T-138: Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

V3 C3 Upper bound confirmed superseded by T-134, T-146, T-174, T-186 and T-224

2026-10-09 published · Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · 16 cases, $n = 84$ to 306

Exact rational witnesses derived here from Mishapolk's decimal centre-and-angle poses, reported on jlevy/squares#470, prove finite upper bounds at the sixteen counts where the side the source prints, rounded up at 12 decimals, is below the case's ceiling: $s (84) \leq 9.697934799017$ , $s (86) \leq 9.820535407499$ , $s (103) \leq 10.679232047362$ , $s (105) \leq 10.789303783751$ , $s (108) \leq 10.904821012324$ , $s (127) \leq 11.810878787590$ , $s (131) \leq 11.951105389418$ , $s (132) \leq 11.986956226066$ , $s (175) \leq 13.767155163551$ , $s (180) \leq 13.916993522482$ , $s (258) \leq 16.563448002138$ , $s (267) \leq 16.838828608296$ , $s (270) \leq 16.936723155038$ , $s (302) \leq 17.881306218091$ , $s (303) \leq 17.920312372919$ and $s (306) \leq 17.963433717497$ .

The source's files are decimal poses, not certificates: the cosines and sines of their angles are not rational and their digits are rounded, and at the printed sides 19 of the 30 poses in the case corpus are not packings, by up to about 1.3e-15. Each witness is this repository's: the printed centres dilated about the box centre to the 12-decimal ceiling, by the exact ratio the packet declares for each count, and each half-angle tangent rounded down at 32 decimals. The dilation moves no centre by as much as 1e-12, and the rounding no corner by as much as 1e-31.

The author reports that Ellsworth's check_packing.py at 40 digits and an independent 60-digit separating-axis check, numerical at multiprecision, accept the poses; neither ran here.

All 90 retained jobs, the 30 in-horizon witnesses with their duplicate-square and outside-container controls, were decided again here on 10 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided every witness with six controls and derived each again from its pinned upstream file by separate arithmetic, square for square. The witnesses are the project's derivation and no route is the source's code, so the replay is independently re-implemented. The 10 October review found no blocking defect in the bounds.

Each side was below the ceiling its case held. On 11 October 2026 the case records adopted a smaller certificate at every one of the sixteen counts, and these stay verified and are not selected. It was the smallest side known only at 103 and 258, where the witnesses are Xu's and SQUISH's packings refined; at 84, 86, 105, 108, 127, 131, 175, 180, 270 and 306 an earlier certificate of Couzo's (T-128, T-130) is smaller, and at 131, 132, 175, 267, 270, 302 and 303 a later one (T-134, T-136, T-140, T-141). Daniel's later regularized certificates (T-146) are smaller at 103 and 258 as well, by 6.94e-13 and 4.21e-12, and at 302 and 303.

Credit Mishapolk (square-packing-records) and, as the issue asks, the discoverers of the packings each started from where it started from the literature: Stenlund, Friedman, Ellsworth, SQUISH (Chaoweeraprasit), Couzo and Xu. The issue does not say which count started from which packing and does not name Gupta; the review measured 103 as Xu's packing and 258 as SQUISH's. It discloses Gemini 3.8 assistance with solver scripting, precision checks and formatting.

Significance, composition and next rung

Significance: Smaller finite construction sides at sixteen counts, 1.15e-12 (258) to 4.37e-03 (132) below the case ceilings; the smallest known when published at 103, 132, 258, 267, 302 and 303, and, until Daniel's regularized certificates of 10 October (T-146), at 103 (1.49e-10 below the case) and 258 (1.15e-12), each an earlier packing refined. No lower bound or optimum.
Next rung: V4 and C4 need a retained human oversight record: two accepted adversarial reviews by distinct AI reviewers are retained, the 10 October review and the 11 October final review (docs/project/reviews/review-2026-10-11-final-upper-bounds.md). No house move is due for this entry: the cases at 103 and 258 hold T-146, whose sides there are smaller, and this entry stays verified and not selected.
Novelty: previously-published Present in an identified source

The cases

This result concerns 16 cases, too many to draw one by one. Each is listed with the film’s bounds, the proved lower bound and the best known side, and links to its case record, where its packing and number line are drawn.

n	Proved lower	Best known	Gap	Status	Records
84	9.422500	9.697935	0.27543479…	open=	frontier n-084.md
86	9.503000	9.820536	0.31753540…		frontier n-086.md
103	10.280000	10.679233	0.39923204…		frontier n-103.md
105	10.280000	10.789304	0.50930378…		frontier n-105.md
108	10.620000	10.904822	0.28482101…		frontier n-108.md
127	11.281929	11.810879	0.52894946…		frontier n-127.md
131	11.465856	11.949660	0.48380348…		frontier n-131.md
132	11.511357	11.985681	0.47432242…		frontier n-132.md
175	13.237739	13.764175	0.52643511…		frontier n-175.md
180	13.432517	13.916994	0.48447636…		frontier n-180.md
258	16.064382	16.563449	0.49906558…		frontier n-258.md
267	16.350867	16.838816	0.48794778…		frontier n-267.md
270	16.445218	16.928163	0.48294353…		frontier n-270.md
302	17.389345	17.870014	0.48066809…		frontier n-302.md
303	17.418924	17.909959	0.49103397…		frontier n-303.md
306	17.507351	17.963434	0.45608236…		frontier n-306.md

Results on these cases

51 results in the register on these cases, oldest first

2005 published T-007 · 16 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
Lower bound incomplete on these cases, superseded by T-073, T-080, T-083, T-090, T-180 (reported), T-206 and T-216
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 16 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-046 · case 86
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
Lower bound recorded superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-27 published T-056 · 9 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-174, T-186 and T-224
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-070 · case 86
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
Lower bound confirmed superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register
2026-09-29 published T-058 · cases 84, 86
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
Method limit confirmed
Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083 · 16 of these cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · case 86
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-02 published T-071 · cases 84, 86
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
Lower bound confirmed superseded by T-090 and T-206
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register
2026-10-02 published T-073 · cases 103, 105
Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-02 published T-075 · case 86
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
Lower bound confirmed on these cases, superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register
2026-10-02 published T-080 · cases 103, 105
Linear-measure lower bound replayed at $n = 101 \dots 105$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-03 published T-082 · case 86
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
Lower bound confirmed on these cases, superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-092 · cases 303, 306
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
Upper bound confirmed superseded by T-174 and T-186
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-04 published T-090 · cases 84, 86
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register
2026-10-05 published T-094 · case 84
Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$
Lower bound confirmed on these cases, superseded by T-206
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register
2026-10-05 published T-098 · 8 of these cases
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
Upper bound confirmed on these cases, superseded by T-174, T-186 and T-224
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 5 of these cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
Upper bound confirmed on these cases, superseded by T-134, T-146, T-174 and T-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · cases 108, 180, 303
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-174 and T-186
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · case 258
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-146
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · cases 108, 180, 302
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-174 and T-186
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-117 · case 105
Exact rational ceiling refinements at $n = 105$ , 292
Upper bound confirmed superseded by T-174
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-119 · case 270
Three exact feasible upper bounds at n=266,270,272 from new source arrangements
Upper bound confirmed on these cases, superseded by T-224
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-07 published T-129 · case 105
Three dated Daniel certificate reports with historical source custody
Upper bound recorded superseded by T-174
Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register
2026-10-08 published T-125 · 10 of these cases
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-134, T-146, T-174 and T-186
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · cases 108, 180
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed on these cases, superseded by T-174
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · 6 of these cases
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-174
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · 5 of these cases
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by T-174, T-186 and T-224
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-131 · case 132
$s (132) \leq 11.987099332245063 \dots$ from Daniel's record hunt
Upper bound confirmed superseded by T-174
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register
2026-10-09 published T-134 · cases 132, 267, 270, 303
Six exact rational certificates from Couzo
Upper bound confirmed
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · cases 131, 270, 302, 303
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed on these cases, superseded by T-174, T-186 and T-224
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 this result · 16 of these cases
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by T-134, T-146, T-174, T-186 and T-224
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-140 · cases 132, 175, 270
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174, T-186 and T-224
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-141 · case 132
Exact rational certificates at 132 and 308 from Daniel's third record hunt
Upper bound confirmed superseded by T-174
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register
2026-10-10 published T-146 · cases 103, 258, 302, 303
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-151 · case 270
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-224
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-11 published T-185 · case 175
A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition
Upper bound confirmed superseded by T-186
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register
2026-10-12 published T-158 · case 175
New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register
2026-10-12 published T-162 · case 303
New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-12 published T-164 · case 302
A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks
Upper bound confirmed superseded by T-186
PingYou Ltd after Xu, Daniel, Levy · PingYou n302 certificate 2026-10-12 · packet · register
2026-10-12 published T-170 · case 105
An exact contact refinement of Xu's packing of 105 squares, after Couzo's
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register
2026-10-13 published T-174 · 16 of these cases
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
Upper bound confirmed
PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register
2026-10-13 published T-180 · 16 of these cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-13 published T-181 · case 302
Exact rational certificates at 211, 241 and 302 from Couzo
Upper bound confirmed superseded by T-186
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 · packet · packet · register
2026-10-14 published T-186 · cases 175, 302, 303
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
Upper bound confirmed
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-16 published T-193 · cases 175, 258, 267, 302
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register
2026-10-16 published T-206 · case 84
Mixed rectangle-measure lower bound $s (84) \geq 3769 / 400$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-208 · case 86
Mixed rectangle-measure lower bound $s (86) \geq 3811 / 400$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-216 · case 108
Mixed rectangle-measure lower bound $s (108) \geq 531 / 50$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-20 published T-224 · case 270
Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly
Upper bound confirmed
Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register
2026-10-22 published T-229 · case 105
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
Upper bound recorded
Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register

Links

On this site: The frontier survey · T-138 in the results table

On GitHub, at main

Register: T-138 in results.yaml, line 14920
Evidence: E-mishapolk-470-pose-report · E-mishapolk-470-exact-feasibility
Proofs and certificates: certificate sources.json · certificate facts
Sources: Mishapolk decimal poses 2026-10-09
Source packet: resources/web/mishapolk-decimal-poses-2026-10-09/README.md
Artifacts: 28 artifacts and controls in 7 directories. Complete artifact list in the result entry. docs/project/reviews packing/devtools packing/resources/web/mishapolk-decimal-poses-2026-10-09 packing/resources/web/mishapolk-decimal-poses-2026-10-09/acquisition packing/resources/web/mishapolk-decimal-poses-2026-10-09/facts packing/resources/web/mishapolk-decimal-poses-2026-10-09/receipts packing/tests
Case file: each case’s file is linked from its row above