there.

• This project’s two maintained exact routes checked the three certificates, and both refused their duplicate-square and outside-container controls. They re-implement the geometry independently of the source’s code, though they share parsing, half-angle conversion and the separating-axis theorem. A separately prompted mathematical review decided them again on its own exact route and refused controls 10−30${10}^{-30}$ past contact. The source’s own checkers did not run here.
• Each case draws Daniel’s regularized view made here at the same side. It is a drawing, not a second bound.
• Smaller later reports exist at both counts, Couzo’s #672 at 227 and his #678 (T-223) at 260; each replaces de Winter’s packing there once it is validated. No optimality, rigidity or priority is claimed.

PRs and supporting records: #704 · The case record at 227 · The case record at 260 · Retained certificates and verification limits · Mathematical review of 20 October · De Winter’s registration request.

Daniel packs 183 unit squares in a square of side 13999580178736252502081470897571/1030<14$13999580178736252502081470897571/{10}^{30}<14$, about 4.20×10−4$4.20\times {10}^{-4}$ below the trivial $14\times 14$$14\times 14$ grid. It is the first packing of 183 squares below side 14 in this register. T-220 is confirmed at V3/C3 and is now the house at 183.

• This project’s two maintained exact routes checked the original certificate, and both refused its duplicate-square and outside-container controls. They re-implement the geometry independently of the source’s code, though the two share parsing, half-angle conversion and the separating-axis theorem with each other. A separately prompted mathematical review accepted the bound.
• The case draws a regularized view made here with Daniel’s regularizer at the same side. It is a drawing, not a second bound. The source’s own regularized certificate and its two checkers did not run here.
• Credit Daniel, after Couzo’s 182-square packing. The source’s KKT, flat-motion and local-minimum statements are numerical evidence that was not checked here, and no optimality, rigidity or priority is claimed.

PRs and supporting records: #687 · The case record at 183 · Retained certificate and verification limits · Mathematical review of 19 October · Daniel’s registration request.

Greff reports that 95 unit squares require a square of side 10, matching the grid construction: $s(95)=10$$s(95)=10$. T-191 remains a reported result at V0/C0, awaiting independent review and adoption.

• Greff constructed a weighted cover of points and line segments using Daniel’s method and checkers. He re-optimised its central cross and reused corners from Hosono’s 59-square cover and the band-insertion construction.
• The project reproduced the source’s two checker runs and retained their receipts. That reproduction has not yet been admitted as evidence with an independent review; the case’s lower bound remains $1993/200=9.965$$1993/200=9.965$.
• The source’s Lean wrapper assumes the covering statement. It has not been built here, and the reported result does not yet establish a completely checked Lean proof.

PRs and supporting records: #627 · Greff’s certificate and source checks · Retained report and verification limits · Greff’s registration request.

Hosono (wand125) reports $s(28)>1149/200=5.745$$s(28)>1149/200=5.745$ using a cover over every square position and angle. T-192 remains at V0/C0; complete local replay and review are pending.

• The certificate uses a new density of 884 weighted rectangles. It applies the continuous-angle method from Guzhou0806’s 40-square result T-145 at a new count, with credit to Guzhou0806, Tokoharu and Levy.
• The reported bound is $1/400$$1/400$ above Hosono’s earlier report T-142 and $1/100$$1/100$ above the standing lower bound $1147/200=5.735$$1147/200=5.735$ in T-107. The standing bound remains unchanged.
• Hosono reports complete checks by two source programs using the same covering method. The project checked exact premises and a diagnostic sample; those checks do not replace a complete replay or establish an independent method.

PRs and supporting records: #627 · Hosono’s certificate and source checks · Retained report and verification limits · Registered lower-bound reports.

Eighteen case records adopt smaller, exactly verified packings from the 14 October intake. Couzo, Ostashev, PingYou Ltd and Polkovskyi supplied the certificates; each case retains credit for earlier designs and refinements.

• The packings cover 105, 129, 146, 153, 154, 156, 176, 204, 207, 211, 237, 241, 261, 271, 272, 302, 303 and 304 squares. The case records give exact sides and certificate links.
• Independent checks and review raised 17 reported results to V3/C3. These establish that the packings fit; they do not prove optimality.
• Smaller certificates from PingYou Ltd’s T-174 await the project owner’s decision on storage cost. Ten cases use the next eligible reviewed certificate. PingYou Ltd’s algebraic certificates for 70, 153 and 207 squares also await a separate owner decision.

PRs and supporting records: #614 · Case records · Reviewed result register · Review of the 14 October upper-bound reports · Review of PingYou Ltd’s PR 574 certificates.

The Explanations page pairs a short introduction to square packing with an advanced tutorial. Beginners can start with bounds and small examples; the companion develops linear programs, exact certificates, regularization and global proofs.

• Existing tutorial links lead to the moved explanations. The two tutorials retain the definitions, notation, worked derivations, figures and sources.
• News has its own page, with individual articles and filters for significance and topic. Reported proof announcements keep verification limits beside the claims.
• The Explanations index links the tutorials, research papers, downloadable PDFs, film and interactive Workbench. Both web Atlas views use the available screen width.

Published artifacts: Current basic tutorial · Current advanced tutorial · Current Explanations index.

Component editions: Site v0.7.0.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #604 · Tutorial preservation and editorial review.

Initial mathematical reviews examine Overbeck’s candidate proof and Fitzharding’s Lean formalization of the 17-square optimum. Both remain unverified proof reports at V0/C1. Sampled checks and component replays do not establish a complete local replay of either proof.

• The reviews distinguish a new mathematical result from a simplification or formalization. A simplification can still have its own proof-assistant check.
• A verification rating for an entire bound or optimum requires evidence covering the stated claim and every square count it includes.
• The review records identify each verifier’s authors, shared components, reading and replay coverage. An independent evaluator in the Fitzharding review agrees on ten native facts, covering only a sample of the full trust boundary.

Published artifacts: Current Overbeck proof report · Current Fitzharding formalization report · Current explainer, including the 17-square reports.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #592 · #595 · Fitzharding mathematical review · Overbeck mathematical review · Evidence and independence rules.

An intake ledger covers every open upper-bound report: 306 claims at 163 counts from 28 reports. It compares each claim with all earlier reports and recommends recording it only when its side is decisively smaller.

• Printed and exact sides are compared rigorously against a frozen record. Undecided comparisons remain visible for the owner to resolve.
• Retained mathematical reviews examine Fang’s direct certificates and excess theorem, and the fine-net, linear and closed-form lower bounds reported by Hosono (wand125).
• Each open result issue has a per-count verdict.

PRs and supporting records: #601 · The intake ledger · Ledger review · Fang theorem and direct-certificate review · wand125 closed-form review.

Fang reports a construction packing k2−k+1$k^2-k+1$ unit squares in a square of side less than $k$$k$, for every integer $k\ge 15$$k\ge 15$. Hosono reports a closed-form lower bound for every non-square count $n\ge 49$$n\ge 49$, raising 154 reported case bounds. The register adds these two families and 22 other results. Fang’s theorem has a mathematical review; Hosono’s complete formula replay remains unfinished.

• The 24 records, T-161 through T-184, preserve report order. No new packing is adopted in this registration.
• Couzo, Ostashev, PingYou Ltd and Polkovskyi supplied smaller packings and tighter certificates at 69 counts. Their source certificates were replayed; independent review and adoption remained pending at this date.
• Hosono also reports lower bounds at 37, 39, 44 and 145 through 149 squares. Greff reports an optimality proof for 76 squares. These reports retain their separate verification limits.
• Independently verified lower bounds are unchanged. Each case record distinguishes reported bounds from independently verified ones.

Published artifacts: Current Fang theorem record · Current closed-form lower-bound record.

Component editions: Site v0.7.1.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #602 · Current result register · The intake ledger.

Forty unit squares require a container side strictly greater than $6.718$$6.718$, following a complete replay of Guzhou0806’s certificate. Smaller exact packings replace the records at 154, 175, 232 and 264 squares. All four reviewed results are V3/C3; earlier packings remain in the history.

• The 40-square result rests on the producer’s kernels and complete continuous cover, with separate finite-step audits. A second independent continuous-cover method has not been established.
• Ostashev’s certificates verify packings of 175, 232 and 264 squares; Daniel’s verifies a packing of 154. They establish feasibility, without proving optimality.
• Calle’s certified 102-square layout is another arrangement of Xu’s design. A better packing was already recorded, so it changes no standing bound.
• The regularized drawings and methods paper include the adopted packings and their sources. The earlier solved 264-square drawing remains available as history.

Published artifacts: Current 40-square lower-bound record · Current Ostashev certificate record · Current Calle layout record · Current Daniel 154-square certificate record · Current methods survey · Current methods survey PDF.

Component editions: Site v0.6.1; How Record Square Packings Are Found v0.2.2.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #598 · Current result register · Certificate adoption workflow.

The public changelog links releases to papers, posters and result pages. Each paper and the site have separate edition histories; the poster pair shares one. Readers can identify which component changed.

• The site and both posters select regularized drawings by default, retain the originals for comparison and include compact attribution.
• A new edition of the lower-bounds explainer covers the reported 17-square proofs. The other papers keep their own editions.
• Each changelog entry has a stable link, source credit, verification limits and known release dates. The audit also accounts for internal work omitted from the public changelog.

Published artifacts: Current About changelog · Current 1–100 packing infographic · Current 1–324 packing poster · Current lower-bounds explainer.

Component editions: Site v0.6.0; Infographics v0.6.0; New Lower Bounds for Square Packing for n = 11 v0.4.6.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #590 · GitHub changelog · Component release rules.

The Atlas and both posters now show exactly checked, regularized versions of existing packings. Readers can compare them with the published originals. The workflow combines Daniel’s regularizer and exact solver with grid-slack cleanup and credits each source packing and modification.

• Reproducible tools re-solve at the same container side, close free grid slack and support comparison, selection and admission of drawings.
• T-155 records a 59-count exact-side batch at V3/C3. T-156 supplies an exact certificate for the 290-square ceiling already implied by T-155, adding no bound beyond that implication.
• The documented improvement rule selects 79 of 90 drawings as regularized defaults; the other 11 keep their house drawings. Regularized / Original controls and the case pages’ “As published” disclosures preserve the comparison.
• Both posters use the selected drawings and compact attribution. The records retain generator, modification and construction credits.

Published artifacts: Current atlas with Regularized / Original comparison · Current 1–100 packing infographic · Current 1–324 packing poster · Current exact-side batch record · Current 290-square exact certificate record.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #511 · #530 · #548 · #549 · #559 · #580 · #585 · #587 · #591 · Regularization policy · Regularization process.

A project audit checked the verification ratings of 27 recent results and examined the evidence behind Lean-backed claims. A separate review assessed 48 draft significance scores and corrected three. Numerical bounds and verification ratings are unchanged.

• The recent-results review replayed four upper-bound receipts and checked 36 certificates from seven entries on a third exact route. It found no rung that needed changing.
• The Lean review distinguishes a statement-closure build from a complete proof build. It identifies the reviews, retained executions and human oversight needed for higher ratings, without establishing a new full Lean replay.
• T-053 moves from S4 to S3, T-063 from S1 to S3 and T-065 from S3 to S2. A maintained announcement tool includes source credit and verification limits in result descriptions.

Published artifacts: Current results, including the corrected significance scores.

First public: 2026-10-14 (publication evidence).

PRs and supporting records: #563 · Lean-backed rung review · Recent-result rung review · Significance review · Announcement drafts and reply rules.

An audit of 92 packing records identifies gaps between illustrations and certified bounds: numerically checked drawings, exact drawings above the verified bound, and reported sides without matching exact certificates. The audit changes no drawing or bound.

• A recorded defect shows that numerical tolerances can distort reported minimum clearances. A small overlap can pass a numerical check at the printed side.
• The review lists the remaining checks and proposes requiring exact drawings at or below the verified upper bound.

The proposed requirement and missing certificates remain follow-up work; the documented gaps are still open.

PRs and supporting records: #581 · Verification-gap review.

Fitzharding reports a Lean 4 proof that Bidwell’s 1998 packing of 17 unit squares is optimal: $s(17)=4.67553009360455\dots$$s(17)=4.67553009360455\dots$. It is the first proof-assistant proof of this value in the project’s records and uses a different argument from Overbeck’s candidate proof. T-154 remains reported at V0/C1; the full Lean build has not been replayed here.

The source comparison traces 32 of the 35 patterns to the project’s certified sub-pattern ledger, including eleven from Hosono (wand125). The review distinguishes those mathematical sources from Fitzharding’s formalization and reported build.

Published artifacts: Current 17-square Lean proof report.

First public: 2026-10-13 (publication evidence).

PRs and supporting records: #588 · Source and premise review.

Overbeck reports that Bidwell’s packing attains the 17-square optimum, $s(17)=L$$s(17)=L$, where $L$$L$ is Bidwell’s container side. T-153 retains the source, review and sampled checks as an unverified proof report at V0/C1. The optimum remains unverified here pending complete capture and tree replays.

Sampled work covers 34 capture rows checked by the producer and independently, three producer tree replays and an independently isolated root. The full capture computation and ten remaining trees have not been replayed. These samples do not confirm the whole proof.

Published artifacts: Current 17-square candidate-proof record.

Publication evidence.

PRs and supporting records: #560 · Candidate-proof review.

A second project AI review independently re-derived the normalized 52-row gate in Overbeck’s candidate proof for 17 squares. It found no blocking defect in the parts examined and retained its checking programs. The complete proof remains unverified at V0/C1.

• The reviewer reconstructed gate coefficients independently of the producer’s code, checked 52 base-gap identities and audited 152 separating alternatives. The review records nine nonblocking findings and the scope left unread.
• The full capture computation and tree certificates were not replayed. The component checks leave the verified 17-square bounds unchanged.
• A separately prompted project AI agent conducted the review, blind to the first review until its findings were written. It is not a human oversight record.

PRs and supporting records: #577 · Second adversarial review and its dated corrections · Candidate proof record.

The case records and drawings adopt smaller certified packings for 42 square counts. Four results from PingYou Ltd, Ostashev and Barnett, T-147 through T-150 at V3/C3, join certificates from the preceding intake. Earlier packings remain in the history.

• The four results retain source packets, construction credit and exact checks, including a third checking method. They establish that the packings fit, without claiming optimality.
• Upper bounds and drawing sources change together. Private-worker custody checks and a repeatable adoption tool preserve earlier certificates.
• Guzhou0806’s reported $s(40)>3359/500$$s(40)>3359/500$ release is also retained, with finite-step audits and an independent near-axis check. The complete 43-slab kernel replay and result registration remain unfinished at this date.

Published artifacts: Current PingYou 207-square admission · Current PingYou 153-square admission · Current Ostashev admission · Current Barnett refinement admission · Current atlas of adopted packings.

Publication evidence.

PRs and supporting records: #512 · #516 · #518 · PingYou certificate review · Barnett and Ostashev certificate review.

Chaoweeraprasit’s SQUISH constructions and Couzo’s refinements improve nine standing upper bounds. Couzo’s refinements are at $n=304$$n=304$ and $305$$305$. T-151 and T-152 are V3/C3; the records also retain nine verified entries beyond the case corpus and preserve earlier bounds.

• Both maintained checking methods and a third exact method accept the certificates. Separate adversarial reviews examine them and record credit for the constructions they build on.
• Each adoption retains the certificate it replaces, so the current certificate and its dated history can be reproduced.

The certificates establish that the packings fit in the stated containers. They give upper bounds, without a lower-bound or optimality claim.

Published artifacts: Current SQUISH admission record · Current Couzo refinement record.

Publication evidence.

PRs and supporting records: #553 · SQUISH certificate review · Couzo certificate review.

Twelve upper-bound entries have complete verification and independent review at V3/C3, alongside Guzhou0806’s two 40-square lower bounds. The continuous-pose result becomes the standing 40-square lower bound. The upper bounds are adopted in a later release.

• Couzo, Daniel, Fang, Chaoweeraprasit, Deleeuw and Polkovskyi contributed the upper-bound work. It includes Daniel’s fifteen regularized certificates, checked for finite feasibility and reviewed before adoption. Three entries were registered in earlier PRs; this intake completes their reviews.
• Guzhou0806’s clipped-corner and continuous-pose certificates reach V3/C3. Both 40-square lower-bound records adopt the continuous-pose bound.
• Hosono (wand125) reports lower bounds at $n=28$$n=28$, $29$$29$, $30$$30$ and $122\le n\le 126$$122\le n\le 126$. These remain at V0/C0, with complete confirmation runs pending.
• Imports driven by source declarations, a third exact feasibility checker and typed records make unfinished confirmations and regularizations explicit beside each claim.

Published artifacts: Current 40-square continuous-pose admission · Current Daniel regularized-certificate admission · Current result register.

Publication evidence.

PRs and supporting records: #487 · Final lower-bound review · Final upper-bound review · Final asymptotic and grading review.

Shared navigation connects homepage packing previews with the Atlas, result records, papers and exports. Readable formulas and consistent layouts present the mathematical record alongside the drawings.

• The homepage previews the Atlas, recent results, papers, visuals and documentation.
• Case and result pages share diagram placement, navigation and responsive controls.
• Red indicators replace stars for recent results. Proof and bound status remain distinct.

Published artifacts: Overview · Atlas · Results · Papers and downloads.

Publication evidence.

PRs and supporting records: #462.

The project retains Daniel’s sources and exact local replays for certificates at 132 and 155 squares. T-131 is V0/C0 pending review and adoption. The 155-square certificate matches T-128’s exact side; comparing the poses establishes neither a connecting motion nor local-minimum equivalence.

Published artifacts: Current 132-square record-hunt record.

Publication evidence.

PRs and supporting records: #478 · Retained record-hunt source packet.

Couzo reports smaller sides at $n=375$$n=375$ and $378$$378$, with earlier source facts preserved as dated history. These printed-value comparisons lie beyond the 1–324 case corpus. The reports remain unreplayed and change no selected bound.

PRs and supporting records: #479 · Beyond-horizon source coverage.

Five attributed examples now illustrate construction, simulated annealing, packing surgery, sequential linear programming and exact algebraic certification in the methods survey. Updated explanations connect the figures to 32 selected upper-bound improvements.

The examples credit Trump, Schadt, Ellsworth, Casson, Chaoweeraprasit and Xu. The revision explains exact arithmetic for Xu’s 51-square witness, records later refinements and distinguishes local rigidity from optimality. It follows the survey’s 9 October publication.

Published artifacts: Current methods tutorial · Current methods tutorial PDF.

Component editions: How Record Square Packings Are Found v0.2.0.

Publication evidence.

PRs and supporting records: #477.

The web Atlas and redesigned 100- and 324-packing posters share legends and complete named credits. Triangle and Grid layouts and Fixed, Row and Global scales provide different views of the same packings on screen and in print.

• Both posters have matching SVG, PNG and PDF exports, compact complete rows and printable project addresses.
• The update corrects names and the 211-square drawing’s orientation, and keeps rigidity distinct from verification.

The published v0.5.0 edition remains available. The redesign establishes no mathematical result or proof.

Published artifacts: Current interactive atlas · Current 100-packing infographic PDF · Current 324-packing infographic PDF · Current 100-packing SVG · Current 324-packing SVG.

Publication evidence.

PRs and supporting records: #474.

Couzo reports packings at 20 counts from 327 to 379, beyond the project’s 1–324 case corpus. Printed sides, centers and angles are retained with dated source references for later checking.

• The counts are 327, 332, 335–342, 364, 369 and 372–379. No local geometry replay or exact certificate establishes the feasibility of these decimal packing reports here.
• The source compares them with border-row or grid constructions, rather than a survey of published best records. They do not enter selected-case bounds or the Atlas.
• Couzo’s subsequent smaller reports at 375 and 378 are recorded in the next update, with these earlier values preserved.

PRs and supporting records: #466 · Twenty reports and their source scope · Beyond-horizon source coverage.

Daniel’s three rational certificates give smaller verified packings at 266, 270 and 272 squares. Exact replays and a separate mathematical review support the upper bounds recorded as T-119.

• The verified side ceilings are $16.8230287507564760$$16.8230287507564760$, $16.9378072284460292$$16.9378072284460292$ and $16.9681101457696006$$16.9681101457696006$, respectively. The case records retain each complete exact pose.
• Each case and the source packet credit earlier constructions and refinements. Daniel discloses Claude assistance under his direction.
• Four exact-form reports and 178 configuration-specific local-minimum assertions are retained separately as T-120–T-124. The feasibility checks do not confirm those stronger assertions.

PRs and supporting records: #441 · Certificates, source credit and replay scope · Independent mathematical review.

Gupta refines Chaoweeraprasit’s SQUISH packings with Daniel’s optimizer, supplying 14 smaller selected upper bounds. All 17 submitted rational certificates pass exact verification, recorded as T-127 at V3/C3.

• The selected improvements are at 88, 130, 153, 154, 179, 180, 199, 207, 208, 209, 236, 237, 238 and 239 squares.
• Xu’s packings had already surpassed the other three submissions, at 108, 123 and 129 squares. Their complete certificates and comparisons remain available.
• The checks establish finite feasibility, without establishing local or global optimality.

PRs and supporting records: #448 · #459 · Refinement certificates and complete credit · Review of the 17 source cases.

Hosono (wand125) reports stronger lower bounds at 19, 20, 26, 27, 28, 29 and 31 squares, using measure certificates on finer direction nets. The sources and exact input-premise checks are retained; complete local geometric replays remain pending.

• The original reported bounds are $4.8275$$4.8275$, $4.9125$$4.9125$, $5.555$$5.555$, $5.6525$$5.6525$, $5.74$$5.74$, $5.815$$5.815$ and $5.97$$5.97$, in that order. A later report raises the 27-square value to $5.655$$5.655$.
• The reports use 832 or 2,073 declared directions. Input-premise checks do not establish the required geometric conclusion at every direction.
• Standing bounds and assurance ratings are unchanged. Later reports at 28 and 29 squares have their own entries and review history.

PRs and supporting records: #450 · Seven reports and the 27-square follow-up.

Twelve sub-pattern certificates from Hosono (wand125) pass complete replays and enter the 17-square research census. They reduce remaining symmetry classes from 4,683 to 3,636, leaving 28,528 states to consider.

• The standing Python verifier accepted all 12 certificates from a clean checkout. The Rust verifier agreed on all 26 compared receipt fields. Complete deciding objects and receipts are retained.
• The known endpoint survives, and the difficult distance-two subset falls from 95 symmetry classes to 94.
• The certificates reduce this finite search; they neither close the global proof nor improve the numerical lower bound.

PRs and supporting records: #475 · Experiment 317 and its acceptance evidence · Hosted certificate manifest.

The first edition of the square-packing methods survey explains how geometric seeds, numerical search, local refinement and exact certificates contribute to record packings. It is available as a web article, Markdown and PDF.

• Eight method families include annealing, basin hopping, packing surgery, sequential linear programming and contact-based refinement. Five construction lineages retain finder credit and source links.
• The survey distinguishes a reported search method from a certificate that proves a packing fits, and both from a proof of optimality.
• Version 0.1.0 has an 8 October evidence cutoff, covering retained releases through 7 October. The later illustrated edition is recorded separately.

PRs and supporting records: #458 · Methods survey · Initial publication.

Ryu reports quarter-power and cube-root lower bounds for the deficiency of packings of closed, pairwise-disjoint unit squares in an integer-sided square. The project retains the papers and a review of their geometric arguments and numerical constants.

• For a container of integer side $k$$k$, deficiency is k2$k^2$ minus the largest number of closed, pairwise-disjoint unit squares it can contain. The reported cube-root bound exceeds 0.06k1/3$0.06k^{1/3}$ for k≥3.5×1017$k\ge 3.5\times {10}^{17}$.
• Interval-arithmetic checks cover 108 numerical consequences, conditional on the stated geometric premises. Independent labeling of every box in an inherited computation remains open.
• The papers disclose extensive Claude assistance and no human peer review. Their claims remain reported and change no bound in the 1–324 case corpus.

PRs and supporting records: #439 · #440 · Asymptotic bound statements · Review of the quarter-power and cube-root arguments.

Xu’s 25 rational packing certificates and a separate 51-square construction pass exact checks and independent review. Seventeen rational certificates and the radical construction replace selected bounds at 18 counts.

• The 51-square packing fits at side (16+52​)/3$(16+5\sqrt{2})/3$. Its axis-aligned squares and squares rotated by 45∘${45}^{\circ }$ can be checked exactly over the number field Q(2​)$\mathbb{Q}(\sqrt{2})$.
• T-125 records feasibility of all 25 rational certificates, including superseded ones. T-126 records the undilated 51-square construction. Both are V3/C3.
• Xu describes an LLM-assisted workflow. The checks establish that the submitted packings fit, without proving optimality or reconstructing the search.

PRs and supporting records: #442 · Rational and radical certificates · Independent review.

Ahmed’s Astra-assisted proof establishes $s(11)=T$$s(11)=T$, where $T=3.8770835900228141773078970601\dots$$T=3.8770835900228141773078970601\dots$ is the exact algebraic side attained by Trump’s construction. It rules out every smaller square container, allowing each unit square to rotate independently and permitting boundary contact.

• The proof builds on work by Levy and Overbeck. A complete cover divides the problem into 2,184 cases, excludes 2,180 and handles the four symmetry-related survivors with an exact argument ruling out every side below $T$$T$.
• The project completed all 2,180 exclusions and ten capture nodes, with an adversarial AI review of their mathematical composition. The computations share arithmetic and construction components with the source; they do not constitute a second independent proof method.
• Status clarification, 1 October 2026: the revised review ladder rates T-060 at V3/C3. A second adversarial AI review and documented human oversight are still needed for V4/C4.

PRs and supporting records: #246 · Current eleven-square result · Current eleven-square optimality explainer · Ahmed’s original proof · Retained verification and reproduction limits · Mathematical audit and final acceptance · Revised verification and confirmation ladder.