$n=84$$n=84$

$s(84)$$s(84)$ — open

Fine-net validation, 19 October 2026 (T-206). The retained candidate mixed_n84_L94225 raises both lower lanes from $9411/1000$$9411/1000$ to $3769/400$$3769/400$. The complete 201-direction maintained native replay and mathematical review confirm it by a re-implementation sharing components with the producer’s checker, Hosono’s adaptation of this repository’s sqverify_fast crate, apart from the declared-net readers. The exact audits check premises and saved poses only. The original 18 source reports and historical native receipts remain retained.

House adoption, 17 October 2026. PingYou Ltd’s refinement of Couzo’s refinement of Xu’s packing (T-174) is now this case’s house: its exact certificate proves $s(84)\le 9.69793479901492131$$s(84)\le 9.69793479901492131$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 9.70×10−20$9.70\times {10}^{-20}$. The earlier ceiling, displayed outward as $9.6979347990149214$$9.6979347990149214$ from Xu’s packing, improved by Couzo (T-130), applied until this adoption and stays retained as history (the certificate).

House adoption, 11 October 2026. Couzo’s refinement of Xu’s packing (T-130) was this case’s house until 17 October 2026: its exact certificate proves $s(84)\le 9.6979347990149214$$s(84)\le 9.6979347990149214$, the reported and the verified upper bound, 1.17×10−4$1.17\times {10}^{-4}$ below the ceiling the case held until then, $9.6980520605096981$$9.6980520605096981$ from Xu’s packing (T-125), which stays retained as history (the certificate).

External intake, 2026-10-05 (certificate of 5 October). Hosono’s mixed-certificate source reports $s(84)\ge 9411/1000=9.411$$s(84)\ge 9411/1000=9.411$ (T-094), from a density of 686 rectangles of total mass $8399999/100000<84$$8399999/100000<84$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the source’s $3763/400$$3763/400$ of the day before, by $0.0035$$0.0035$, and the verified $47/5$$47/5$ by $0.011$$0.011$. It supersedes the source’s mixed_n84_L94075 (T-090). sqverify-fast, this repository’s clean-room verifier, decided it here on 5 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-05 (certificate of 4 October). Hosono’s mixed-certificate source reports the mixed certificate mixed_n84_L94075 at side $3763/400=9.4075$$3763/400=9.4075$ (T-090), since superseded (T-094), from a density of 569 rectangles of total mass $8399999/100000<84$$8399999/100000<84$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the $47/5$$47/5$ this case reported, by $0.0075$$0.0075$. It supersedes the source’s mixed_n84_L940. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions: confirmed, independently re-implemented. The verified lower bound is unchanged, the certificate above being higher. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-02. Hosono’s mixed-certificate source reports the mixed certificate mixed_n84_L940 at side $47/5=9.4$$47/5=9.4$ (T-071), since superseded (T-090), from a density of 661 rectangles of total mass $8399999/100000<84$$8399999/100000<84$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above Green’s reported $9.2667335\dots$$9.2667335\dots$, which this case held by monotonicity, by more than $0.1332$$0.1332$. Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, so it was the verified lower bound until 5 October. Hosono’s README says parts of the work were produced with AI assistance under human direction.

Previously, the packing gave $s(84)\le 9.70710679$$s(84)\le 9.70710679$, and the verified lower bound then was $s(84)\ge 9411/1000=9.411$$s(84)\ge 9411/1000=9.411$, from Hosono’s mixed rectangle-density certificate of 5 October, leaving a gap of $0.2961$$0.2961$.

The packing

Found by Göbel in 1979, via a diagonal-strip construction — the strip family at $a=8$$a=8$, whose side is 9+2​/2$9+\sqrt{2}/2$ exactly. cases/gobel_strip builds it and verifies it exactly over Q(sqrt 2), which is what moved verified_upper_bound from the grid ceiling onto the exact side. The retained witness declares that side rounded down at its own digits — about 4.85×10−30$4.85\times {10}^{-30}$ below the exact value — and a different slack choice for the strip, so the certificate carries the construction’s coordinates rather than the witness’s (D-398).

The lower bound

Before 2 October 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(84)$$s(84)$ (approximately $9.266733533246$$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). Inherited at n=84 by monotonicity. That changed the reported source field only; the verified field now holds Hosono’s replayed mixed certificate. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete decision here (E-n084-wand125-mixed-9411-sqverify-fast-replay, V3/C3) of Hosono’s mixed certificate of 5 October by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 5 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here.

From 2 October 2026 until this decision the verified field rested on a complete replay here (E-n084-wand125-mixed-940-source-replay) of the source’s certificate mixed_n84_L940, at $47/5$$47/5$: the source’s unchanged checker and per-angle functions on the pinned bundle, all 201 directions, each returning the certificate’s own record.

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this $n$$n$, applies to every nonsquare $N\ge 8$$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.

The Xu construction

Xu reports (T-125) a construction at exact side $19396104121019396104121/2000000000000000000000$$19396104121019396104121/2000000000000000000000$, displayed upward as $9.6980520605096981$$9.6980520605096981$. The pinned factual packet retains the entire source input and complete actual deciding outcomes.

The complete rational source certificate already includes the source’s stated dilation. No additional dilation or coordinate rounding is applied here.

Complete native replay, required full-roster controls and scoped independent review confirm finite feasibility at V3/C3. The source discloses LLM assistance. No global optimum, local-minimum theorem, rigidity or novelty is established. The source’s 191-touching-pair assertion for n51 is unconfirmed and is not admitted.

Francisco Couzo’s certificate (T-130)

Couzo’s refinement of Xu’s packing, reported in a comment on pull request #460, at exact side $9697934799014921307163820128651/1000000000000000000000000000000$$9697934799014921307163820128651/1000000000000000000000000000000$, displayed upward as $9.6979347990149214$$9.6979347990149214$ (T-130). The certificate gives a rational side and, for each of the 84 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The review of 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(84)\le 9.6979347990149214$$s(84)\le 9.6979347990149214$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The house it replaces, $9.6980520605096981$$9.6980520605096981$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $9.697934799017$$9.697934799017$, stay verified and are not selected. T-174 at $9.6979347990149214$$9.6979347990149214$, 9.70×10−20$9.70\times {10}^{-20}$ below this house, is confirmed here and accepted by the review of 14 October 2026. From 15 October 2026 until the owner’s house rule of 16 October 2026 lifted the hold, PR #574’s houses were held for the owner, since their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker’s snapshot.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Couzo’s refinement of Xu’s packing, reported in pull request #574, at an exact rational side, displayed upward as $9.69793479901492131$$9.69793479901492131$ (T-174). The certificate gives a rational side and, for each of the 84 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. The source ran both maintained routes itself on its corners witnesses; those runs are its own, and the replay here is independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(84)\le 9.69793479901492131$$s(84)\le 9.69793479901492131$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is PingYou Ltd’s tightening of the packing T-130 records, that packing by the source’s statement, not decided here (T-174’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by ‘PY’, an unreleased agentic AI offering by PingYou Ltd.

The house it replaces, $9.6979347990149214$$9.6979347990149214$ from Xu’s packing, improved by Couzo (T-130), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $9.697934799017$$9.697934799017$ and T-125 at $9.6980520605096981$$9.6980520605096981$, stay verified and are not selected.