$n=82$$n=82$

$s(82)$$s(82)$ — open

External intake, 2026-10-02 (afternoon). Hosono’s linear-certificate source reports $s(82)\ge 233/25=9.32$$s(82)\ge 233/25=9.32$ (T-076), from a measure of points, segments and rectangles of total mass $8199999/100000<82$$8199999/100000<82$ on the support of its $n=83$$n=83$ linear certificate scaled by $932/935$$932/935$, accepted there at every net angle by the linear verifier the 2 October reviews read with no blocking defect. It is above Green’s reported $9.2667335\dots$$9.2667335\dots$, the value this case reported before it, by $0.0532664\dots$$0.0532664\dots$, and does not rest on Green’s argument, which jlevy/squares#308 reports the published material does not establish for $k\ge 4$$k\ge 4$. Its complete replay here on 3 October 2026 returned the certificate’s own record at all 201 directions, so it is also the verified lower bound. Hosono’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives $s(82)\le 9.53553391$$s(82)\le 9.53553391$, and the verified lower bound is $s(82)\ge 233/25=9.32$$s(82)\ge 233/25=9.32$, from Hosono’s linear certificate, leaving a gap of $0.2155$$0.2155$.

The packing

Found by Göbel in 1979, via a hand construction: the $n=65$$n=65$ centred-block family pose with one L of seventeen unit squares around it, exactly as [Friedman DS7] section 2 states. cases/gobel82 builds it and verifies it exactly over Q(sqrt 2), which is what moved verified_upper_bound from the grid ceiling onto 6+(5/2)2​$6+(5/2)\sqrt{2}$. The retained witness declares that side — the exact value correctly rounded up at its thirty-two digits — but a different layout: its centre set matches none of the construction’s eight dihedral images, so the certificate is about the construction, not the witness’s drawing.

The lower bound

Before Hosono’s certificate above, the selected external report was [Friedman DS7], which gives the lower-bound expression 942​/41+247/41$94\sqrt{2}/41+247/41$ for $s(82)$$s(82)$ (approximately $9.266733533246$$9.266733533246$). Friedman’s DS7 survey, Theorem 9, k=9, reports this bound at n=82; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at $k=9$$k=9$ that point pattern leaves a unit square empty (review). That changed the reported source field only; the verified field now holds Hosono’s replayed linear certificate. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete replay here (E-n082-wand125-linear-932-source-replay, V3/C3) of Hosono’s linear certificate: the source’s unchanged checker and replay functions on the pinned bundle, all 201 directions on 3 October 2026, each returning the certificate’s own record. It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way.

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below), and from then until this replay it was 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)≥min(⌈N​⌉,N−2⌊N​⌋+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.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

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.