T-079: $s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted

V3 C3 lower bound confirmed superseded by T-095

2026-10-02 established · Levy after Daniel · $n = 12$

$s (12) \geq 15680000 / 3949423$ = 3.97020020..., by re-weighting Evan Daniel's 1,736 points. It raises the verified lower bound by 33774720/31204391123, about 0.0010824, over squarepacker's $s (12) \geq 31360 / 7901$ (T-078), and leaves a gap of 117692/3949423, about 0.0298, to the grid's 4.

The certificate keeps the points of Daniel's $s (12)$ certificate (T-049), every coordinate scaled by 3951000/3949423, and solves for new weights by linear programming over their 223 D4 orbits, with rows from the near-tight cells Daniel's verifier reports; the total is 14970347/1250000 = 11.9762776 < 12. Every closed unit square in [0, 15680000/3949423]^2, at every angle, captures weight at least one, decided over the rational angle net 2 arctan(k/96000) with Daniel's per-bin shrink.

It was decided here three ways on 2 and 3 October 2026: by Daniel's verifier, built unmodified with overflow checks on, in the producing run and again by a review lane that did not produce it, both VERIFIED over all 39,765 bins, least captured weight 10000045/10^7; and by this repository's parent-core interval branch and bound over all 39,765 rows, which shares no code with the sweep or with the LP and gives the strict $s (12) > 15680000 / 3949423$ . Both checkers refuse two mutated certificates.

This project's re-weighting, after Evan Daniel's certificate and verifier (T-049), whose method is Burns's and Massaccesi's, and after squarepacker's rescaling of that certificate (T-078), which showed a finer net leaves room for a larger container. The same work rescaled Daniel's certificate whole to $s (12) \geq 1568000 / 395039$ , which this bound implies.

Significance, composition and next rung

Significance: Raises the verified lower bound at $n = 12$ again, by 0.0016 over Daniel's T-049 and 0.0011 over T-078, to within 0.0298 of 4, the best verified bound at $n = 12$ this repository knows of. T-049, which raised it by 0.0086, is S3, and the step here is new weights on Daniel's geometry, not a rescaling alone. Not S4: the method, a weighted cover re-optimised at a finer net, is Daniel's, Burns's and Massaccesi's. Not S5: pure point covers are capped well short of 4, so the central question at $n = 12$ does not move.
Composition: Primary at $n = 12$ : one certificate, no monotonicity. Two methods decide it: Daniel's arrangement sweep (exact-algebraic), run by the producing lane and replayed by the review lane, and this repository's directed-rounding interval coverage of every row (interval-certified); that is C3, with two methods shown beside the rung. The exact audit of the file is a premise check, well-formedness only, and decides nothing. All rest on the certificate, the counting step and the shrink lemma on the net $N = 96000$ , and the native rows are the sweep's bins and sigma_k by design.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. Rung 5 needs a proof-assistant formalization reviewed by human experts; Daniel's Lean pose-box-tree checker, which kernel-checks T-049's certificate, could in principle take this one. The same 1,736 points carried the bound further: the LP total was 0.024 below 12 at this side, and squarepacker's re-weighting of them reached 7943/2000 (T-095). The exact value remains open; the conjecture is $s (12) = 4$ .
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The case

Case record

3.974

345

3.4644.464

\sqrt{n}

\sqrt{n} + 1

Proven

$3.97150$ $\leq s (12) \leq$ $4$

exact

Citation record n-012

lowersquarepacker after Daniel, Levy 2026, GitHub (confirmed T-095)

Open

optimality

The case record

LowerUpper

Verified

\frac{7943}{2000}

Reported

\frac{7943}{2000}

Gap

\frac{57}{2000}

= 0.0285

Results on the case

10 results in the register on $n = 12$ , oldest first, each with what it established and how it stands now.

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-095
Nagamochi · Nagamochi 2005 · source · register
2026-08-25 published T-049
$s (12) \geq 15680 / 3951 = 3.9686155 \dots$
V3 C3 lower bound confirmed superseded by T-095
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-04 established T-017
$s (12) \geq 99 / 25 = 3.96$
V3 C3 lower bound confirmed superseded by T-095
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
2026-09-29 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-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-095
Karakuş · Karakuş 2026 · source · register
2026-10-02 published T-078
$s (12) \geq 31360 / 7901 = 3.9691178 \dots$ , Daniel's certificate rescaled by $7902 / 7901$
V3 C3 lower bound confirmed superseded by T-095
squarepacker after Daniel · squarepacker s12 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-02 established T-079 this result
$s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted
V3 C3 lower bound confirmed superseded by T-095
Levy after Daniel · source 1 · source 2 · review · register
2026-10-05 published T-095
$s (12) \geq 7943 / 2000 = 3.9715$ , Daniel's points dilated and re-weighted
V3 C3 lower bound confirmed
squarepacker after Daniel, Levy · squarepacker s12 2026-10-05 · packet · packet · source 1 · source 2 · review 1 · review 2 · 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

T-079: s(12)≥15680000/3949423=3.9702002…s(12)≥15680000/3949423=3.9702002…, Daniel's points re-weighted

The case

Results on the case

Links

T-079: $s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted