T-008: $s (46) = 7$

V3 C3 optimality confirmed

2010-09-13 published · Bentz · $n = 46$

$s (46) = 7$ : the lower half by T-004's audited unavoidable set, the upper half by the exact $7 \times 7$ grid packing of 46 squares.

Significance, composition and next rung

Significance: The first equality from the literature whose both halves are machine-confirmed here end-to-end.
Composition: Compound: lower = T-004 (exact-algebraic audit), upper = the grid bound replayed exactly by the basic-bounds checker. Under epistemics.md's rule for an equality whose upper half is an exactly replayed packing, the lower half sets the rung: it is machine-replayed here, so the equality is C3.
Next rung: Rises with T-004: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A method-independent second confirmation of the lower half would be shown beside the rung.
Novelty: previously-published Present in an identified source

The case

Case record

678

6.7827.782

\sqrt{n}

\sqrt{n} + 1

Proven

$s (46) =$ $7$

optimal
exact

Citation record n-046

lowerBentz 2010, Electron. J. Combin. 17 (confirmed T-004, T-008)

The case record

LowerUpper

Verified77

Reported77

Gap0 solved: the verified bounds meet

Results on the case

8 results in the register on $n = 46$ , 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V0 C1 lower bound incomplete on this case, superseded by T-008
Nagamochi · Nagamochi 2005 · source · register
2010-09-13 published T-004
Bentz 2010, Theorem 8 ( $s (46) \geq 7$ ) is correct as printed, machine-audited in full
V3 C3 audit confirmed
Bentz · Bentz 2010 · source · register
2010-09-13 published T-008 this result
$s (46) = 7$
V3 C3 optimality confirmed
Bentz · Bentz 2010 · source · 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-008
Karakuş · Karakuş 2026 · source · register
2026-09-30 published T-064
$s (k^{2} - 3) = k$ for every integer $k$ from 6 up; $k = 6 \dots 18$ are the cases held here
V3 C3 optimality confirmed on this case, second certificate
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · 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-008: s(46)=7s(46)=7

The case

Results on the case

Links

T-008: $s (46) = 7$