T-084: s(k2−1)=k$s(k^2-1)=k$ for every integer $k\ge 3$$k\ge 3$; $k=3\dots 18$$k=3\dots 18$ are the cases held here

S3 V3 C3 optimality confirmed

2026-09-29 published · Karakuş · 16 cases, $n=8$$n=8$ to 323

s(k^2 - 1) = k for every integer k >= 3: Karakuş 2026, Corollary 1.2. The lower half is his rectangle bound, Theorem 1.1, at a = b = k, through Corollary 6.1, and is the value of Corollary 6.2 at N = k^2 - 1; the upper half is the k x k grid with one square removed. k = 2, $s(3)=2$$s(3)=2$, is classical and is not this entry's.

This entry's scope is the cases this record holds, k = 3 to 18: $s(8)=3$$s(8)=3$, $s(15)=4$$s(15)=4$, $s(24)=5$$s(24)=5$, $s(35)=6$$s(35)=6$, $s(48)=7$$s(48)=7$, $s(63)=8$$s(63)=8$, $s(80)=9$$s(80)=9$, $s(99)=10$$s(99)=10$, $s(120)=11$$s(120)=11$, $s(143)=12$$s(143)=12$, $s(168)=13$$s(168)=13$, $s(195)=14$$s(195)=14$, $s(224)=15$$s(224)=15$, $s(255)=16$$s(255)=16$, $s(288)=17$$s(288)=17$ and $s(323)=18$$s(323)=18$. Nagamochi 2005 stated the family first (T-007); Nagamochi's Lemma 1, on which that proof rests, is false (T-085), and Karakuş's proof uses neither that lemma nor the paper's technical lemmas.

The proof was read in full and re-derived here. Its Proposition 5.1 is machine-checked here, independently re-implemented (devtools.check_karakus_strip_measure), by its own decomposition rather than the paper's Lemma 5.2. Hakan Karakuş, arXiv:2609.37410, 29 September 2026.