T-098: Exact optima of 48 known-best packings: `s(n) ≤ S'`, `3.5e-13` to `5.0e-11` below each printed side

V3 C3 Upper bound confirmed

2026-10-05 published · Daniel after Couzo, de Winter, Ellsworth, Levy · 48 cases, $n = 68$ to 307

For each of 48 counts n from 68 to 307, $s (n)$ <= S', where S' is the side of an exact rational packing Daniel published on 5 October 2026: this register's own known-best packing at that count, solved to its exact optimum. The counts are 68, 102, 103, 106, 110, 123, 126, 131, 132, 152, 154, 155, 156, 172, 177, 180, 181, 182, 199, 206 to 211, 228, 236 to 241, 259, 263, 268 to 273, 297 and 301 to 307, and each S' is the verified upper bound its case record carries, written out in full, from 8.798795237218283902664668919394 at $n = 68$ to 17.981030548633310696277454165157 at $n = 307$ .

The packings are their finders': Couzo's at 46 counts (T-056, and at seven of them his revision of T-092) and de Winter's at $n = 126$ , from the Kingbird catalogue, and $n = 211$ (T-057). Each S' lies below the side its finder prints, by 3.5e-13 at $n = 211$ to 5.0e-11 at $n = 270$ . The source's solver moves the binary64 pose to a nearby exact KKT point of minimizing the side, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational tan(theta/2).

Matched square for square with the known-best witness, every pose lies within 1.5e-3 of it, and every square the source does not list as free within 4.2e-5. At $n = 126$ the verified ceiling had been the $12 \times 12$ grid, and at $n = 206$ , 259, 305 and 306 a rounding above a printed side the printed pose did not certify.

Every certificate is decided here exactly: converted without rounding, by sqpack's exact separating-axis test and by devtools.check_rational_witness_independent, which share no code with each other or with the source, so independently re-implemented. The source's own checkers, verify_cert.py and verify_cert2.py, run here as retained, accept every one as well, reproduced with the producer's code. All four refuse each certificate with its tightest pair moved one unit of the side's denominator past touching, and each with its box shrunk that unit past its least clearance. The source's report that 44 of the 48 exact points are KKT local minima rests on numerical checks at those points and is not verified here; it bears on the packings, not on $s (n)$ .

Daniel after Couzo, de Winter and Ellsworth's analytic minimisation, evand/square-packing, from this register's witnesses. Registration was requested on jlevy/squares#375. Its author says the solver, its checkers and the batch were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by him.

Significance, composition and next rung

Significance: Forty-eight best known sides lowered by 3.5e-13 to 5.0e-11 by solving the same packings exactly, and the verified ceiling with them: a citable detail of each case that moves no theorem, S2 as T-088's optimization of a packing already reported is. At $n = 126$ it moves the verified ceiling off the $12 \times 12$ grid by 0.225, as T-056 did at 49 counts (S3), but there by certifying a packing this record already reported rather than by a new one; at four counts it removes a ceiling that trailed its report. The reusable part, an exact solver with certificates, is the source's method and not a result here.
Composition: Two machine entries decide all 48 bounds, each at its case's verified value: the exact decision here by two checkers of this repository that share no code with each other or the source (exact-algebraic, independently re-implemented), and the source's own two checkers run here (exact-algebraic, reproduced with the producer's code). Both are exact rational decisions of the same certificates, one method, so C3 with one machine method beside it. The comparisons with the earlier sides and poses are recorded values and set no rung, and the source's numerical KKT claims are outside the claim.
Next rung: V4 and C4 need a second adversarial AI review of the complete claim by a reviewer distinct from the one of 2026-10-06, and a human oversight record. That review could run no code and decided no certificate itself; the second should, with a decider of its own. A method-distinct second route, such as an interval decision of each certificate's pose, would be shown beside the rung. Verifying that each exact point is a local minimum needs an interval enclosure of the KKT root, which the source names as its next step; it bears on the packings, not on $s (n)$ .
Regularized drawings: n = 106: a regularization of T-098’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; unmodified; solved to its side; 1 free square straightened; identity: measured here; nearest centres one to one; 28 squares moved, at most 0.411604; turns up to 2.31046°).

n = 152: a regularization of T-098’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; the project’s modifications: retry rung force-kkt; solved to its side; 1 free square straightened; identity: measured here; nearest centres one to one; 16 squares moved, at most 0.262686; turns up to 10.8334°).

n = 177: a regularization of T-098’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; unmodified; solved to its side; 2 free angles moved; identity: measured here; nearest centres one to one; 28 squares moved, at most 0.118246; turns up to 5.64624°).

n = 301: a regularization of T-098’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; unmodified; solved to its side; 163 free squares straightened, 1 free angle moved; identity: measured here; nearest centres one to one; 189 squares moved, at most 0.440369; turns up to 1.34112°).
Novelty: previously-published Present in an identified source

The cases

This result concerns 48 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
68	8.530000	8.798796	0.26879523…	open=	frontier n-068.md
102	10.280000	10.605829	0.32582869…		frontier n-102.md
103	10.280000	10.679233	0.39923204…		frontier n-103.md
106	10.310708	10.822909	0.51219960…	open≈	frontier n-106.md
110	10.512492	10.996784	0.48429119…	open=	frontier n-110.md
123	11.094810	11.591379	0.49656809…		frontier n-123.md
126	11.235455	11.742641	0.50718541…		frontier n-126.md
131	11.465856	11.949660	0.48380348…		frontier n-131.md
132	11.511357	11.985681	0.47432242…		frontier n-132.md
152	12.342719	12.830719	0.48799951…	open≈	frontier n-152.md
154	12.426860	12.921101	0.49424049…	open=	frontier n-154.md
155	12.468709	12.952261	0.48355174…		frontier n-155.md
156	12.510412	12.979166	0.46875370…		frontier n-156.md
172	13.119429	13.618989	0.49955949…		frontier n-172.md
177	13.316005	13.822980	0.50697411…	open≈	frontier n-177.md
180	13.432517	13.916994	0.48447636…	open=	frontier n-180.md
181	13.471121	13.950820	0.47969775…		frontier n-181.md
182	13.509611	13.974091	0.46447888…		frontier n-182.md
199	14.110657	14.617573	0.50691458…		frontier n-199.md
206	14.365424	14.860159	0.49473403…		frontier n-206.md
207	14.401438	14.880961	0.47952159…		frontier n-207.md
208	14.437359	14.924519	0.48715890…		frontier n-208.md
209	14.473188	14.946224	0.47303461…		frontier n-209.md
210	14.508925	14.973002	0.46407539…		frontier n-210.md
211	14.544571	14.993550	0.44897772…		frontier n-211.md
228	15.103081	15.604602	0.50151960…		frontier n-228.md
236	15.374474	15.861935	0.48745960…		frontier n-236.md
237	15.408051	15.902604	0.49455212…		frontier n-237.md
238	15.441552	15.924107	0.48255398…		frontier n-238.md
239	15.474979	15.948444	0.47346419…		frontier n-239.md
240	15.508331	15.969686	0.46135431…		frontier n-240.md
241	15.541608	15.987118	0.44550813…		frontier n-241.md
259	16.096473	16.591379	0.49490418…		frontier n-259.md
263	16.224185	16.733167	0.50898081…		frontier n-263.md
268	16.382380	16.878815	0.49643464…		frontier n-268.md
269	16.413830	16.901514	0.48768311…		frontier n-269.md
270	16.445218	16.928163	0.48294353…		frontier n-270.md
271	16.476545	16.948948	0.47240242…		frontier n-271.md
272	16.507810	16.965587	0.45777588…		frontier n-272.md
273	16.539014	16.983926	0.44491103…		frontier n-273.md
297	17.240669	17.740418	0.49974824…		frontier n-297.md
301	17.359715	17.846668	0.48695189…	open≈	frontier n-301.md
302	17.389345	17.870014	0.48066809…	open=	frontier n-302.md
303	17.418924	17.909959	0.49103397…		frontier n-303.md
304	17.448451	17.929428	0.48097650…		frontier n-304.md
305	17.477926	17.950467	0.47253963…		frontier n-305.md
306	17.507351	17.963434	0.45608236…		frontier n-306.md
307	17.536725	17.980863	0.44413774…		frontier n-307.md

Results on these cases

70 results in the register on these cases, oldest first

2005 published T-007 · 48 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-180 (reported) and T-205
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 48 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
2026-09-16 published T-057 · case 211
$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record
Upper bound confirmed superseded by T-188
de Winter · de Winter n211 2026-09-16 · packet · register
2026-09-22 published T-044 · case 68
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
Lower bound confirmed superseded by T-205
Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-27 published T-046 · case 68
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
Lower bound recorded superseded by T-205
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 · 46 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-098, T-127, T-136, T-146, T-173, T-174, T-182, T-186, T-190, T-197, T-218, T-219, T-222 and T-224
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-070 · case 68
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
Lower bound confirmed superseded by T-205
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 · case 68
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 · 48 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 68
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · case 68
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-073 · cases 102, 103
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-080 · cases 102, 103
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-092 · 7 of these cases
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
2026-10-05 published T-098 this result · 48 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
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · 7 of these cases
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-139, T-174, T-186, T-197 and T-222
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · 9 of these cases
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-139, T-146, T-174, T-190, T-197 and T-222
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · 6 of these cases
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-127, T-173, T-174, T-186 and T-218
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-118 · case 68
Exact rational ceiling refinements at $n = 68$
Upper bound confirmed superseded by T-174
Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register
2026-10-07 published T-119 · cases 270, 272
Three exact feasible upper bounds at n=266,270,272 from new source arrangements
Upper bound confirmed on these cases, superseded by T-174 and T-224
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-07 published T-120 · case 102
Reported s(102) ≤ alpha_102 with a degree-8 exact side form
Upper bound recorded superseded by T-146
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-121 · case 106
Reported s(106) ≤ alpha_106 with a degree-32 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-122 · case 152
Reported s(152) ≤ alpha_152 with a degree-40 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-123 · case 177
Reported s(177) ≤ alpha_177 with a degree-32 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-08 published T-125 · 7 of these cases
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-139, T-146, T-174 and T-218
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · 11 of these cases
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · 5 of these cases
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-174 and T-197
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · case 270
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by 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 · 5 of these cases
Six exact rational certificates from Couzo
Upper bound confirmed on these cases, superseded by T-174, T-186, T-190 and T-224
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · 14 of these cases
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · 8 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-174, T-186 and T-224
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-09 published T-139 · case 126
A more precise certificate of Xu's 126-square arrangement, from Daniel
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register
2026-10-10 published T-140 · 5 of these cases
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174, T-190 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-144 · cases 123, 126
Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported
Lower bound recorded
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register
2026-10-10 published T-146 · 8 of these cases
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-147 · case 207
A packing of 207 squares from two placed patches of Xu's packing of 70
Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register
2026-10-11 published T-149 · cases 155, 156, 211
New packings of 155, 156 and 211 squares, each with an exact rational certificate
Upper bound confirmed superseded by T-186, T-188 and T-197
Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register
2026-10-11 published T-151 · 8 of these cases
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-174, T-186, T-219 and T-224
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-11 published T-152 · cases 304, 305
Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares
Upper bound confirmed superseded by T-174 and T-186
Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register
2026-10-11 published T-159 · case 102
Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout
Upper bound confirmed superseded by T-146
Calle after Xu, Levy · Calle n102 2026-10-12 · packet · register
2026-10-12 published T-160 · case 154
A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly
Upper bound confirmed superseded by T-222
Daniel after Couzo, Levy · Daniel n154 2026-10-12 · packet · register
2026-10-12 published T-161 · cases 156, 237, 271
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Upper bound confirmed superseded by T-186, T-190 and T-219
Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 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-163 · cases 211, 241, 273, 307
s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang
Upper bound reviewed superseded by T-174, T-182 and T-188
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · 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-167 · case 207
A tighter rational realization of PingYou Ltd's packing of 207 squares
Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register
2026-10-12 published T-171 · case 272
An exact contact refinement of SQUISH's packing of 272 squares
Upper bound confirmed superseded by T-174
PingYou Ltd after Chaoweeraprasit, Daniel, Levy · PingYou n272 refinement 2026-10-12 · packet · register
2026-10-12 published T-172 · case 154
A packing of 154 squares from Couzo, continuing Daniel's arrangement
Upper bound confirmed superseded by T-222
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · packet · register
2026-10-12 published T-173 · case 207
Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there
Upper bound confirmed
Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register
2026-10-13 published T-174 · 44 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-175 · cases 211, 241, 304
New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly
Upper bound confirmed superseded by T-182, T-186 and T-188
Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register
2026-10-13 published T-180 · 48 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 · cases 211, 241, 302
Exact rational certificates at 211, 241 and 302 from Couzo
Upper bound confirmed superseded by T-182, T-186 and T-188
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
2026-10-13 published T-182 · cases 154, 211, 241
Exact rational certificates at 154, 211 and 241 from Couzo
Upper bound confirmed
Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register
2026-10-13 published T-183 · case 237
Polkovskyi's certificate for 237 squares, the smallest known side there when registered
Upper bound confirmed superseded by T-190
Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · Mishapolk records 2026-10-14 · packet · register
2026-10-14 published T-186 · 6 of these cases
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
2026-10-14 published T-188 · case 211
A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangement
Upper bound confirmed
Couzo after Ostashev, Daniel, Levy · Couzo verified bounds 2026-10-14 · packet · register
2026-10-14 published T-189 · case 237
Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds
Upper bound confirmed superseded by T-190
Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-14 · packet · register
2026-10-15 published T-190 · case 237
A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate
Upper bound confirmed
Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register
2026-10-16 published T-193 · 7 of these cases
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-205 · case 68
Mixed rectangle-measure lower bound $s (68) \geq 853 / 100$ 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-214 · case 106
Mixed rectangle-measure lower bound $s (106) \geq 10503 / 1000$ 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-217 · case 110
Mixed rectangle-measure lower bound $s (110) \geq 1073 / 100$ 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-218 · case 236
Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactly
Upper bound confirmed
Daniel after Chaoweeraprasit, Levy · Daniel n236 2026-10-16 · packet · register
2026-10-17 published T-197 · case 155
Daniel's smaller n155 packing, started from Ostashev's packing
Upper bound confirmed
Daniel after Ostashev, Levy · Daniel n155 explorer 2026-10-17 · packet · register
2026-10-17 published T-219 · case 271
Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly
Upper bound confirmed
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel n271 2026-10-17 · packet · register
2026-10-20 published T-222 · case 154
Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly
Upper bound confirmed
Daniel after Couzo, Ostashev, Levy · Daniel n154 sweep 2026-10-20 · packet · packet · 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

Links

On this site: The frontier survey · T-098 in the results table

On GitHub, at main

Register: T-098 in results.yaml, line 10036
Evidence: E-evand-exact-optima-2026-10-05-report · E-evand-exact-optima-2026-10-05-exact-replay · E-evand-exact-optima-2026-10-05-source-replay
Proofs and certificates: certificate certs
Sources: evand exact optima 2026-10-05 (its own site, retained copy)
Source packet: resources/web/evand-square-packing-2026-10-05/README.md
Artifacts: 13 artifacts and controls
resources/web/evand-square-packing-2026-10-05/README.md resources/web/evand-square-packing-2026-10-05/acquisition/sources.json resources/web/evand-square-packing-2026-10-05/receipts/first-party-check.json resources/web/evand-square-packing-2026-10-05/receipts/source-replay.json resources/web/evand-square-packing-2026-10-05/receipts/register-comparison.json resources/web/evand-square-packing-2026-10-05/receipts/reproduction-sample.json devtools/evand_exact_certificates.py devtools/apply_exact_optima.py devtools/check_rational_witness_independent.py docs/project/reviews/review-2026-10-06-evand-exact-optima.md docs/project/reviews/review-2026-10-06-evand-exact-optima-fix-check.md resources/web/evand-square-packing-2026-10-05/receipts/negative-controls.json tests/test_evand_exact_certificates.py
Case file: each case’s file is linked from its row above

T-098: Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

The cases

Results on these cases

Links

T-098: Exact optima of 48 known-best packings: `s(n) ≤ S'`, `3.5e-13` to `5.0e-11` below each printed side