$n=70$$n=70$

$s(70)$$s(70)$ — open

House adoption, 11 October 2026. Daniel’s display regularization of Xu’s packing (T-146) is now this case’s house: its exact certificate proves $s(70)\le 8.8809603715555796$$s(70)\le 8.8809603715555796$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.53×10−10$1.53\times {10}^{-10}$. The earlier ceiling, displayed outward as $8.8809603717088810$$8.8809603717088810$ from Xu’s packing (T-125), applied until this adoption and stays retained as history (the certificate).

Exact certificate, 2026-10-06. Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it previously proved $s(70)\le 8.88166675700901$$s(70)\le 8.88166675700901$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $9$$9$.

External intake, 2026-10-05. Hosono’s mixed-certificate source reports $s(70)\ge 3463/400=8.6575$$s(70)\ge 3463/400=8.6575$ (T-091), from a density of 513 rectangles of total mass $6999999/100000<70$$6999999/100000<70$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the $3459/400$$3459/400$ this case reported, by $0.01$$0.01$. It supersedes the source’s mixed_n70_L86475. 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, 2026-10-03. Hosono’s mixed-certificate source reports the mixed certificate mixed_n70_L86475 at side $3459/400=8.6475$$3459/400=8.6475$ (T-082), since superseded (T-091), from a density of 495 rectangles of total mass $6999999/100000<70$$6999999/100000<70$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the $3451/400$$3451/400$ this case reported, by $0.02$$0.02$. 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 rectangle-density source reports the rectangle certificate rect_n70_L86275 at side $3451/400=8.6275$$3451/400=8.6275$, since superseded (T-082), with total mass $6999/100=69.99<70$$6999/100=69.99<70$, accepted by Tokoharu’s unchanged interval checker. Its complete coverage replay has not yet run here; the verified lower bound was $s(70)\ge 69/8=8.625$$s(70)\ge 69/8=8.625$ until 5 October, from the source’s certificate for this count in the 2026-10-01 intake, whose complete 201-direction replay passed here. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. Hosono’s rectangle-density source reports a direct $69/8=8.625$$69/8=8.625$ certificate for this case, whose reported bound the 2026-10-02 intake above raises, with total mass $6999/100=69.99<70$$6999/100=69.99<70$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. Hosono’s rectangle-density source reports a direct $431/50=8.62$$431/50=8.62$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $6999/100=69.99<70$$6999/100=69.99<70$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. Hosono’s rectangle-density source reports a direct $861/100=8.61$$861/100=8.61$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $6999/100=69.99<70$$6999/100=69.99<70$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged.

External intake, 2026-09-22. Hosono’s retained point-certificate source reports $s(70)\ge 171/20=8.55$$s(70)\ge 171/20=8.55$, which was the verified lower bound until 2026-10-02. The complete exact replay verifies the retained certificate.

Previously, the packing gave $s(70)\le 8.88166676$$s(70)\le 8.88166676$, and the verified lower bound is $s(70)\ge 3463/400=8.6575$$s(70)\ge 3463/400=8.6575$, from Hosono’s mixed rectangle-density certificate of 4 October, leaving a gap of about $0.2242$$0.2242$.

The exact certificate

Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register then listed as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 70 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $8.8816667570090046250\dots$$8.8816667570090046250\dots$, 4.6×10−15$4.6\times {10}^{-15}$ above the side the Kingbird catalogue prints, $8.88166675700900$$8.88166675700900$. Square for square, its pose lies within 4.5×10−16$4.5\times {10}^{-16}$ of the binary64 pose the atlas then pictured for this count. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s(70)\le 8.88166675700901$$s(70)\le 8.88166675700901$, the verified upper bound at that intake: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The verified upper bound and the printed side at that intake agreed to one unit of the printed side’s last place, the precision at which the record compares them. The printed side itself was not certified by that rational certificate: the certificate’s side lay above it.

The packing

Found by DeVincentis in 2014, via a hand construction. Its side length is algebraic of degree 4 over $\mathbb{Q}$$\mathbb{Q}$. Being a quartic it is solvable in radicals — the Galois group is S4 — so a closed form exists in principle. It is not recorded because the radical expansion runs to many lines and carries no insight the quartic does not already carry; the empty exact_form here means “impractical”, not “impossible”, which is the opposite of the situation at $n=28$$n=28$ and $n=39$$n=39$.

The lower bound

The verified field rests on a complete decision here (E-n070-wand125-mixed-86575-sqverify-fast-replay, V3/C3) of Hosono’s mixed certificate of 4 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. Until this decision the verified lower bound was $69/8$$69/8$ (E-wand125-rectangle-2026-10-01-source-replay).

The exact replay verifies all five covering-certificate conditions over the complete rational direction net. A second interval implementation or generalized native importer would add confirmation and reuse, but it is not a missing premise of this fixed bound.

Corrected 2 October 2026: Nagamochi’s Lemma 1 is false (Karakuş 2026), so his closed form below is now a reported bound whose published proof is incomplete (review). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s earlier general closed form remains part of the evidence history and applies to every $N\ge 4$$N\ge 4$:

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 $88809603717088809603717/10000000000000000000000$$88809603717088809603717/10000000000000000000000$, displayed upward as $8.8809603717088810$$8.8809603717088810$. 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.

Evan Daniel’s certificate (T-146)

Daniel’s display regularization of Xu’s packing, reported in his regularized record lists, on no issue, at exact side $8880960371555579594785501178639/1000000000000000000000000000000$$8880960371555579594785501178639/1000000000000000000000000000000$, displayed upward as $8.8809603715555796$$8.8809603715555796$ (T-146). The certificate gives a rational side and, for each of the 70 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. No route here is the source’s code, so the replay is independently re-implemented. The review of 11 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(70)\le 8.8809603715555796$$s(70)\le 8.8809603715555796$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate regularizes for display the packing T-125 records, at the side the source reports as that packing’s exact optimum, so the packing stays with its finder and Daniel is credited with the regularization and the certificate. That the two are one packing is the source’s claim, measured here and not decided (T-146’s regularizations).

The source’s credits say the repository’s work was produced by Claude (Anthropic) under human direction.

The house it replaces, $8.8809603717088810$$8.8809603717088810$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-137 at $8.8809603715662510$$8.8809603715662510$, stay verified and are not selected. T-174’s algebraic certificate at this count, a root of a degree-5 integer polynomial 8.88×10−20$8.88\times {10}^{-20}$ below this house, is decided here on one exact route. The rule of 15 October left it out for the owner (RP-3 of the review of 14 October 2026), and the owner’s house rule of 16 October 2026 adopts it: the case keeps this house until its adoption, staged under think-ot5c, moves it to T-174’s side, once the writer takes an algebraic side.