Case record

The best packing known for 29 squares, side

5.93383346 …

, Schadt 2025

$n = 29$

=open

$\frac{581}{100} \leq s (29) \leq 5.93383346 …$

5.815.934

567

5.3856.385

\sqrt{n}

\sqrt{n} + 1

Proven

$5.81000$ $\leq s (29) \leq$ $5.933834$

exact

Citation record n-029

lowerHosono after Tokoharu, Levy et al. 2026, GitHub (confirmed T-108)

upperSchadt et al., GitHub (confirmed T-195)

Open

optimality

Bounds

Best known packing

5.93383346…

5.9338334626769292

Found by: Schadt 2025
Improved by: Ellsworth, PingYou Ltd
Construction: annealing
Tilt angles: $0^{\circ}$ , $25.25865530 …^{\circ}$ , $20.80012676 …^{\circ}$ , $- 17.50626847 …^{\circ}$ , $24.96258798 …^{\circ}$ , $24.30835840 …^{\circ}$
Source: [PingYou PR 622 certificate 2026-10-15]
Evidence: E-pingyou-622-certificate-report

Verified upper bound

5.93383346…

5.9338334626769292

The reported value, verified here.

Evidence: E-pingyou-622-exact-feasibility

Reported lower bound

$\frac{581}{100}$

Proved by: Hosono 2026
Kind: counting
Scope: Unrestricted unit-square packing with independent rotations and disjoint interiors.
Note: Hosono's square-packing-bounds (6 October 2026) reports $s (29) \geq 581 / 100$ from a density of 505 uniform rectangles of total mass 2899999/100000, on a net the certificate declares: core side 999/1000 and 416 half-angle tangents of step 1/1001, accepted there at every angle by its research copy of Tokoharu's verify.cpp at threshold one. It is above the 2319/400 the record reported (T-068). sqverify-fast, this repository's clean-room measure verifier, decided it here at all 416 directions on 6 October 2026.
Source: [wand125 mixed bounds check2 2026-10-06]
Evidence: E-n029-wand125-mixed-581-report

Verified lower bound

$\frac{581}{100}$

The reported value, verified here.

Evidence: E-n029-wand125-mixed-581-sqverify-fast-replay

Gap

0.12383346…

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-009 V3 C3 Levy · 2026-08-31 · n = 29
$s (29) \leq 5.93383346267692918974 \dots$ , by a Krawczyk interval certificate
Claim and records
Claim
$s (29) \leq 5.93383346267692918974379895098$ , by a Krawczyk interval certificate over the retained rational 29-square witness at a declared relaxation of 1e-20.
Significance
As far as the archived corpus shows, the first interval certificate for a square-in-square bound; 5.2337e-5 tighter than the robust rational route on the same packing.
Novelty
apparently-novel Not found in the recorded search, subject to its stated gaps
Records
n = 29 register evidence 1
n = 29 register evidence 1
T-044 V3 C3 Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 17 cases
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
T-046 V0 C0 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 48 cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
T-047 V3 C3 Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 7 cases
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
T-058 V3 C3 Hosono after Tokoharu, Daniel · 2026-09-29 · 100 cases
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
T-068 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
T-070 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 25 cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
T-074 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 31 cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; Vlasenko · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
T-108 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-06 · n = 29
Mixed rectangle-measure lower bound verified at $n = 29$ , on a declared net
Claim and records
Claim
A rectangle density of wand125/square-packing-bounds, checked at coverage one on a net it declares and published on 6 October 2026, proves $s (29) \geq 581 / 100$ = 5.81.

The certificate is a density of 505 uniform rectangles in D4 orbits, and no point mass, of total mass 2899999/100000, on the net T-096's certificate declares: core side 999/1000 and 416 half-angle tangents of step 1/1001. Since 999/1000 (1 + 1/1001) = 500499/500500 < 1, and the last tangent 415/1001 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior.

The source reports every such core capturing mass at least one: at all 415 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, run by the same driver as T-096's certificate, and at the axis by exact integer tables.

The value is above 2319/400 = 5.7975, the source's rectangle certificate rect_n29_L57975 and the reported (T-068) and verified (T-074) lower bound at $n = 29$ before it, by 0.0125.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 416 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, independently re-implemented. The verifier shares no code with the source's checker and runs the same net-and-shrink method, so it is a second implementation and not a second method. The source's own checker was replayed here at nodes 0, 1, 364 and 415, the axis and the least-bound node among them, each returning the certificate's own record, and not in full: a partial reproduction with the producer's code.

Hosono after Tokoharu and Levy, square-packing-bounds. No registration was requested: an intake pass of the source found it. The source says parts of the work were produced with AI assistance under human direction.
Significance
The strongest verified lower bound on record at $n = 29$ , by 0.0125 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net with the C++ checker of T-069 and later; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty
previously-published Present in an identified source
Records
n = 29 register evidence 1 evidence 2 source review
n = 29 register evidence 1 evidence 2 source review
T-132 V0 C0 Hosono after Tokoharu and Levy · 2026-10-10 · n = 29
Mixed rectangle-measure lower bound $s (29) \geq 291 / 50$ reported, on a 2,073-direction declared net
Claim and records
Claim
A rectangle measure of wand125/square-packing, published on 10 October 2026 and reported on jlevy/squares#446, is reported to prove $s (29) \geq 291 / 50$ = 5.82.

The certificate, mixed_n29_L582, is a density of 650 uniform rectangles in D4 orbits, and no point mass, of total mass 2899999/100000, on a net its candidate declares: core side 4999/5000 and 2,073 half-angle tangents of step 1/5002, the net of T-102 and T-103. Since 4999/5000 (1 + 1/5002) = 25009997/25010000 < 1, and the last tangent 1036/2501 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior.

The source reports every such core capturing mass at least one at all 2,073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and a control with every mass scaled by 197/200 refused at 32 sampled directions.

The value is above T-108's 581/100 = 5.81, the verified and reported lower bound at $n = 29$ , by 0.01, and above the 1163/200 the issue reported on 8 October by 1/200.

Here the maintained readers accepted its exact mass and declared-net premises, and sqverify-fast, this repository's clean-room measure verifier, verified it at eight sampled directions with the source's node counts and least certified bounds: a diagnostic, not a replay. No complete replay has run here, so the value stands as reported.

Hosono after Tokoharu and Levy, square-packing. The source says parts of the work were produced with AI assistance under human direction.
Significance
Scored as a claim: if a complete replay passes, it is the strongest lower bound on record at $n = 29$ , 0.01 above T-108, by the certificate kind and declared net of T-102 and T-103, with no new technique, at S3 as T-102 to T-111 are. V0/C0 says how far it stands, and the score gates nothing.
Novelty
previously-published Present in an identified source
Records
n = 29 register evidence 1
n = 29 register evidence 1
T-155 V3 C3 Levy after Daniel · 2026-10-12 · 59 cases
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
T-195 V3 C3 PingYou Ltd after Schadt, Ellsworth, Daniel, Levy · 2026-10-16 · n = 29
A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155
Claim and records
Claim
One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (29) \leq 24534865107674036688168549876397025785476132874112370099771467927504056975875987776681455985875024244254213572886097261458084288767455559197653938689488649902066228662684853391396663327014033932373723477506069710425226218750526706135891585179797517343019488443339867249715360890013756666854628979176900944993305442261602772842180838440053667420917480395180220066666602334511722453879594972998191138612834034189666510059905050426568116428378640731543810807747891542284864038369907643508459843509205620695587608368557533498173488519528447981793482358600152308521631459565592767014183052501199151931076562879250106887616015278050875978119717926899218680011808702390993996536195830670260972350387443818823426662369027496193945575599138014543670320983369783863929251334778825936608589267984001537304179324158789246196108802236193481133294548709295335474346256527544613659984746868470460857727484675678537346741164019856884567986913934704956927321686209550808604138457927403334411832210594036476801938126308790969489032002021963023705206483679471872149687890566611443899435653214191799667499764111469459385033095127931344323459448933275284125181277683793104256050531452804725334403790564783668180540557925744070464684545649032408833632838438910661813484030778674222303675402958374138153085439248296679730701126691111630742632653637849625799540379878830607499958400806445626950768552900180049729020839822237918053913692758374445410695986201178837254745307861725778970345267181802138568964810374228407899464496680162178116797606104631259924089248154437042373925033307272942769766239315424518195846205980008361535060213644205099149816040655058140387176369175347126651923289738983179736034453072062298517733929444973959326041866213931560671058617171889088120835739271365353588567878531477697526577302776627625153488856099874787185474253160414731224686151501675402728352580934099397733963238715051627968686911414459275184683424784163644191544707377182384601048542578683951517983159198963572073487749781532348390081793066311053790977197895164959450212581707715858563399258411304003599006319077167663800875019623826094756719222309800908141246359094069406089611981416987407570927946930756422932236652442311952880289753229889653917790471125660627044950535988555435869596551538259114765832766790488411832484772360835487671284213135964427186102124863000233796141278872662965770217771690365276248657719966519254396208 / 4134741101514774825128795556716042227961798785923906182461358460239918424782567413226407211403203065887739199609118084844533471798679416900735032687473222139680760065590940944078234018015370334443128772607933060444000826617769429646882194583220599963273124225726071777932738764595777987721482365783510337019481236506878836287721256689347799807863951388092667648197563925993938390556531505228608954855292930046726399665713751905730833652017210190663827825661322365063724262883246743380492604927786523721825931831886303155063551654710154578169238652367383102078792000130468658641539801741146889224943442810013209072949461457217339392927864030591169892883726839884880594167460487749243589056331690253699842043479578088884954988951340936860924955322743194977380476502898903410854524818203652027795198144692092594657450033887027432586825094110442844464606853681664319033486389387445736214034648895232429305200383123921501470838824817050243022153810427067912956942772219391824845560443747547400843195049198930533146285574982530233459642803642285852596410633979360461440217163898024968451883257550199336829150031495720132797637319132645987063404501212699321705824766887053681261752487880911742084111713948715586118826460029866243412955813075255128613682606937176468037935097555448704569843358216916775243025949351743902174698612665846213152609786378184952030306093106808751354257494437791937397557320503668105995208209305535886191424772995183160168155795973488716123209693421756294653489941963987963470670675939442599780713172144359901210230692152439814696253554319377531586894873990381499235730768339867996867328907448233810242342899234366597370788054658934954410215490093061628299835361415650502735274254384722944822121189889481733221480599225541033150521791602097236911676058943766150551365114305649138572343660857535228979083917905543053139843750484912743933307704134563365828870127440166022411140550453711645377040805715862280799298958039516958431092020836360669755785671136319313346695819974050474113917337370449084604146353831484738751871298385136659525010746969348827942534339711929337559777205045248731333573416596885461197747132145825993130411956264953672744857178743458111168300805276706096833515134514702136577188472029912623847531283031692405241550695989903352952233750694266905747073315282264735651376143483435574343954773982361734961798348911999812513553974602740493739669761924137585813615$ .

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided here on 16 October 2026 by both maintained exact routes, and decided again equal to their retained rows: the positive passes both, and both refuse both controls. The source reports running those two routes itself on its corners witness, so on them the replay reproduces the source's own runs.

Two checks that share no code with any check the source ran decided the certificate again: the third exact route, half-extent containment and exact intersection area, with six controls, two of them 1e-40 and 1e-30 past exact contact; and the 16 October review's clean-room exact checker, from the release asset, with four controls, two of them 1e-3000 past contact. On those two the confirmation is independently re-implemented. The review found no blocking defect. The side is 1.57e-36 below T-155's, the case's verified upper bound, and 7.32e-90 below PR 574's N29 certificate, which T-174 does not register.

The author reports that standard-library exact routines of its own, by separating axes and by rational polygon clipping, accept all 29 squares and 406 pairs of the certificate and of its three comparators and refuse duplicate-square and outside-container controls, that the conversion to its corners Witness/v2 is lossless, and that this repository's sqpack exact_verify and check_rational_witness_independent, unchanged at the pull request's base, accept that witness. Those statements are the author's; no source program ran here.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with this certificate refinement, as the submission asks: an 800-digit realization of the packing's contact system from Daniel's source tools and data, scaled about a common centre and cropped exactly. The packing is Schadt's, improved by Ellsworth, as the Kingbird catalogue credits it, and the submission says it is no new construction; the count is a regularization of that packing. Measured by the review against T-009's witness, it pairs with that packing one to one, no square moved beyond 1e-9. On 16 October 2026 the author stated on the pull request that all of its submissions are 'PingYou Ltd, assisted by the PY agentic offering'.
Significance
A padding refinement 1.57e-36 below the case ceiling, of a packing the record already holds, as T-155's and T-174's are; no lower bound or optimum.
Novelty
previously-published Present in an identified source
Records
n = 29 register evidence 1 evidence 2
n = 29 register evidence 1 evidence 2

Verification

upper: replayed here; lower: replayed here

1 conflict

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 4 of the retained witness (witness id 5) translates 0.385694 along (0, -1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 5 of its 29 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Regularization

Drawn as published: the layer's regularized view does not improve on the packing as published (it ties the house drawing on every goal of the improvement rule), so the house drawing is drawn (regularization policy, section 12). The view was regularized by the Squares Project with its own axis tool (regularize_axis_components/2). A drawing of T-195's packing at the witness's own side: a pure regularization, which proves no new bound, so not a result. Also regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT), with the project's modifications: same-side exact re-solve (devtools.regularize_same_side); whether its solve climbed a retry rung is not recorded. A regularization of T-009's packing at the witness's own side, not drawn here; its exact side is registered as T-155, the upper bound it proves. The house is itself its source's regularization of T-009's packing (PingYou Ltd after Schadt, Ellsworth, Daniel, Levy), registered as T-195 for the upper bound its exact side proves.

Open questions

Conflict (scope ambiguity): The source describes an exact analytic solution, but the public SVG serializes a numerical FindRoot result and no formal certificate. E-n029-kingbird-report, E-n029-kingbird-numerical
Blocker (source evidence): The finer-net lower-bound report awaits complete maintained native capture, whole-net controls and independent result review. E-wand125-fine-net-lower-bounds-446-report
Blocker (source evidence): The later 291/50 report awaits a complete maintained capture, whole-net controls and a review that reads it; an eight-direction sample is diagnostic. E-wand125-fine-net-n29-582-followup-report
Priority: Earlier house 296691673133846459484473030817600957/50000000000000000000000000000000000 from T-155 (reported as 5.93383346267692 from [Kingbird]), the case’s ceiling until 17 October 2026; its certificate and replay remain retained. (Schadt, Ellsworth)

Evidence and sources

27 evidence entries

E-regularized-bounds-2026-10-12-exact-decision, E-n029-wand125-mixed-581-sqverify-fast-replay, E-n029-wand125-mixed-581-report, E-n029-wand125-rect-57975-sqverify-fast-replay, E-wand125-rectangle-2026-10-01-source-replay, E-wand125-rectangle-2026-09-28-source-replay, E-wand125-rectangle-2026-10-01-report, E-wand125-rectangle-2026-09-28-report, E-wand125-rectangle-report, E-tokoharu-density-report, E-wand125-point-bounds-report, E-tokoharu-density-source-replay, E-kingbird-upper-register, E-n029-kingbird-report, E-nagamochi-lower, E-basic-grid-upper, E-n029-kingbird-numerical, E-n029-orientation-classes, E-n029-schadt-report, E-n029-schadt-numerical, E-n029-schadt-rational-upper, E-n029-interval-certified-upper, E-green-ds7-theorem10-reported-lower, E-wand125-fine-net-lower-bounds-446-report, E-wand125-fine-net-n29-582-followup-report, E-pingyou-622-certificate-report, E-pingyou-622-exact-feasibility

[wand125 mixed bounds check2 2026-10-06] lower bound proof
[wand125 rectangle bounds 2026-10-01] lower bound proof
[wand125 rectangle bounds 2026-09-28] lower bound proof
[wand125 rectangle bounds 2026] lower bound proof
[Tokoharu density 2026] lower bound proof
[wand125 point bounds 2026] lower bound proof
[Kingbird] record catalogue
[Kingbird n=29 SVG] numerical witness
[Nagamochi 2005] lower bound proof
[Schadt n=29 repository] upper bound report
[Schadt n=29 decimal witness] numerical witness
[Schadt n=29 rational witness] formal certificate
[Friedman DS7] survey
[wand125 fine-net lower bounds 2026-10-08] lower bound proof
[wand125 fine-net n29 2026-10-10] lower bound proof
[PingYou PR 622 certificate 2026-10-15] upper bound report

In the frontier survey On GitHub

$s (29)$ — open

House adoption, 17 October 2026. PingYou Ltd’s tighter certificate of Schadt’s packing, as Ellsworth improved it (T-195) is now this case’s house: its exact certificate proves $s (29) \leq 5.9338334626769292$ , 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 $1.57 \times 10^{- 36}$ . The earlier exact ceiling, $296691673133846459484473030817600957 / 50000000000000000000000000000000000$ from Schadt’s packing, improved by Ellsworth (T-155), applied until this adoption and stays retained as history; the reported upper bound it replaces, $5.93383346267692$ from [Kingbird], is a print $9.19 \times 10^{- 15}$ below the new side, within one unit of its last place, so it does not order them (the certificate).

Exact regularized certificate, 2026-10-12. This project’s regularization pass, Daniel’s exact solver and regularizer run here on this register’s witness, produced an exact rational packing of this arrangement, decided here over $ℚ$ by two exact checkers: it proves $s (29) \leq 5.93383346267692918968946061635201914$ , the verified upper bound (T-155), which the record still prints as $5.93383346267692918974379895098$ . Until then the verified upper bound was exactly $5.93383346267692918974379895098$ , the upper endpoint of this project’s interval certificate of the same packing (T-009).

External intake, 2026-10-06. Hosono’s mixed-certificate source reports $s (29) \geq 581 / 100 = 5.81$ (T-108), from a density of 505 rectangles of total mass $2899999 / 100000 < 29$ , on a net the certificate declares: core side $999 / 1000$ and 416 half-angle tangents of step $1 / 1001$ . Since $999 / 1000 \cdot (1 + 1 / 1001) < 1$ , 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 $2319 / 400$ (T-068) by $0.0125$ . 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. 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 $s (29) \geq 2319 / 400 = 5.7975$ , since superseded (T-108), with total mass $2899 / 100 = 28.99 < 29$ , 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. 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 $579 / 100 = 5.79$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $2899 / 100 = 28.99 < 29$ , 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 $1157 / 200 = 5.785$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $2899 / 100 = 28.99 < 29$ , 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 $s (29) \geq 571 / 100 = 5.71$ , which superseded Hosono’s $557 / 100 = 5.57$ 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 $571 / 100$ . Tokoharu’s README says parts of the work were produced with AI assistance under human direction.

Open. The best reported packing gives $s (29) \leq 5.93383347$ , and this repository’s own interval certificate verified $s (29) \leq 5.93383346267692918974379895098$ until 12 October 2026, and its exact regularized certificate now verifies $s (29) \leq 5.93383346267692918968946061635201914$ , while Hosono’s mixed certificate on a declared net proves $s (29) \geq 581 / 100 = 5.81$ (T-108, 2026-10-06, V3/C3), which superseded its rectangle-density certificate of 1 October at $2319 / 400 = 5.7975$ (T-074, replayed here on 2026-10-02), that one its replayed $579 / 100$ (T-070), and that one Tokoharu’s $571 / 100$ . The verified interval has width about $0.1238$ . The verified upper value is not the reported one: it sits above it by $9.18974379895098 \times 10^{- 15}$ , which is what proving the construction costs against reporting it. An earlier reading of this paragraph named $5.93388579981302587863645209$ , the weaker verified ceiling this case held before T-009, and it outlived that result (D-442).

The earlier verified ceiling

Until 17 October 2026 the verified upper bound here was $5.93383346267692918968946061635201914$ , exactly, proved by the exact regularized certificate of this packing (E-regularized-bounds-2026-10-12-exact-decision, T-155), and the record prints it as $5.93383346267692918974379895098$ . As printed it was larger than the catalogue’s printed side $5.93383346267692$ , then the best known, by $9.18974379895098 \times 10^{- 15}$ .

It was not the value of $s (29)$ and not a different packing: it is the certificate’s own exact side. It lies $9.2 \times 10^{- 15}$ above the side the catalogue prints, more than half a unit of the printed side’s last place, so the two disagree at the printed precision and the printed side itself is not certified here. A mathematics blocker in the frontmatter recorded the difference until PingYou Ltd’s certificate (T-195) replaced both lanes.

Until 12 October 2026 that printed decimal was the bound exactly, the upper endpoint of this project’s interval certificate (E-n029-interval-certified-upper, T-009) at a declared relaxation of $1 \times 10^{- 20}$ , which on 30 August 2026 had replaced an exact rational replay at $5.93388579981302587863645209$ .

The regularized exact certificate

This project’s regularization pass of 11 and 12 October 2026 ran Daniel’s exact solver and regularizer (evand/square-packing, MIT, vendored here) on this register’s witness for this count, with the conservative angle policy and coincide placement, and this project’s same-side re-solve then fixed the result’s tight contacts over $ℚ$ at the witness’s own side. The result is an exact rational packing of the same 29 squares in a square of side $5.93383346267692918968946061635201914$ , $5.4 \times 10^{- 20}$ below the endpoint of the interval certificate. Measured against the witness under the box’s eight symmetries, its squares pair one to one with the witness’s; 3 move by more than $1 \times 10^{- 3}$ , the largest by $1.3 \times 10^{- 2}$ , and none turns by more than $1 \times 10^{- 6}$ as a sine. It redraws the packing visibly, and is drawn only once the regularized layer admits it under the regularization policy.

Each pair and each wall is decided over $ℚ$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s (29) \leq 5.93383346267692918968946061635201914$ , the verified upper bound (T-155), the exact form of the case’s verified_upper_bound, which prints it as $5.93383346267692918974379895098$ , the decimal it printed before and an upper display of it. It says nothing about optimality, and the packing stays its finders’.

The packing

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

The high-precision Kingbird packing of twenty-nine unit squares.

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

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 ${s, a, b, c, d, i}$ . $s (29)$ 's reported value is the root of that system near $5.9339$ .

The six unknowns are $s$ 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:

Quantity	Value
$s$ at 60 digits	$5.93383346267692918968946061635201913843383418107788697463883$
Agreement with the reported $5.93383346267692$	all 15 published digits
Maximum equation residual at 420 digits	$8.85 \times 10^{- 421}$
Maximum equation residual at 1200 digits	$1.11 \times 10^{- 1200}$
Wall-clock for a 420-digit solve	about 2 seconds

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 solve did not move a bound. A high-precision root is not a certificate: no outward-rounded interval and no exact algebraic form is produced, so at the time of the 28 August measurement, verified_upper_bound remained the Schadt rational and the $5.23 \times 10^{- 5}$ gap to the reported record stood. The later interval and exact regularized certificates above superseded that ceiling. What the solve does establish is that precision at this $n$ 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 $1 \times 10^{- 80}$ , and replayed the source’s nine derived offsets and six defining equations. The orientations are $0^{\circ}$ , ${25.258655}^{\circ}$ , ${20.800127}^{\circ}$ , $- {17.506268}^{\circ}$ , ${24.962588}^{\circ}$ , and ${24.308358}^{\circ}$ 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

Schadt’s 2025 repository reports the earlier $5.933885799813025878636452 \dots$ construction as valid at 300 decimal digits with tolerance $1 \times 10^{- 100}$ . 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 $- 8.81 \times 10^{- 102}$ , 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 $4.93 \times 10^{- 31}$ , 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 proved the earlier Schadt rational ceiling. 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 $6 \sqrt{5} / 5 + 2 \sqrt{2}$ for $s (29)$ (approximately $5.511708697746$ ). 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 $571 / 100$ later owned the reported source field and was the verified lower bound, after the complete interval replay and exact premise checks, until Hosono’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 $N \geq 4$ :

s (N) \geq min {⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1}

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.

bound	evidence	run	code	programs
verified lower	`E-n029-wand125-mixed-581-sqverify-fast-replay`	replayed here	independent	`V-sqverify-fast` (first-party)
verified upper	`E-pingyou-622-exact-feasibility`	replayed here	independent	`V-sqpack-verify`, `V-check-rational-witness-independent`, `V-check-half-angle-area`, `V-pingyou-622-exact-review` (first-party); `V-evand-arrangement-receipts`, `V-upper-bound-reports`, `V-evand-exact-certificates` (first-party, premises)

PingYou Ltd’s certificate (T-195)

PingYou Ltd’s tighter certificate of Schadt’s packing, as Ellsworth improved it, reported in pull request #622, at an exact rational side, displayed upward as $5.9338334626769292$ (T-195). The certificate gives a rational side and, for each of the 29 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 witness, so on them the replay reproduces its runs; the third route and the review’s clean-room checker, which share no code with any check it ran, are independently re-implemented. The review of 16 October 2026 accepted the replay for finite feasibility. That proves $s (29) \leq 5.9338334626769292$ , 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-009 records, measured here pose by pose as that packing (T-195’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The author says all of its submissions are PingYou Ltd’s, assisted by the PY agentic offering.

The house it replaces, $296691673133846459484473030817600957 / 50000000000000000000000000000000000$ from Schadt’s packing, improved by Ellsworth (T-155), stays retained with its own replay as history.