\(s (135)\)$s(135)$ — open

Open. The best known packing gives \(s(135) \le 12\)$s(135)\le 12$, and the strongest lower bound independently verified here is \(11.646748\)$11.646748$ from Karakuş’s general theorem, leaving a gap of \(0.3533\)$0.3533$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The packing

The record catalogue does not picture \(n = 135\)$n=135$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{135}\rceil = 12\)$\lceil \sqrt{135}\rceil =12$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which applies to every nonsquare \(N \ge 8\)$N\ge 8$:

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, \(s(N) \ge \min(\lceil\sqrt{N}\rceil, \sqrt{N - 2\lfloor\sqrt{N}\rfloor + 1} + 1)\)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, which is stronger at this \(n\)$n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

Source-reported bounds are recorded separately from this independently verified theorem.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.