\(s (29)\)$s(29)$ — open

External intake, 2026-10-06. wand125’s mixed-certificate source reports \(s(29) \ge 581/100 = 5.81\)$s(29)\ge 581/100=5.81$ (T-108), from a density of 505 rectangles of total mass \(2899999/100000 < 29\)$2899999/100000<29$, on a net the certificate declares: core side \(999/1000\)$999/1000$ and 416 half-angle tangents of step \(1/1001\)$1/1001$. Since \(999/1000 \cdot (1 + 1/1001) < 1\)$999/1000·(1+1/1001)<1$, every unit square contains a core at a net angle strictly in its interior. The source accepts it at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the reported \(2319/400\)$2319/400$ (T-068) by \(0.0125\)$0.0125$. This repository’s clean-room verifier sqverify-fast decided it here on 6 October 2026 at all 416 directions of its net, and refused two mutants scaled below coverage one: confirmed, independently re-implemented, so it is also the verified lower bound. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. wand125’s rectangle-density source reports \(s (29) \ge 2319 / 400 = 5.7975\)$s(29)\ge 2319/400=5.7975$, since superseded (T-108), with total mass \(2899/100 = 28.99 < 29\)$2899/100=28.99<29$, 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one and checked the mass and net premises, so it was also verified until 2026-10-06, when the certificate above superseded it in both lanes. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(579/100 = 5.79\)$579/100=5.79$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(2899/100 = 28.99 < 29\)$2899/100=28.99<29$, 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. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(1157/200 = 5.785\)$1157/200=5.785$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(2899/100 = 28.99 < 29\)$2899/100=28.99<29$, 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.

The Tokoharu density source, retained on 2026-09-22, reports \(s (29) \ge 571 / 100 = 5.71\)$s(29)\ge 571/100=5.71$, which superseded wand125’s \(557/100 = 5.57\)$557/100=5.57$ point certificate and was the verified lower bound until 2026-10-02. The mathematical audit records the complete local interval replay and exact premise checks. That evidence verifies the retained fixed certificate at \(571/100\)$571/100$. Tokoharu’s README says parts of the work were produced with AI assistance under human direction.

Open. The best reported packing gives \(s(29) \le 5.93383347\)$s(29)\le 5.93383347$, and this repository’s own interval certificate verifies \(s(29) \le 5.93383346267692918974379895098\)$s(29)\le 5.93383346267692918974379895098$, while wand125’s mixed certificate on a declared net proves \(s(29) \ge 581/100 = 5.81\)$s(29)\ge 581/100=5.81$ (T-108, 2026-10-06, V3/C3), which superseded its rectangle-density certificate of 1 October at \(2319/400 = 5.7975\)$2319/400=5.7975$ (T-074, replayed here on 2026-10-02), that one its replayed \(579/100\)$579/100$ (T-070), and that one Tokoharu’s \(571/100\)$571/100$. The verified interval has width about \(0.1238\)$0.1238$. The verified upper value is not the reported one: it sits above it by \(9.18974379895098 \times 10^{-15}\)$9.18974379895098\times {10}^{-15}$, which is what proving the construction costs against reporting it. An earlier reading of this paragraph named \(5.93388579981302587863645209\)$5.93388579981302587863645209$, the weaker verified ceiling this case held before T-009, and it outlived that result (D-442).

The verified upper bound is a ceiling

verified_upper_bound for this case is \(5.93383346267692918974379895098\)$5.93383346267692918974379895098$, proved by E-n029-interval-certified-upper on outward-rounded enclosures at a declared relaxation of \(1 \times 10^{-20}\)$1\times {10}^{-20}$. It is larger than the best known \(5.93383346267692\)$5.93383346267692$ two fields above it, by \(9.18974379895098 \times 10^{-15}\)$9.18974379895098\times {10}^{-15}$.

It is not a tighter reading of the same packing and it is not the value of \(s(29)\)$s(29)$: it is the strongest ceiling this repository can certify from its own evidence.

That is a much smaller gap than this section usually reports — most trailing cases sit at the integer grid bound, half a unit or more away — and it is still a gap. The bound moved here on 2026-08-30 from an exact rational replay at \(5.93388579981302587863645209\)$5.93388579981302587863645209$, a tightening of about \(5.2 \times 10^{-5}\)$5.2\times {10}^{-5}$, because the assurance contract says an interval certificate carrying a certificate and a replay is formal evidence and this one is. What it does not say, and what a reader should not infer, is that the reported value is certified: an enclosure of positive width decides strict inequalities and never an equality, and this one closes on a value just above the report rather than on it.

exact_form is the enclosure’s upper endpoint as a rational, which is exact: an outward-rounded enclosure closes on a terminating decimal and that decimal is a rational number. It is the exact form of the ceiling, never of \(s(29)\)$s(29)$ — the enclosure proves a bound and does not name the value, and an enclosure of positive width decides strict inequalities and never an equality. status stays open and the mathematics blocker in the frontmatter names what remains. Read reported_upper_bound for the best known side length.

The packing

Found by Thomas Schadt in 2025 via simulated annealing, then improved by David Ellsworth with the analytic six-equation construction retained in the primary SVG.

The document-ready view evaluates the retained roughly 100-digit numerical source at 160 decimal digits of working precision and tolerance \(1 \times 10^{-80}\)$1\times {10}^{-80}$.

The analytic characterization

This case has no minimal polynomial anywhere — in the literature or here — which is why exact_form, algebraic_degree and minimal_polynomial are all null. It does not follow that nothing exact is known. The retained SVG publishes a complete closed system, and that system is the characterization: nine slide scalars r1, r2, r3, r4, r5, r8, rB, rC, rD, each given in closed form, and six equations f1 … f6 in the six unknowns \(\lbrace s, a, b, c, d, i\rbrace\)$\{s,a,b,c,d,i\}$. \(s(29)\)$s(29)$'s reported value is the root of that system near \(5.9339\)$5.9339$.

The six unknowns are \(s\)$s$ together with the five tilted orientation classes; the sixth class is the axis class, which holds fifteen squares at zero degrees. cases.kingbird29.verify_svg transcribes every scalar and every equation, and uses them to check residuals at the serialized pose.

Solving that same transcription, rather than only evaluating it, was done on 2026-08-28 as a design-discussion measurement — recorded in X-004, which spends no experiment budget and asserts no verdict. It is not yet an in-repository capability: verify_svg still only evaluates. Turning it into one is agenda-005’s BC-047, whose entry condition names this transcription and which is ready for that reason. The measurement:

The solved angles agree with the orientation classes below — which are derived independently, from the parsed <use> transforms rather than from the equations — to every digit compared. Two separate readings of the same source therefore agree.

This does not move any bound. A high-precision root is not a certificate: no outward-rounded interval and no exact algebraic form is produced, so verified_upper_bound remains the Schadt rational and the \(5.23 \times 10^{-5}\)$5.23\times {10}^{-5}$ gap to the reported record stands. What the solve does establish is that precision at this \(n\)$n$ is available to any depth on demand, which is a precondition for an exact promotion rather than the promotion itself. See X-004.

Six Numerically Checked Orientation Classes

Exp-012 first reconstructed all 29 squares from the retained SVG, checked all 406 pairs numerically at 160 decimal digits and tolerance \(1 \times 10^{-80}\)$1\times {10}^{-80}$, and replayed the source’s nine derived offsets and six defining equations. The orientations are \(0^\circ\)$0^{\circ }$, \(25.258655^\circ\)${25.258655}^{\circ }$, \(20.800127^\circ\)${20.800127}^{\circ }$, \(-17.506268^\circ\)$-{17.506268}^{\circ }$, \(24.962588^\circ\)${24.962588}^{\circ }$, and \(24.308358^\circ\)${24.308358}^{\circ }$ modulo quarter turns. Exp-037 repeats that check under H-042’s explicitly numerical criterion and rejects its three-class bound. H-024 remains formally unresolved because the SVG supplies no exact or rigorous feasibility certificate.

The check is deliberately described as high-precision numerical reconstruction. The SVG serializes a FindRoot solution rather than supplying an interval or symbolic certificate, so this does not independently certify exactness or optimality of the record value.

What the Schadt repository establishes

Thomas Schadt’s 2025 repository reports the earlier \(5.933885799813025878636452 \ldots\)$5.933885799813025878636452\dots$ construction as valid at 300 decimal digits with tolerance \(1 \times 10^{-100}\)$1\times {10}^{-100}$. Replaying its complete 29-square input through the generic witness tool reproduces that numerical acceptance. It also exposes why this is not formal: 13 pairs have slightly negative best separation, with the worst about \(- 8.81 \times 10^{-102}\)$-8.81\times {10}^{-102}$, inside the checker’s tolerance. The source checker also accepts an incomplete file, so its output alone does not establish the quantified 29-square claim.

The generic promotion command, run at its default --rational-digits 36, rationalizes the orientations, adds an explicit side relaxation of about \(4.93 \times 10^{-31}\)$4.93\times {10}^{-31}$, and writes a complete rational-corner witness. A separate exact checker, sharing no geometry code with the promoter, verifies all 29 unit squares, all 406 pairs, and the container constraints. This proves the formal upper bound shown above. It remains weaker than the newer Kingbird report, and neither construction says anything about global optimality.

The lower bound

The superseded external report, [Friedman DS7], gives the lower-bound expression \(6 \sqrt{5} / 5 + 2 \sqrt{2}\)$6\sqrt{5}/5+2\sqrt{2}$ for \(s(29)\)$s(29)$ (approximately \(5.511708697746\)$5.511708697746$). Friedman’s DS7 survey, Theorem 10, k=5, reports this bound at n=28; reference [8] is Green’s private communication (2000). The source proof has not been recovered. Inherited at n=29 by monotonicity. Tokoharu’s \(571/100\)$571/100$ later owned the reported source field and was the verified lower bound, after the complete interval replay and exact premise checks, until wand125’s replayed rectangle-density certificate superseded it on 2026-10-02. The historical source audit compares Green’s exact theorem expression separately from opaque table decimals.

The replay uses the retained source coverage implementation. A method-distinct coverage implementation would add confidence and improve native support for future rectangle-density certificates, 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.