$n=11$$n=11$

Bounds

Results in the register

• T-007 S3 V0 C1 Nagamochi · 2026-08-31 · 321 cases

  s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

• T-010 S4 V3 C3 Levy after Stromquist · 2026-08-31 · n = 11

  s(11)≥2+4/5​$s(11)\ge 2+4/\sqrt{5}$, by a repair of Stromquist 2003's Figure 14 point set

  Claim and records

  Claim

  s(11)≥2+4/5​$s(11)\ge 2+4/\sqrt{5}$, by a source-distinct repair of Stromquist 2003's Figure 14 point set: the replacement G' = (79/100, 37/20) restores the complete Figure 13 localization, A-triple forcing, repaired unavoidability, and 3+9 capacity chain, certified exactly. The printed argument does not close (exp-016) and is not relied on.

  Significance

  Resolves the proof status of the subject's flagship case: the $n=11$$n=11$ lower bound was stated in 1979 and cited as proved since, its printed argument was falsified here, and this restores it with a source-distinct repair.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1

  n = 11registerevidence 1

• T-011 S2 V3 C3 Trump · 2026-08-31 · n = 11

  Trump's 1979 packing is exactly valid, so $s(11)\le 3.877083590022814\dots$$s(11)\le 3.877083590022814\dots$

  Claim and records

  Claim

  Trump's 1979 packing is exactly valid: 11 unit squares in a square of side the published degree-8 algebraic number 3.877083590022814..., with 14 of 55 pairs in exact zero-separation contact and 20 corner coordinates exactly on the boundary, so $s(11)$$s(11)$ <= that side.

  Significance

  Confirms the record witness exactly, deciding the 14 zero-gap contacts no finite numerical evaluation can: real and citable, changing no theorem.

  Novelty

  previously-published Present in an identified source

  Records

  n = 11registerevidence 1

  n = 11registerevidence 1

• T-018 S5 V3 C3 Levy after Burns, Massaccesi · 2026-09-04 · n = 11

  $s(11)\ge 381/100=3.81$$s(11)\ge 381/100=3.81$

  Claim and records

  Claim

  $s(11)\ge 381/100$$s(11)\ge 381/100$, by this project's weighted fractional unavoidable-set certificate at container side 381/100 = 3.81. This improves Stromquist's 2 + 4/sqrt(5) = 3.788854..., stated in 1984 (Memo III, p. 10) and published in 2003.
  
  No intervening improvement was found by the recorded search, and $n=11$$n=11$ was then the smallest open case. The movement was +0.021146, and the case stayed open against Trump's 1979 packing at 3.877084: the interval narrowed from 0.088230 to 0.067084. Two rungs are retained below it -- 19/5, the value that first passed Stromquist, and the 189/50 calibration rung below him.

  Significance

  Scored against the rubric's own anchor for S5, "movement on a central open case". $s(11)$$s(11)$ is the smallest open case and a central one for this project; the recorded public search found no movement beyond Stromquist's value, stated in 1984 and published in 2003, and this displaces a bound from the refereed literature rather than an inherited or closed-form one.
  
  What S5 does not claim: the weighted-resource lineage predates this project and the pure-atomic direction-net architecture follows Burns and Massaccesi, so what is new is an instance and the generator that found it; apparent novelty is not absolute priority; and the remaining gap to Trump's 1979 packing is 0.067, which this approach does not close. The calibration rung at 189/50, below Stromquist's bound, was run first by design and is retained.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2

  n = 11registerevidence 1evidence 2

• T-022 S5 V3 C3 Levy after Burns, Massaccesi · 2026-09-06 · n = 11

  s(11)≥381008100042893309449​/899996306539=3.8100257…

  Claim and records

  Claim

  $s(11)\ge 38100$$s(11)\ge 38100$*sqrt(8100042893309449)/899996306539 = 3.810025723614703, proved by an exact dilation-limit corollary of T-018's retained certificate. For every positive rational q with q^2 below 810004289330944900000000/809993351783841654158521, common scaling preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side qL. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.

  Significance

  The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0000257236147034 beyond T-018. The movement derives from an exact support identity and unused strictness margin in an existing certificate; it is not a new certificate architecture and leaves a gap of about 0.0670578664 to Trump's packing.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3source

  n = 11registerevidence 1evidence 2evidence 3source

• T-023 S3 V3 C3 Levy · 2026-09-08 · n = 11

  Conditional exclusion: no eleven-square packing in the four-owner branch at $q=96/25$$q=96/25$

  Claim and records

  Claim

  At q = 96/25, if four distinct unit squares have selected strict cores of side B = 9977/10000 containing, respectively, the four closed rational patches in arms.endpoint.footprint_union of the retained exp143 receipt, at most five further unit squares fit. Thus the specified four-owner branch contains no eleven-square packing. The five saved rational dots meet every admissible remaining core on the complete 361-direction net; audited containment and boundary arguments transfer this exact check to arbitrary physical angles. This is a conditional exclusion and changes no global bound for $s(11)$$s(11)$.

  Significance

  A substantive case exclusion at a central open size, demonstrating that forced occupied area can make a five-dot counting argument sufficient. It excludes one owner combination and does not move the global bound.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3source

  n = 11registerevidence 1evidence 2evidence 3source

• T-024 S5 V3 C3 Levy after Burns, Massaccesi · 2026-09-09 · n = 11

  s(11)≥3175000518400042893309449​/598960960743657=3.8166095…

  Claim and records

  Claim

  $s(11)\ge 3175000$$s(11)\ge 3175000$*sqrt(518400042893309449)/598960960743657 = 3.816609502788862, proved by an exact dilation-limit corollary of T-018's retained atoms re-certified on a finer direction net.
  
  The 1121 atoms of the retained 381/100 certificate, on the same D4-symmetric sites and with every weight multiplied by 200000/198931, form a certificate of the retained form at side 381/100 on the 1440-step net with shrunken side 2494953/2500000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell mass exactly 1.
  
  For every positive rational q with q^2 below 3240000268083184056250000000000/3228788092454681596330191602841, common scaling preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 381/100. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.

  Significance

  The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0065838 beyond T-022 and by about 0.0066095 beyond the T-018 certificate side, for about 0.0277551 beyond Stromquist in total. The movement comes from the shrink that a finer net admits under Condition 4, measured on the frozen T-018 atoms; it is not a new certificate architecture, no atom was re-optimised, and the gap to Trump's packing is still about 0.0604741. The same measurement shows where this route ends: the exact depth-one ceiling family at 191/50 caps every one-body point certificate at unit side 3.8288, and these certificates sit about 0.011 below it.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

• T-025 S5 V3 C3 Levy · 2026-09-09 · n = 11

  $s(11)\ge 191/50=3.82$$s(11)\ge 191/50=3.82$, by a threshold certificate

  Claim and records

  Claim

  $s(11)\ge 191/50$$s(11)\ge 191/50$ = 3.82, by a threshold certificate: 584 point atoms of mass 271052551/31250000 and 320 threshold atoms, every one 2-of-3, of budget 143352577/62500000, on the D4-symmetric site set at shrunken side 9977/10000 and the 181-direction net.
  
  A threshold atom (S, k, w) charges w to every admissible core holding at least k points of S and costs w floor(|S| / k) of the budget, so the family's total budget is 685457679/62500000 = 10.967322864, strictly below 11, while every event cell at every net direction carries charge at least 100000203/100000000. The counting argument then rules out eleven unit squares with pairwise disjoint interiors in a container of side 191/50.

  Significance

  The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by 0.0033905 beyond T-024 and by 0.0311456 beyond Stromquist, leaving the interval [3.82, 3.877084] and a gap of 0.057084. What earns the score beyond the arithmetic is the architecture: the exact depth-one ceiling family retained at this same side proves that no D4-symmetric point-atom measure of mass below eleven exists here, so no certificate of the earlier one-body form could have reached it, and the threshold atoms carry 2.293641 of this budget that the point method cannot have. It is the first certificate of a new kind in this project, decided from frozen bytes by two routes and reviewed adversarially before registration.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2source

  n = 11registerevidence 1evidence 2source

• T-026 S5 V3 C3 Levy · 2026-09-09 · n = 11

  s(11)≥955000518400042893309449​/179696714646249=3.8264474…

  Claim and records

  Claim

  $s(11)\ge 955000$$s(11)\ge 955000$*sqrt(518400042893309449)/179696714646249 = 3.826447410572939, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on a finer direction net.
  
  The 584 point atoms and 320 threshold atoms of the retained 191/50 threshold certificate, on the same D4-symmetric sites and with every weight multiplied by 500000000/498684619, form a threshold certificate of the retained form at side 191/50 on the 1440-step net with shrunken side 249507/250000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell charge exactly 1. The rescaled total budget is 5483661432/498684619 = 10.996251384, still strictly below eleven.
  
  For every positive rational q with q^2 below 32400002680831840562500000000/32290909254655439869209770001, common scaling of the point atoms, the threshold atoms' points, the container side and the shrunken side preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 191/50. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.

  Significance

  The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0064474 beyond T-025's 191/50 and by about 0.0375930 beyond Stromquist's 2 + 4/sqrt(5), leaving the interval [3.826447410572939, 3.877084] and a bound gap of about 0.0506362.
  
  The movement is the shrink a finer net admits under Condition 4, measured on the frozen T-025 atoms; no atom was re-optimised and no new theorem was needed, since Conditions 1, 1', 2' and 3 are untouched by common scaling and Condition 5' is preserved by inverse dilation of placements exactly as Condition 5 is.
  
  For this fixed certificate, the rescaled budget is 5483661432/498684619 = 10.996251384 against the eleven a certificate may not reach, leaving about 0.0037 of budget. The measured 720- and 1440-step nets share the same crossing shrink and least unscaled charge. These finite measurements leave the effect of another net, a different core side, re-optimized weights, changed sites, and richer charge atoms unresolved.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

• T-031 S2 V3 C3 Levy · 2026-09-20 · n = 11

  The octagon corner class (threshold $1/2$$1/2$) holds no eleven-square packing at side $96/25$$96/25$

  Claim and records

  Claim

  At L = 96/25 and B = 9977/10000 on the 181-direction net (half-tangents k*207107/90000000, k = 0..180), the D4-symmetric point measure of total mass 10868617/1000000 = 10.868617 in cases/n11_corner_class_certificate/certificate.json, the retained exp-220 covering, charges at least 2000013/2000000 to every closed B-square at a net direction whose minimum of x + y is at least 1/2 in each of the four corner frames, decided by the exact event-cell sweep and by the interval branch and bound, which agree at that value.
  
  Consequently every packing of eleven unit squares in [0, 96/25]^2 contains a unit square whose minimum of x + y in some corner frame is strictly below 1/2, that is, one that meets the open corner triangle x + y < 1/2 at that corner; equivalently, the all-free (octagon) class of lane-a Theorem B at threshold 1/2 contains no eleven-square packing at side 96/25.
  
  This is a conditional exclusion of one corner-bin class and changes no bound on $s(11)$$s(11)$; the all-deep class is separately known to be outside this language's reach (BC-366), so the corner tree cannot close at 96/25 by clipping alone.

  Significance

  The first decided item of the corner-conditioned point language at a central open size: one D4 class of the corner tree is excluded at 3.84 by a certificate on a clipped domain, with the structural consequence that every eleven-square packing in a square of side 96/25 has a square meeting an open corner triangle of depth 1/2. It moves no bound, and the registration review shows the tree it belongs to cannot close at this side in the point language alone, since the all-deep class is a Lemma-D theorem for every site set (BC-366); the mixed classes are a bounded follow-up.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2

  n = 11registerevidence 1evidence 2

• T-033 S3 V3 C3 Levy · 2026-09-22 · n = 11

  s(11)≥9550002073600042893309449​/359341754646249=3.8269975…

  Claim and records

  Claim

  $s(11)\ge 955000$$s(11)\ge 955000$*sqrt(2073600042893309449)/359341754646249 = 3.826997548829543624, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on the 2880-step direction net.
  
  The 584 point atoms and 320 threshold atoms of the retained 191/50 threshold certificate, on the same D4-symmetric sites and with every weight multiplied by 500000000/498684619, form a threshold certificate of the retained form at side 191/50 on the 2880-step net with shrunken side 249507/250000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell charge exactly 1. The rescaled total budget is 5483661432/498684619 = 10.996251384, still strictly below eleven.
  
  The shrink is T-026's and does not move under refinement; what this net changes is the largest half-gap tangent, 207107/1440000000 over the 2881 directions of the net, exactly half the 1440-step value, and the bound rises with it.
  
  For every positive rational q with q^2 below 129600002680831840562500000000/129126496632245014779129770001, common scaling of the point atoms, the threshold atoms' points, the container side and the shrunken side preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 191/50. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.
  
  What that conclusion does not carry is stated with it. No individual certificate is supplied at the supremum, because the sharpened containment test is equality there; no strict inequality is established there; and 191/50 remains the largest side at which the retained T-025/T-026/T-033 fixed-core family holds an individual-side certificate, which is T-025's and not this result's.

  Significance

  This is a substantive controlled case result and machine audit of this project's retained threshold family. It moves T-026 from 3.826447410572939744 to 3.826997548829543624 by halving D inside T-022's sharpened containment factor sqrt(1 + D^2)/(B(1 + D)), at T-026's own crossing shrink, which does not rise under refinement; Condition 4's own coarse test B(1 + D) < 1 has the lower ceiling 152800000000000/39926861627361 = 3.826997509248, which this value exceeds. The sharpened factor is what reaches; no atom was re-optimised and no new theorem was needed, since Conditions 1, 1', 2' and 3 are untouched by common scaling and Condition 5' is preserved by inverse dilation of placements exactly as Condition 5 is.
  
  The certificate is the same measure T-026 registered: the rescaled budget is unchanged at 5483661432/498684619 = 10.996251384, still strictly below the eleven a certificate may not reach, and only the net and the half-gap tangent differ.
  
  The unchanged family cannot reach the stronger current verified bound 31/8: its refinement ceiling 955000/249507 is about 3.82755, already below 3.875. The result therefore records controlled method and calibration evidence, rather than a current public-bound advance. Changed weights, sites, parent domains, and richer charge atoms remain unresolved.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

  n = 11registerevidence 1evidence 2evidence 3evidence 4evidence 5source

• T-035 S3 V3 C3 Levy · 2026-09-24 · n = 11

  Six-plus-five packings near Trump's tilt with side ≤Uhi​$\le U_{hi}$ lie within rho of his pose

  Claim and records

  Claim

  Every packing of eleven unit squares in a square container, six at orientation 0 and five sharing one orientation modulo pi/2 with half-tangent in [91442076901/250000000000, 73154061521/200000000000] (an interval containing [t* - 10^-6, t* + 10^-6] for Trump's exact half-tangent t* = 0.365769307604677...), whose side is at most the rational U_hi of the certificate header (U_hi - U = 2.03e-45), lies, after the quarter turn that puts its tilted centroid in the closed upper-right quadrant and the relabelling that orders each class by x + y/4, strictly within rho = 808514697/200000000000 of Trump's labelled image (rotation 1, labels [3,4,2,5,0,1,8,10,6,9,7]) in every centre coordinate, with every tilted orientation within 2.0e-6 radians of Trump's; every other packing in the family has side greater than U_hi.
  
  The statement is a machine-verified reduction, not optimality: a closed exact cell tree of 139,441,005 records, replayed in full by an independent reader, puts every small packing in the family near Trump's pose. It does not invoke the BC-240 local theorem. T-036 composes it with that theorem's first clause.

  Significance

  A substantive case result and machine audit at the smallest open case: one certificate tree, closed with no unresolved leaf, reduces a whole restricted family to a neighbourhood of Trump's pose. It moves no bound on $s(11)$$s(11)$, its family is one box of half-tangents 2e-6 wide, and its cost, about 1.7e8 nodes for that one box, argues against S4's reusable technique. The 2026-09-24 closed-tree review scored the composed theorem S3 on the same grounds.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1

  n = 11registerevidence 1

• T-036 S3 V3 C2 Levy · 2026-09-24 · n = 11

  Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry

  Claim and records

  Claim

  Every packing of eleven unit squares in a square container, six at orientation 0 and five sharing one orientation modulo pi/2 with half-tangent in [91442076901/250000000000, 73154061521/200000000000] (an interval containing [t* - 10^-6, t* + 10^-6] for Trump's exact half-tangent t*), has container side at least U = 3.877083590022814..., the exact side of Trump's packing, and a packing in the family has side exactly U only if it is a quarter-turn image of Trump's pose with the squares relabelled within the two classes.
  
  Reflections leave the family, since they send the tilt theta* to pi/2 - theta*, whose half-tangent 0.464 is outside the box, so that Z/4 x S6 x S5 orbit is the whole equality set. The statement is T-035's reduction composed with the first clause of the BC-240 local theorem at Trump's pose. It says nothing about any other tilt, about packings whose orientation classes are not six plus five, or about $s(11)$$s(11)$.

  Significance

  A substantive case result and machine audit: the first optimality statement with an equality case for a family containing Trump's packing. Stromquist 2003's bound for squares at 0 and 45 degrees is an earlier restricted-orientation statement, and it is not attained and its family does not contain Trump's packing. It moves no bound on $s(11)$$s(11)$, and its cost, about 1.7e8 nodes for one box, argues against S4's reusable technique.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1evidence 2evidence 3

  n = 11registerevidence 1evidence 2evidence 3

• T-037 S5 V3 C3 Overbeck after Levy, Guzhou0806, Mira-acc · 2026-09-29 · n = 11

  $s(11)>31/8=3.875$$s(11)>31/8=3.875$

  Claim and records

  Claim

  $s(11)>31/8$$s(11)>31/8$ = 3.875, by Overbeck's 11-squares-certified-bound v1.0.2 release of 22 September 2026. It raised the case's verified lower bound by 0.048002 over T-033 and leaves 0.0020836 to Trump's 3.8770835900.
  
  The certificate is an exact nonnegative D4-invariant system of point and k-of-m threshold charges over 12,028 contiguous parent-angle intervals, each with a closed core strictly inside every parent of the interval, parents of side 764/775 in a container of side 191/50. Eleven disjoint cores would carry at least 11 x 999962528 against a budget of 10999479944, which they exceed by 107864 integer units, and compactness makes the bound strict.
  
  It was replayed here in full by the source's two exact event-cell sweeps, in Python and JavaScript, and decided a second time by this repository's native parent-core interval branch and bound (devtools.verify_kleddamag_n11_native), which accepted all 12,028 rows with no stalled box.
  
  Overbeck, 11-squares-certified-bound, building on Squares Project (Levy): the release's ATTRIBUTION.md says the work builds on this project and that global-certificate.json was developed from T-026's certificate.

  Significance

  Scored at S5 by the rubric's anchor, movement on a central open case. $n=11$$n=11$ is a central open case of this project, and T-019 gave not being one as its first reason for staying below S5. The bound closes about 96 per cent of the gap the register held, from 0.0501 to 0.0021 below Trump's packing, and it is this project's T-026 certificate adopted and carried further outside it. The mathematics is Overbeck's; this repository's part is the replay and the method-distinct second decision.
  
  Confirmed on 11 October against its neighbours: T-018, T-022 and T-024 to T-026 are S5 for smaller movements at the same case, and T-061, the 3.9e-9 step above this bound, is S2 because it changes no stated digit of the gap; the step here is the largest in the case's history, and who made it does not enter the score.

  Novelty

  previously-published Present in an identified source

  Records

  n = 11registerevidence 1evidence 2evidence 3evidence 4source 1source 2review 1review 2

  n = 11registerevidence 1evidence 2evidence 3evidence 4source 1source 2review 1review 2

• T-047 S4 V3 C3 Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 7 cases

  $s(11)\ge 381/100$$s(11)\ge 381/100$; $s(n)\ge 1377/250$$s(n)\ge 1377/250$ for $n=26\dots 28$$n=26\dots 28$; $s(n)\ge 571/100$$s(n)\ge 571/100$ for $n=29\dots 31$$n=29\dots 31$

• T-058 S2 V3 C3 Hosono after Tokoharu, Daniel · 2026-09-29 · 100 cases

  Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

• T-059 S2 V3 C3 Hosono after Tokoharu, Daniel · 2026-09-29 · n = 11

  Reported equality of 12028 n11 row minima reproduced by a complete bound replay

  Claim and records

  Claim

  wand125/square-packing-tools reports that its exact general_pose_tree checker reproduces all 12028 per-row core minima of Overbeck's n11 certificate, release v1.0.2, with global row minimum 999962528 units and all witnesses replayed. This is the source's row-scan equality claim only, not a new proof of the global counting theorem or independent rectangle-density coverage.
  
  Hosono, square-packing-tools, 29 September 2026, building on Tokoharu's rectangle method, solver and verifier and on Daniel's point verifier. The source discloses Codex and Claude assistance.

  Significance

  A reusable independent row-checking implementation worth replaying; the existing n11 bound does not change. Confirmed on 11 October: a same-implementation replay of row minima of a certificate the record had already decided in full (T-037) is a citable detail; T-004 (S3) is a machine audit of a published proof that had none.

  Novelty

  previously-published Present in an identified source

  Records

  n = 11registerevidence 1evidence 2

  n = 11registerevidence 1evidence 2

• T-060 S5 V3 C3 Ahmed after Levy, Overbeck · 2026-09-29 · n = 11

  Trump's eleven-square packing is globally optimal

  Claim and records

  Claim

  s(11)=T=(6u+4)/(1+2u−u2)=3.8770835900228141773078970601…$s(11)=T=(6u+4)/(1+2u-u^2)=3.8770835900228141773078970601\dots$, where $u$$u$ is the unique root in $(9/25,37/100)$$(9/25,37/100)$ of 5u8−10u7−2u6+14u5+12u4−6u3+2u2+2u−1=0$5u^8-10u^7-2u^6+14u^5+12u^4-6u^3+2u^2+2u-1=0$. Independent rotations and boundary contact are allowed.
  
  The proof, published by Ahmed on 29 September 2026, is confirmed here by a re-implementation sharing components with the source -- complete repository geometric executions on kernels the source also bundles -- and a mapped mathematical audit.
  
  Ahmed, 11SquaresOptimal, Astra-assisted work building on Squares Project and Overbeck. The matching construction is Trump's.

  Significance

  Resolves global optimality for $n=11$$n=11$, a central case, by closing the gap to Trump's exact construction, beyond the previous strict T-037 lower bound.

  Novelty

  previously-published Present in an identified source

  Records

  n = 11registerevidence 1evidence 2evidence 3source 1source 2review 1review 2

  n = 11registerevidence 1evidence 2evidence 3source 1source 2review 1review 2

• T-061 S2 V3 C3 Wang, Li after Overbeck, Levy · 2026-09-30 · n = 11

  $s(11)>3875000000/999999999=3.875000003875\dots$$s(11)>3875000000/999999999=3.875000003875\dots$, 3.9e-9 above $31/8$$31/8$

  Claim and records

  Claim

  $s(11)>3875000000/999999999$$s(11)>3875000000/999999999$ = 3.875000003875000003875..., by Wang and Li's Zenodo record of 29 September 2026, reported on jlevy/squares#247. The step over 31/8 is exactly 31/7999999992, about 3.9e-9.
  
  The certificate is Overbeck's v1.0.2 global-certificate.json (T-037) with thirteen threshold-charge orbits raised, which puts the minimum core charge at 1000047559 on every one of the 12,028 parent-angle intervals, and with the parent and every core side scaled by 999999999/1000000000, so that the container over the parent is (31/8) times 1000000000/999999999. Eleven disjoint cores exceed the budget 11000095024 by 428125 integer units.
  
  One more step of the scale factor, 1 - 2*10^-9, already fails at row 615, so these data admit at most about twice the step. Scaling Overbeck's certificate alone already passes a full sweep, so the reweighting enlarges the surplus but is not needed for the bound.
  
  It was replayed here in full by the authors' two verifiers and by Overbeck's own Python and JavaScript event-cell sweeps, and decided a second time by this repository's native parent-core interval branch and bound, which accepted all 12,028 rows with no stalled box.
  
  Wang and Li, Zenodo record 23038546, building on Overbeck and this project. The record makes no statement about AI assistance.

  Significance

  It strictly raises the verified lower bound at a central case, which the S5 anchor names, but by 31/7999999992, about 3.9e-9, spending the slack of T-037's certificate after a reweighting; the release itself calls it a proof-of-slack result, and these data admit at most about twice the step. The case's gap to Trump's packing, 0.0020836, does not change at any printed digit, so it is scored as a citable detail rather than movement; T-042 is the precedent for scoring a true, fully replayed step by what it changes.
  
  Confirmed on 11 October against T-022 (S5), which also spends an existing certificate's slack at this case: T-022's step, 2.6e-5, changed the stated gap and proved the dilation-limit theorem that T-024, T-026 and T-033 reuse; this step changes no stated digit and proves nothing new, so it is S2 with T-078 and T-093, the other slack-spending steps. The line between them is what the step changes at the precision the record states the gap, not the case.

  Novelty

  previously-published Present in an identified source

  Records

  n = 11registerevidence 1evidence 2evidence 3source 1source 2review

  n = 11registerevidence 1evidence 2evidence 3source 1source 2review

• T-083 S3 V3 C3 Karakuş · 2026-10-02 · 301 cases

  s(n)≥1/2+n−⌊n​⌋+1/4​$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$$8\le n\le 324$

• T-085 S3 V3 C3 Karakuş; Vlasenko · 2026-10-02 · 315 cases

  Nagamochi 2005, Lemma 1 is false for every container with $a>3$$a>3$ and $b>2$$b>2$

• T-112 S3 V3 C2 Levy after Ahmed · 2026-10-06 · n = 11

  Trump's packing is the only optimal packing of eleven squares, up to symmetry

  Claim and records

  Claim

  Every packing of eleven unit squares in a square of side T = 3.8770835900228141773…, the least side T-060 proves possible, is Trump's 1979 packing after one of the eight symmetries of the container and a relabelling of the squares. Independent rotations and boundary contact are allowed. A quarter-turn reparametrization of one square is the same physical square and does not count as another packing. No symmetry of the container maps Trump's packing to itself, so there are exactly eight optimal packings as unlabelled configurations, the images of one another.
  
  This is a direct corollary of T-060 and adds no computation. T-060's proof embeds any packing of side S <= T concentrically in the rational cap U > T, where its 2,180 exclusions, the D4 reduction to case 438, the closed capture partition and pose inclusion all apply, and maps it rigidly into the fixed side-T frame, where the local isolation lemma forces the exact construction. For S < T the construction's span T is a contradiction, which is T-060. For S = T nothing is contradicted: the packing is the construction after the alignment, a D4 element composed with the case-438 quarter turn, which is this result.
  
  It is T-036's equality clause without T-036's restriction to six squares at angle 0 and five near Trump's tilt, and with reflections, which leave that family.

  Significance

  A substantive case result: it completes the classification at eleven squares, the optimum being a single rigid packing up to symmetry, with no sliding family and no second contact type at T, in the case T-060 settled. It moves no bound and adds no method, since it is T-060's argument read at equality, which is what an S4 would need; it is one step above T-036's restricted equality case.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 11registerevidence 1

  n = 11registerevidence 1

• T-124 S2 V0 C0 Daniel after Couzo · 2026-10-07 · 178 cases

  Reported non-strict local minima for 178 source configurations

In the frontier survey On GitHub