T-061: $s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$

V3 C3 lower bound confirmed superseded by T-060

2026-09-29 published · Wang, Li after Kleddamag, Levy · $n = 11$

$s (11) > 3875000000 / 999999999$ = 3.875000003875000003875..., by Ke Wang and Can 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 Kleddamag'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 Kleddamag'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 Kleddamag'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.

Ke Wang and Can Li, Zenodo record 23038546, building on Kleddamag and this project. The record makes no statement about AI assistance.

Significance, composition and next rung

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.
Composition: Primary at $n = 11$ : one certificate and the bound L/A', with no monotonicity step. Two evidence entries whose methods differ both passed on the retained certificate, which is C3 with the two methods shown beside the rung: the event-cell sweeps (exact-algebraic: the authors' primary, Kleddamag's Python and JavaScript, and the authors' re-implementation are one method) and this repository's directed-rounding interval coverage of every row (interval-certified). Both rest on the certificate and on T-037's counting and transfer theorem, whose hypotheses were re-checked at the scaled sides.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the complete claim and a human oversight record; the 2026-09-30 review is a same-project read and is not recorded in this entry's reviews. Rung 5 needs a proof-assistant formalization of the transfer theorem and the counting argument, reviewed by human experts. The reweighting is not needed for the bound: Kleddamag's own weights at the scaled sides pass Kleddamag's full event-cell sweep with the same minimum 999962528 and surplus 107864 (the packet's scale-only receipt), so the scaling alone carries the step and the thirteen raised orbits only enlarge the surplus to 428125.
Novelty: previously-published Present in an identified source

The case

Case record

3.877

345

3.3174.317

\sqrt{n}

\sqrt{n} + 1

Proven

$s (11) =$ $3.877084$

optimal
exact
rigid

Citation record n-011

lowerAhmed after Levy, Kleddamag 2026, GitHub (confirmed T-060)

upperTrump 1979, Squares in Squares (confirmed T-011)

The case record

LowerUpper

Verified3.87708359…3.87708359…

Reported3.87708359…3.87708359…

Gap0 solved: the verified bounds meet

Results on the case

23 results in the register on $n = 11$ , oldest first, each with what it established and how it stands now.

1979 published T-011
Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$
V3 C3 upper bound confirmed
Trump · Trump 2023 · register
2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete on this case, superseded by T-060
Nagamochi · Nagamochi 2005 · source · register
2026-08-24 established T-010
$s (11) \geq 2 + 4 / \sqrt{5}$ , by a repair of Stromquist 2003's Figure 14 point set
V3 C3 lower bound confirmed superseded by T-060
Levy after Stromquist · register
2026-09-04 established T-018
$s (11) \geq 381 / 100 = 3.81$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 correction confirmed
Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-06 established T-022
$s (11) \geq 38100 \sqrt{8100042893309449} / 899996306539 = 3.8100257 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-08 established T-023
Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$
V3 C3 case exclusion confirmed superseded in part by T-060
Levy · source · register
2026-09-09 established T-024
$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-09 established T-025
$s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-09 established T-026
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-20 established T-031
The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$
V3 C3 case exclusion confirmed superseded by T-060
Levy · register
2026-09-22 established T-033
$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-22 published T-037
$s (11) > 31 / 8 = 3.875$
V3 C3 lower bound confirmed superseded by T-060
Kleddamag after Levy, Guzhou0806, Mira · Kleddamag n11 2026 · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-22 published T-047
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
V3 C3 lower bound confirmed superseded by T-060
Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register
2026-09-24 established T-035
Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within rho of his pose
V3 C3 case exclusion confirmed
Levy · register
2026-09-24 established T-036
Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry
V3 C2 restricted optimality confirmed superseded in part by T-060 and T-112
Levy · register
2026-09-29 published T-058
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
V3 C3 method limit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-059
Reported equality of 12028 n11 row minima reproduced by a complete bound replay
V3 C3 audit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-060
Trump's eleven-square packing is globally optimal
V3 C3 optimality confirmed
Ahmed after Levy, Kleddamag · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-29 published T-061 this result
$s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$
V3 C3 lower bound confirmed superseded by T-060
Wang, Li after Kleddamag, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed on this case, superseded by T-060
Karakuş · Karakuş 2026 · source · register
2026-10-06 established T-112
Trump's packing is the only optimal packing of eleven squares, up to symmetry
V3 C2 uniqueness confirmed
Levy after Ahmed · packet · register
2026-10-07 published T-124
Reported non-strict local minima for 178 source configurations
V0 C0 restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

Links

On this site: Case record, $n = 11$ · Frontier row, $n = 11$ · T-061 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-061 in results.yaml, line 5344
Evidence: E-n011-wang-li-report · E-n011-wang-li-source-replay · E-n011-wang-li-native-parent-core
Proofs and certificates: certificate n11_wang_li_zenodo_release_2026-09-29_stage10_doi_23038546-1.zip · proof n11_lower_bound_wang_li.pdf · audit review-2026-09-30-issue-247-wang-li-n11.md · proof parent_core.py
Sources: Wang Li n11 2026 (its own site, retained copy)
Source packet: resources/web/wang-li-n11-2026-09-29/README.md
Artifacts: 11 artifacts and controls
resources/web/wang-li-n11-2026-09-29/README.md resources/web/wang-li-n11-2026-09-29/receipts/certificate/diff.json resources/web/wang-li-n11-2026-09-29/receipts/authors/fresh-full-replay/RESULT.json resources/web/wang-li-n11-2026-09-29/receipts/kleddamag/full-replay/RESULT.json devtools/audit_wang_li_n11.py devtools/verify_n11_parent_core_native.py docs/project/reviews/review-2026-09-30-issue-247-wang-li-n11.md resources/web/wang-li-n11-2026-09-29/receipts/controls/source-checkers.json resources/web/wang-li-n11-2026-09-29/receipts/controls/native.json resources/web/wang-li-n11-2026-09-29/receipts/controls/cliff.json tests/test_wang_li_n11.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-061: s(11)>3875000000/999999999=3.875000003875…s(11)>3875000000/999999999=3.875000003875…, 3.9e-9 above 31/831/8

The case

Results on the case

Links

T-061: $s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$