Every Result

Date	S	Result	n	Credit	Rungs	Status
2026-10-06 established		Trump's packing is the only optimal packing of eleven squares, up to symmetry	11	Levy after Queuingtheorydotcom	V3 C2uniqueness	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 41$ , on a declared net	41	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 39$ , on a declared net	39	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 30$ , on a declared net	30	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 29$ , on a declared net	29	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 28$ , on a declared net	28	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 27$ , on a declared net	27	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 26$ , on a declared net	26	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 20$ , on a declared net	20	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 19$ , on the finest declared net yet	19	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published	★	Mixed rectangle-measure lower bound verified at $n = 18$ , on the finest declared net yet	18	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-06 published		Mixed rectangle-measure lower bound verified at $n = 19$ , past the best 18-square packing	19	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-103
2026-10-06 published		Mixed rectangle-measure lower bound verified at $n = 18$ , on a finer declared net	18	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-10-05 published		Exact certificates of 77 catalogue packings: `s(n) ≤ S'`, `1.2e-16` to `9.8e-15` above each side	28, 37, 39, 41, 50, 51, 53–55, 69–71, 83, 87, 88, 101, 104, 107–109, 122, 124, 125, 127–129, 145–151, 153, 170, 171, 173–176, 178, 179, 197, 198, 200–205, 226, 227, 229–235, 257, 258, 260–262, 264–267, 290, 291, 293–296, 298–300	Daniel after Couzo, de Winter, Ellsworth, Levy	V3 C3upper bound	confirmed
2026-10-05 published		Exact optima of 48 known-best packings: `s(n) ≤ S'`, `3.5e-13` to `5.0e-11` below each printed side	68, 102, 103, 106, 110, 123, 126, 131, 132, 152, 154–156, 172, 177, 180–182, 199, 206–211, 228, 236–241, 259, 263, 268–273, 297, 301–307	Daniel after Couzo, de Winter, Ellsworth, Levy	V3 C3upper bound	confirmed
2026-10-05 published	★	Mixed rectangle-measure lower bound verified at $n = 66$	66	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-05 published		Mixed rectangle-measure lower bound replayed at $n = 18$ , on a net the certificate declares	18	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-10-05 published	★	$s (12) \geq 7943 / 2000 = 3.9715$ , Daniel's points dilated and re-weighted	12	squarepacker after Daniel, Levy	V3 C3lower bound	confirmed
2026-10-05 published	★	Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$	67, 84	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-04 published	★	Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$	53, 54, 58, 70–73, 76, 87–95	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-04 published	★	Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$	42–44, 51, 56, 57, 67, 69, 72, 75, 84, 86, 88, 93–96	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-03 published		Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways	208, 209, 228, 263, 272, 303, 306	Couzo	V3 C3upper bound	confirmed superseded by T-098
2026-10-03 published	★	Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$	51, 52, 55, 58, 69–71, 73–76, 86–96	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-03 published		$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here	21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320	Daniel after Burns, Massaccesi	V0 C1optimality	reviewed
2026-10-02 published	★	Linear-measure lower bound replayed at $n = 101 \dots 105$	101–105	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-02 established		$s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted	12	Levy after Daniel	V3 C3lower bound	confirmed superseded by T-095
2026-10-02 published		$s (12) \geq 31360 / 7901 = 3.9691178 \dots$ , Daniel's certificate rescaled by $7902 / 7901$	12	squarepacker after Daniel	V3 C3lower bound	confirmed superseded by T-095
2026-10-02 published		Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70	20, 42, 70	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-090, T-091 and T-104
2026-10-02 published	★	Linear-measure lower bound replayed at $n = 82$	82	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-02 published	★	Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$	83, 85–88, 91–93, 96	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-02 published		Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$	83, 101–105	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-02 published		Mixed rectangle-measure lower bound replayed at $n = 76$	76	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-091
2026-10-02 published		Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$	84–87	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-075, T-090, T-091 and T-094
2026-10-01 published	★	Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$	19, 20, 26–31, 38–44, 53–58, 68–70, 74, 75, 88, 89, 93–95	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-01 published	★	Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$	37, 65, 66, 90, 92	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-01 published		Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$	19, 20, 26–31, 38–44, 53–56, 66, 68–70, 74–76, 86–90, 93–95	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-10-01 published	★	$s (77) = 9$ , by a mixed cover extended from Daniel's $s (60)$ cover	77, 78	wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3optimality	confirmed
2026-10-01 published	★	$s (59) = 8$ , by a mixed cover of points and grid-line segments	59	wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-30 published	★	$s (17) > 18641771 / 4000000 = 4.66044275$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed
2026-09-30 published	★	$s (k^{2} - 3) = k$ for every integer $k$ from 6 up; $k = 6 \dots 18$ are the cases held here	33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-29 published	★	$s (k^{2} - 1) = k$ for every integer $k \geq 3$ ; $k = 3 \dots 18$ are the cases held here	8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323	Karakuş	V3 C3optimality	confirmed
2026-09-29 published	★	$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$	8, 10–15, 17–24, 26–35, 37–48, 50–63, 65–80, 82–99, 101–120, 122–143, 145–168, 170–195, 197–224, 226–255, 257–288, 290–323	Karakuş	V3 C3lower bound	confirmed
2026-09-29 published		$s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$	11	Wang, Li after Kleddamag, Levy	V3 C3lower bound	confirmed superseded by T-060
2026-09-29 published	★	Trump's eleven-square packing is globally optimal	11	Queuingtheorydotcom after Levy, Kleddamag	V3 C3optimality	confirmed
2026-09-29 published		Reported equality of 12028 n11 row minima reproduced by a complete bound replay	11	wand125 after Tokoharu, Daniel	V3 C3audit	confirmed
2026-09-29 published		Rectangle-certificate ceiling `α·UB(n)` proved for $n = 1$ ..100; `B·UB(n)` on 64 grid rows	1–100	wand125 after Tokoharu, Daniel	V3 C3method limit	confirmed
2026-09-28 published		Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$	29, 38, 39, 41–44, 51–55, 57, 59, 60, 67–74, 86, 95	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-062, T-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110 and T-111
2026-09-28 published	★	$s (61) = 8$ , by monotonicity from T-062	61	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-28 published	★	$s (60) = 8$ , by a mixed cover of points and grid-line segments	60	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-28 published		$s (21) = 5$ by a point-only route	21	wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3simplification	confirmed
2026-09-28 published		$s (45) = 7$ by a second, point-only route	45	wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3simplification	confirmed
2026-09-28 published	★	$s (50) \geq 37 / 5 = 7.4$	50, 51	wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed
2026-09-28 published		$s (17) > 116511 / 25000 = 4.66044$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093
2026-09-28 published		$s (17) > 233009 / 50000 = 4.66018$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093
2026-09-27 published		Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways	68, 102, 103, 105, 106, 110, 123, 130–132, 152, 154–156, 172, 177, 180–182, 199, 206–210, 228, 236–241, 259, 263, 268–273, 292, 297, 301–307	Couzo	V3 C3upper bound	confirmed
2026-09-27 published	★	$s (45) = 7$ , by a mixed cover of points and grid-line segments	45	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-27 published	★	$s (21) = 5$ , by a mixed cover of points and grid-line segments	21	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-27 published		Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$	18–20, 26–31, 37–44, 51–61, 66–78, 86, 88–91, 94, 95	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V0 C0lower bound	recorded superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110 and T-111
2026-09-27 published		Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$	18–20, 26–28, 30–32, 40, 61, 75–78	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-051, T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-102, T-103, T-104, T-105, T-106, T-107 and T-109
2026-09-27 published		$s (17) > 466001 / 100000 = 4.66001$	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3lower bound	confirmed superseded by T-093
2026-09-26 published	★	$s (32) = 6$	32	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2026-09-26 published		$s (17) > 232001 / 50000 = 4.64002$	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3lower bound	confirmed superseded by T-093
2026-09-25 published		$s (17) > 231001 / 50000 = 4.62002$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093
2026-09-24 published		$s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , Chang's packings, certified exactly	83, 87	Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis	V3 C3upper bound	confirmed superseded by T-101
2026-09-24 published		$s (69) \leq 8.82719465572975$ , Ellsworth's degree-38 packing, certified exactly from its picture	69	Ellsworth after hmbelvedere, Cantrell, Schadt, Morandi	V3 C3upper bound	confirmed superseded by T-101
2026-09-24 established		Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry	11	Levy	V3 C2restricted optimality	confirmed superseded in part by T-060 and T-112
2026-09-24 established		Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within `rho` of his pose	11	Levy	V3 C3case exclusion	confirmed
2026-09-23 published		$s (21) \geq 5000 / 1001 = 4.995004995 \dots$	21	Daniel after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-052
2026-09-23 established		$s (21) \geq 122 / 25 = 4.88$	21	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-052
2026-09-22 published		$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$	11, 26–31	Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-060, T-068, T-074, T-105, T-106, T-107, T-108 and T-109
2026-09-22 published		Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files	26, 29, 39–41, 52–57, 68–73	wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110 and T-111
2026-09-22 published		$s (11) > 31 / 8 = 3.875$	11	Kleddamag after Levy, Guzhou0806, Mira	V3 C3lower bound	confirmed superseded by T-060
2026-09-22 established		$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$	11	Levy	V3 C3lower bound	confirmed superseded by T-060
2026-09-21 published		$s (17) \leq 4.6755300936045509 \dots$ , Bidwell's packing certified exactly	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3upper bound	confirmed
2026-09-21 published		$s (17) > 461300 / 99853 = 4.6197910929 \dots$	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3lower bound	confirmed superseded by T-093
2026-09-20 published		$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , and beneath it Mira's $s (17) \geq 4613 / 1000$	17	Guzhou0806, Mira after Levy, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-093
2026-09-20 established		The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$	11	Levy	V3 C3case exclusion	confirmed superseded by T-060
2026-09-19 established		$s (18) \geq 4679 / 1000 = 4.679$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-09-19 established		$s (18) \geq 1871 / 400 = 4.6775$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-09-19 established		$s (18) \geq 187 / 40 = 4.675$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-09-18 established		$s (18) \geq 467 / 100 = 4.67$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102
2026-09-16 published		$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record	211	de Winter	V3 C3upper bound	confirmed superseded by T-098
2026-09-09 established		$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$	11	Levy	V3 C3lower bound	confirmed superseded by T-060
2026-09-09 established		$s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate	11	Levy	V3 C3lower bound	confirmed superseded by T-060
2026-09-09 established		$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$	11	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-060
2026-09-08 established		Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$	11	Levy	V3 C3case exclusion	confirmed superseded in part by T-060
2026-09-06 established		$s (11) \geq 38100 \sqrt{8100042893309449} / 899996306539 = 3.8100257 \dots$	11	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-060
2026-09-05 established		$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$	20, 21	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-052 and T-104
2026-09-04 published	★	$s (k^{2} - 2) = k$ for every integer $k \geq 2$ ; $k = 3 \dots 18$ are the cases held here	7, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322	chelokot	V3 C3optimality	confirmed
2026-09-04 published		Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$	10–324	Karakuş; chelokot	V3 C3correction	confirmed
2026-09-04 established		$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$	19–21	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-052, T-103 and T-104
2026-09-04 established		$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$	17–19	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-093, T-102 and T-103
2026-09-04 established		$s (11) \geq 381 / 100 = 3.81$	11	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-060
2026-09-04 established		$s (12) \geq 99 / 25 = 3.96$	12	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-095
2026-09-03 established		Goebel's $n = 5$ optimum is rigid at fixed side: its pose is an isolated feasible point	5	Levy	V3 C3rigidity	confirmed
2026-08-31 established		Bentz 2010, Lemma 10 is false as printed and true as corrected to $(1.74, 1)$	13	Levy after Bentz	V3 C3correction	confirmed
2026-08-31 established		The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$	17, 18	Levy after Bentz	V3 C3method limit	confirmed
2026-08-31 established		$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001	18	Levy after Bentz	V3 C3lower bound	confirmed superseded by T-102
2026-08-31 established		$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set	17	Levy after Bentz	V3 C3lower bound	confirmed superseded by T-093
2026-08-30 established		Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order	40	Levy	V3 C3rigidity	confirmed
2026-08-30 established		Goebel's $n = 5$ packing is second-order rigid at fixed side	5	Levy	V3 C3rigidity	confirmed
2026-08-29 established		$s (29) \leq 5.933833 \dots$ , by a Krawczyk interval certificate	29	Levy	V3 C3upper bound	confirmed
2026-08-25 published		$s (12) \geq 15680 / 3951 = 3.9686155 \dots$	12	Daniel after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-095
2026-08-24 established		$s (11) \geq 2 + 4 / \sqrt{5}$ , by a repair of Stromquist 2003's Figure 14 point set	11	Levy after Stromquist	V3 C3lower bound	confirmed superseded by T-060
2026-08-21 published		$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015	18, 19	Massaccesi after Burns	V3 C3lower bound	confirmed superseded by T-102 and T-103
2026-08-21 published		$s (17) \geq 22529 / 5000 = 4.5058$	17	Massaccesi after Burns	V3 C3lower bound	confirmed superseded by T-093
2018 published		$s (37) \geq 5 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ and $s (61) \geq 7 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ , by optimal piercing	37, 61	Bašić, Slivková	V3 C1lower bound	reviewed superseded by T-063 and T-069
2010-09-13 published		$s (46) = 7$	46	Bentz	V3 C3optimality	confirmed
2010-09-13 published		$s (13) = 4$	13	Bentz; Daniel after Burns, Massaccesi	V3 C3optimality	confirmed
2010-09-13 published		Bentz 2010, Theorem 8 ( $s (46) \geq 7$ ) is correct as printed, machine-audited in full	46	Bentz	V3 C3audit	confirmed
2005 published		$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$	4–324	Nagamochi	V0 C1lower bound	incomplete
1979 published		Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$	11	Trump	V3 C3upper bound	confirmed

T-112

Claim: Every packing of eleven unit squares in a square of side T = 3.8770835900228141773…, the least side T-060 proves possible, is Walter Trump's 1979 packing after one of the eight symmetries of the container and a relabelling of the squares. Independent rotations and boundary contact are allowed. A quarter-turn reparametrization of one square is the same physical square and does not count as another packing. No symmetry of the container maps Trump's packing to itself, so there are exactly eight optimal packings as unlabelled configurations, the images of one another.

This is a direct corollary of T-060 and adds no computation. T-060's proof embeds any packing of side S <= T concentrically in the rational cap U > T, where its 2,180 exclusions, the D4 reduction to case 438, the closed capture partition and pose inclusion all apply, and maps it rigidly into the fixed side-T frame, where the local isolation lemma forces the exact construction. For S < T the construction's span T is a contradiction, which is T-060. For S = T nothing is contradicted: the packing is the construction after the alignment, a D4 element composed with the case-438 quarter turn, which is this result.

It is T-036's equality clause without T-036's restriction to six squares at angle 0 and five near Trump's tilt, and with reflections, which leave that family.
Composition: One entry, E-n011-optimum-uniqueness, proof-audited, which cites the completed exact executions of E-n011-global-optimality-independent unchanged and adds one prose step, that every premise of T-060's endpoint is stated for side S <= T and at S = T yields the construction rather than a contradiction. Proof-audited with a proof block supports V3. The step is prose and not mechanised, so the declared confirmation is C2, as for T-036's composing step, though every quantity it consumes is T-060's and re-implemented sharing named components.

T-060's evidence, E-n011-global-optimality-independent, is a premise of the entry and is not cited here beside it: it claims the exact value, which would give this result a bound's standing that it does not have.
Next rung: C3 needs the step at S = T mechanised, or reviewed again by a distinct reviewer with the oversight record the ladder asks for. The Lean theorem ElevenSquare.optimality in 11SquaresFormalized, whose statement the project's audit of 6 October reads as $s (11)$ = T, states the lower bound and the construction and no equality case; a Lean statement of the equality case would be the route to rung 5, once a formalization is reviewed here. Rung 4 needs what T-060's rung 4 needs: the owner's oversight record and a second adversarial review of the global chain.
Significance: A substantive case result: it completes the classification at eleven squares, the optimum being a single rigid packing up to symmetry, with no sliding family and no second contact type at T, in the case T-060 settled. It moves no bound and adds no method, since it is T-060's argument read at equality, which is what an S4 would need; it is one step above T-036's restricted equality case.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1

T-111

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 $s (41) \geq 271 / 40$ = 6.775.

The certificate is a density of 376 uniform rectangles in D4 orbits, and no point mass, of total mass 4099999/100000, 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 416 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 169/25 = 6.76, the source's rectangle certificate rect_n41_L676 and the reported (T-068) and verified (T-074) lower bound at $n = 41$ before it, by 0.015.

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, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 409 and 415, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 416 directions (devtools.audit_wand125_declared_net cpp-sample, about 37 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 41$ , by 0.015 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 41 register evidence 1 evidence 2 source review

T-110

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 $s (39) \geq 133 / 20$ = 6.65.

The certificate is a density of 600 uniform rectangles in D4 orbits, and no point mass, of total mass 3899999/100000, 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 416 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 1327/200 = 6.635, the source's rectangle certificate rect_n39_L6635 and the reported (T-068) and verified (T-074) lower bound at $n = 39$ before it, by 0.015.

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, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 394 and 415, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 416 directions (devtools.audit_wand125_declared_net cpp-sample, about 83 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 39$ , by 0.015 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 39 register evidence 1 evidence 2 source review

T-109

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 $s (30) \geq 11767 / 2000$ = 5.8835.

The certificate is a density of 402 uniform rectangles in D4 orbits, and no point mass, of total mass 2999999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 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 832 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 47/8 = 5.875, the source's rectangle certificate rect_n30_L5875 and the reported (T-068) and verified (T-074) lower bound at $n = 30$ before it, by 0.0085.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 347 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 832 directions (devtools.audit_wand125_declared_net cpp-sample, about 59 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 30$ , by 0.0085 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 30 register evidence 1 evidence 2 source review

T-108

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.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an independent implementation, C3, on the census route. The source's checker was replayed at sampled directions and its axis tables at the axis only, so no second method and no complete reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete replay of the source's own checker, the bundle's driver over all 416 nodes (42 CPU-hours by the source's own seconds, about 30 at the sample's rate here), would add a reproduction with the producer's code beside the rung.
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

T-107

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 $s (28) \geq 1147 / 200$ = 5.735.

The certificate is a density of 531 uniform rectangles in D4 orbits, and no point mass, of total mass 2799999/100000, 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 416 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 2289/400 = 5.7225, the source's rectangle certificate rect_n28_L57225 and the reported (T-068) and verified (T-074) lower bound at $n = 28$ 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, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 48 and 415, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 416 directions (devtools.audit_wand125_declared_net cpp-sample, about 29 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 28$ , by 0.0125 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 28 register evidence 1 evidence 2 source review

T-106

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 $s (27) \geq 11287 / 2000$ = 5.6435.

The certificate is a density of 439 uniform rectangles in D4 orbits, and no point mass, of total mass 2699999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 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 832 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 1127/200 = 5.635, the source's rectangle certificate rect_n27_L5635 and the reported (T-068) and verified (T-074) lower bound at $n = 27$ before it, by 0.0085.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 632 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 832 directions (devtools.audit_wand125_declared_net cpp-sample, about 76 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 27$ , by 0.0085 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 27 register evidence 1 evidence 2 source review

T-105

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 $s (26) \geq 1109 / 200$ = 5.545.

The certificate is a density of 477 uniform rectangles in D4 orbits, and no point mass, of total mass 2599999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 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 832 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above 2213/400 = 5.5325, the source's rectangle certificate rect_n26_L55325 and the reported (T-068) and verified (T-074) lower bound at $n = 26$ 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 832 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 241 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 832 directions (devtools.audit_wand125_declared_net cpp-sample, about 60 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 26$ , by 0.0125 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 26 register evidence 1 evidence 2 source review

T-104

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 $s (20) \geq 981 / 200$ = 4.905.

The certificate is a density of 327 uniform rectangles in D4 orbits, and no point mass, of total mass 1999999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 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 832 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above the 49/10 = 4.9 the record reported and verified at $n = 20$ , the source's rectangle certificate rect_n20_L49 (T-077), by 0.005.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 679 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 832 directions (devtools.audit_wand125_declared_net cpp-sample, about 46 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 20$ , by 0.005 above T-077, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 20 register evidence 1 evidence 2 source review

T-103

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 $s (19) \geq 193 / 40$ = 4.825.

The certificate is a density of 341 uniform rectangles in D4 orbits, and no point mass, of total mass 1899999/100000, on a net it declares itself, finer than any it declared before: core side 4999/5000 and 2073 half-angle tangents of step 1/5002. 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 2073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above the 48229/10000 = 4.8229 of T-100, the source's earlier certificate and the reported and verified lower bound at $n = 19$ before it, by 0.0021, and supersedes it.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 2073 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 1021 and 2072, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 2073 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 2073 directions (devtools.audit_wand125_declared_net cpp-sample, about 97 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 19$ , by 0.0021 above T-100, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 19 register evidence 1 evidence 2 source review

T-102

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 $s (18) \geq 941 / 200$ = 4.705.

The certificate is a density of 324 uniform rectangles in D4 orbits, and no point mass, of total mass 1799999/100000, on a net it declares itself, finer than any it declared before: core side 4999/5000 and 2073 half-angle tangents of step 1/5002. 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 2073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.

The value is above the 588/125 = 4.704 of T-099, the source's earlier certificate and the reported and verified lower bound at $n = 18$ before it, by 0.001, and supersedes it.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 2073 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 1234 and 2072, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.

wand125 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.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 2073 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 2073 directions (devtools.audit_wand125_declared_net cpp-sample, about 95 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.
Significance: The strongest verified lower bound on record at $n = 18$ , by 0.001 above T-099, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.
Novelty: previously-published Present in an identified source
Records: n = 18 register evidence 1 evidence 2 source review

T-100

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 $s (19) \geq 48229 / 10000$ = 4.8229.

The certificate is a density of 313 uniform rectangles in D4 orbits, and no point mass, of total mass 1899999/100000, on the net T-096's certificate declares: core side 999/1000 and 416 half-angle tangents of step 1/1001, with 999/1000 (1 + 1/1001) = 500499/500500 < 1. The source reports every core at a net angle 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 the 1927/400 = 4.8175 of the source's rectangle certificate rect_n19_L48175 (T-074), the reported and verified lower bound at $n = 19$ before it, by 0.0054. It is also above (7 + sqrt 7)/2 = 4.8228757..., the side of Hämäläinen's packing of 18 squares and the verified upper bound at $n = 18$ , by about 2.43e-5, so $s (18)$ < $s (19)$ , as the source notes.

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 6 October review of the certificate found no defect. The source's own checker was replayed here at 10 of the 416 directions, each returning the certificate's own record, and not in full.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 6 October 2026 on jlevy/squares#366. The source says parts of the work were produced with AI assistance under human direction.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an independent implementation, C3, on the census route, whose per-certificate conditions for a declared net the soundness review of the declared-net change set and this certificate meets. The source's checker was replayed at 10 directions and its axis tables were not run in full here, so no second method and no complete reproduction with the producer's code stands beside it.

The separation $s (18)$ < $s (19)$ is cited from $n = 18$ 's record, whose verified upper bound is E-lifted-q7-upper, the exact verification of Hämäläinen's packing; this entry adds no evidence for it beyond its own bound.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record. One is retained, the review of 6 October that read the certificate. A complete replay of the source's own checker, the bundle's driver over all 416 nodes by devtools.audit_wand125_declared_net replay (13.4 CPU-hours priced here), would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bound on record at $n = 19$ , by 0.0054, and the first to pass the side of the best packing of 18 squares, which separates $s (18)$ from $s (19)$ (Evan Daniel's question on jlevy/squares#281), by about 2.43e-5. The certificate kind and checker of T-069 and later on T-096's declared net; no new technique, at S3 as T-074 is, the separation a corollary of this bound and Hämäläinen's packing rather than a method or a family of bounds. The 6 October review confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 19 register evidence 1 evidence 2 source review

T-099

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 $s (18) \geq 588 / 125$ = 4.704.

The certificate is a density of 209 uniform rectangles in D4 orbits, and no point mass, of total mass 1799999/100000, on a net it declares itself: core side 1999/2000 and 832 half-angle tangents of step 1/2006, finer than the 416 of step 1/1001 at core 999/1000 that T-096's certificate declares. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 is past tan(pi/8), every unit square contains a concentric core of side 1999/2000 at a net angle strictly in its interior.

The source reports every such core capturing mass at least one: at all 831 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 changed only to allow more workers, and at the axis by exact integer tables.

The value is above T-096's 47/10, the source's earlier certificate and the reported and verified lower bound at $n = 18$ before it, by 0.004, and supersedes it.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 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 6 October review of the certificate re-derived the 832-node net in exact arithmetic and found no defect. The source's own checker was also replayed here in full, the bundle's own driver over all 832 nodes with its assertions on, each returning the certificate's own record: confirmed, reproduced with the producer's code.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 6 October 2026 on jlevy/squares#366. The source says parts of the work were produced with AI assistance under human direction.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an independent implementation, C3, on the census route, whose per-certificate conditions for a declared net the soundness review of the declared-net change set and this certificate meets. Beside it, the bundle's own driver replayed the source's checker over all 832 nodes, the axis tables among them, each returning the shipped record: a complete reproduction with the producer's code, not a second method.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record. One is retained, the review of 6 October that read the certificate. A second machine method would be a method-distinct decision of rotated coverage; the complete replay of the source's own checker already stands beside the rung (E-n018-wand125-mixed-4704-source-replay).
Significance: The strongest verified lower bound on record at $n = 18$ , by 0.004 above T-096, and the strongest reported. The certificate kind and checker of T-069 and later on a net twice as fine as T-096's, a parameter choice the same driver reads from the candidate; no new technique, at S3 as T-096, T-074 and T-045 are. The 6 October review confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 18 register evidence 1 evidence 2 evidence 3 source 1 source 2 review

T-101

Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

Claim: For each of 77 counts n from 28 to 300, $s (n)$ <= S', where S' is the side of an exact rational packing Evan Daniel published on 5 October 2026: this register's own known-best packing at that count, the one the Kingbird catalogue prints, solved to a nearby exact point of minimizing the side. The counts are 28, 37, 39, 41, 50, 51, 53 to 55, 69 to 71, 83, 87, 88, 101, 104, 107 to 109, 122, 124, 125, 127 to 129, 145 to 151, 153, 170, 171, 173 to 176, 178, 179, 197, 198, 200 to 205, 226, 227, 229 to 235, 257, 258, 260 to 262, 264 to 267, 290, 291, 293 to 296 and 298 to 300.

Each S' lies 1.2e-16 to 9.8e-15 above the side the catalogue prints, and each case record carries S' rounded up at the fourteen decimals printed as its verified upper bound, from 5.82444461667406 at $n = 28$ to 17.82412338847855 at $n = 300$ .

Before this the verified upper bound was the integer grid at 74 of the counts, above the printed side by up to 0.464 at $n = 101$ , 122, 145, 170, 197, 226, 257, 290 and 291, and at $n = 69$ , 83 and 87 the rounded-up sides of exact certificates of a binary64 parse of the catalogue's pictures (T-088, T-089), 2e-14 to 8.9e-13 above it. At the 22 counts whose printed side is a decimal alone, the verified upper bound now agrees with the report to one unit of its last place.

At the other 55 the catalogue also gives the side as a closed form, and S' lies 7.6e-20 to 1.8e-19 above it, the certificate's outward rounding, so the verified upper bound trails the report by one unit of the fourteenth decimal. Forty-nine of those closed forms are irrational, which no rational certificate reaches; the other six, at $n = 50$ , 171, 198, 230, 261 and 293, are rational, and an exact certificate of the packing at that side would reach them; at $n = 50$ a rational one appears to exist.

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 its separating-axis gap, and each with its box shrunk that unit past its least clearance.

Matched square for square with the known-best witness, every pose lies within 7.1e-4 of it, and every square the source does not list as free within 1.2e-7; four of the 77 were solved from the witness after the source's unpublished local squeeze ( $n = 127$ , 129, 260 and 299). The source's report that 75 of the exact points are KKT local minima rests on numerical checks and is not verified here; it bears on the packings, not on $s (n)$ .

Evan Daniel after the catalogue's finders and David Ellsworth's analytic minimisation, evand/square-packing, from this register's witnesses. On jlevy/squares#375 the author offered these certificates as an independent exact replay of the existing upper bounds and asked for nothing to be registered from them; the owner decided on 6 October 2026 to act on them where they lower a verified ceiling. The 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.
Composition: Two machine entries decide all 77 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. Each case record's verified value is S' rounded up at the printed precision, which $s (n)$ <= S' implies. The comparisons with the printed sides, the closed forms and the witnesses' poses are recorded values and set no rung, and the source's numerical KKT reports are outside the claim.
Next rung: V4 and C4 need two adversarial AI reviews of the complete claim by distinct reviewers and a human oversight record. A method-distinct second route, such as an interval decision of each certificate's pose, would be shown beside the rung. No certificate here has yet been decided by code of a reviewer's own. At the 49 counts whose report is an irrational closed form, the verified upper bound reaches the report only through an exact algebraic certificate of the packing at that side. At the six rational ones, $n = 50$ , 171, 198, 230, 261 and 293, an exact certificate at that side would reach it, rational where the packing's exact point is; at $n = 50$ the source's own exact point, before its outward rounding, appears to be a rational one (the review's EC-1, think-l8gt). This entry gives neither.
Significance: Seventy-seven verified ceilings moved onto packings the record already reported, off the integer grid at 74 counts by up to 0.464, and to within one unit of the report's last place at 22: citable details of each case that move no reported side and no theorem, S2 as T-098's certificate of $n = 126$ , which took that count's ceiling off the grid the same way. No new packing and no reusable technique here; the exact solver is the source's, as in T-098.
Novelty: previously-published Present in an identified source
Records: 77 cases register evidence 1 evidence 2 evidence 3

T-098

Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

Claim: For each of 48 counts n from 68 to 307, $s (n)$ <= S', where S' is the side of an exact rational packing Evan 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': Francisco Couzo's at 46 counts (T-056, and at seven of them his revision of T-092) and Joost 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 $s (n)$ .

Evan Daniel after Francisco Couzo, Joost de Winter and David 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.
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 $s (n)$ .
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.
Novelty: previously-published Present in an identified source
Records: 48 cases register evidence 1 evidence 2 evidence 3

T-097

Claim: A rectangle density of wand125/square-packing-bounds, checked at coverage one and published on 5 October 2026, after those of T-094, proves $s (66) \geq 843 / 100$ = 8.43.

The certificate is a density of 713 uniform rectangles in D4 orbits, and no point mass, of total mass 6599999/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-069 and later. The source reports it accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. It supersedes the source's mixed_n66_L842 (T-069).

The value is above T-069's 421/50 = 8.42, the reported and verified lower bound at $n = 66$ before it, by 0.01. A packing of 67 squares contains one of 66, but at $n = 67$ the record already holds more in both lanes (212/25), so it does not carry further.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, 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 12 of the 201 directions, each returning the certificate's own record, and not in full.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 5 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3, on the census route whose per-certificate conditions the review of 5 October of that route set and this certificate meets. The source's checker was replayed at 12 directions and its axis tables were not run here, so no second method and no complete reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record. One is retained, the review of 5 October that read the certificate; the review of the census route that day read T-094's certificates and not this one. A complete replay of the source's own checker, devtools.audit_wand125_point_and_mixed mixed-replay n66-L843 over the 189 directions not yet replayed and mixed-merge (think-0fkt, about 14 CPU-hours), would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bound on record at $n = 66$ , above the replayed mixed certificate that held it (T-069) by 0.01. A further size from the generator and checker of T-069 and later, with no new technique, at S3 as those are. The review of 5 October that read the certificate confirmed S3.
Novelty: previously-published Present in an identified source
Records: n = 66 register evidence 1 evidence 2 source review

T-096

Claim: A rectangle density of wand125/square-packing-bounds, checked at coverage one on a net it declares and published on 5 October 2026, proves $s (18) \geq 47 / 10$ = 4.7.

The certificate is a density of 136 uniform rectangles in D4 orbits, and no point mass, of total mass 1799999/100000, on a net it declares itself: core side 999/1000 and 416 half-angle tangents of step 1/1001. The source's earlier mixed certificates, from T-069 on, use core side 9977/10000 on 201 tangents of step 83/40000. Since 999/1000 (1 + 1/1001) = 500499/500500 < 1, every unit square contains a concentric core of side 999/1000 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-069's certificates, and at the axis by exact integer tables.

The value is above the 939/200 = 4.695 of the source's rectangle certificate rect_n18_L4695 (T-045), the reported and verified lower bound at $n = 18$ , by 0.005. Its mass is below 19 too, but the record holds more there.

The certificate passed a complete replay here on 5 October 2026 of the bundle's own driver and the source's unchanged checker on the pinned tarball: all 416 net nodes, each regenerated record equal to the certificate's own. The replay runs the source's own algorithm and is not an independent decision of coverage. Its exact premises, those of the declared net among them, were recomputed here, every shipped record and input was bound to the declared net and the expanded candidate, and the 5 October review found no defect. This repository's clean-room verifier sqverify-fast, extended to read a declared net, also decided all 416 directions here, a second implementation that is not yet a registered verifier and on which no rung rests.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#366. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim on one certificate, on its own reported entry and its own replay entry. E-n018-wand125-mixed-470-source-replay replays it in full with the bundle's own driver and the source's checker: one interval-certified method, reproduced with the producer's code, C3. Coverage is decided by the source's C++ checker alone, byte for byte the checker the 28 September and 2, 3 and 5 October reviews read; the axis tables are a second implementation for one direction and not a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A second machine method would be a method-distinct decision of rotated coverage; the replay here runs the source's own checker. The checker's controls were made on the $n = 37$ certificate of the same kind. A sqverify-fast evidence entry (think-3ok2) would show beside the rung as independently re-implemented.
Significance: Raised the verified lower bound at $n = 18$ by 0.005, to within 0.123 of Hämäläinen's packing, by the certificate kind and checker of T-069 and later on a finer net that the same driver reads from the candidate: a parameter choice within the framework, no new technique, at S3 as those are. The 5 October review confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 18 register evidence 1 evidence 2 source review

T-095

Claim: $s (12) \geq 7943 / 2000$ = 3.9715, by squarepacker (Ryu Sungjoon) after Evan Daniel and this project's Route B (T-079), published on 5 October 2026 as release v1.1 of squarepacker/s12-lower-bound and reported on jlevy/squares#363. It raises the verified lower bound by 10266889/7898846000, about 0.0013, over T-079's 15680000/3949423, and leaves 57/2000 = 0.0285 to the grid's 4.

The certificate keeps Daniel's 1,736 points of T-049, dilated by 31382793/31360000 to the container 7943/2000 and rounded to the grid 1/4000000 one D4 orbit at a time, every coordinate within 11/39200000 of its exact image, with new weights found by linear programming: all 223 orbits positive, total 119974808/10^7 = 11.9974808 < 12. Every closed unit square in [0, 7943/2000]^2, at every angle, captures weight at least one, decided over the rational angle net 2 arctan(k/96000) with Daniel's per-bin shrink. The nets $N = 24000$ and 48000 refuse it, which a pass at one net does not need.

It was decided here on 5 October 2026 by Daniel's verifier, reproduced with the producer's code: built from the retained source with overflow checks on, VERIFIED over all 39,765 bins, least captured weight 10000050/10^7 at bin 0, every line as the source's log; and independently re-implemented by this repository's parent-core interval branch and bound over all 39,765 rows, which shares no code with the sweep or with the search that made the weights and gives the strict $s (12) > 7943 / 2000$ . squarepacker's own indep_check.cpp, replayed here at $N = 96000$ and 192000 with the source's minimum, is the producer's evidence: the search stopped when its range-restricted copy found no violation. All three refuse the source's two altered certificates.

squarepacker (Ryu Sungjoon), s12-lower-bound, archived as Zenodo 10.5281/zenodo.23157015, after Evan Daniel's points and verifier (T-049) and this project's re-weighting of them (T-079), whose tool the source says guided its own scripts, written from scratch. The source says the rescaling, the re-weighting, the verification runs and its tools were prepared with the help of Claude (Anthropic).
Composition: Primary at $n = 12$ : one certificate, no monotonicity. Two methods decide it: Daniel's arrangement sweep (exact-algebraic), the producer's verifier replayed here in full with overflow checks, and this repository's directed-rounding interval coverage of every row (interval-certified), which shares no code with it or with the search that made the weights; that is C3, with two methods shown beside the rung.

squarepacker's indep_check.cpp, replayed here too, is a second implementation of the sweep's method and the producer's own, and its range-restricted copy was the search's stopping rule, so it is producer evidence and confirms nothing beyond the other two (review F3). The exact audit of the file is a premise check and decides nothing. All rest on the certificate, the counting step and the shrink lemma on the net $N = 96000$ , and the native rows are the sweep's bins and sigma_k by design (review F9).
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; the one retained review read the argument and computed nothing (its F1), so a second one that runs its own exact checks would add most. Rung 5 needs a proof-assistant formalization reviewed by human experts; Daniel's Lean pose-box-tree checker, which kernel-checks T-049's certificate, could in principle take this one, as the source notes. The source's own analysis puts re-weighting of these points near its end below 560/141 = 3.97163; a higher bound needs new points or another method, and point covers stop short of 3.99. The exact value remains open; the conjecture is $s (12) = 4$ .
Significance: Raises the verified lower bound at $n = 12$ by 0.0013 over T-079, the largest step since T-049. New weights on Daniel's points by T-079's recipe, Route B, with a complete scan in every cycle of the search, so no new technique and not S4; T-078, a rescaling, is S2, and T-079, new weights, is S3. Its source shows these points nearly exhausted for re-weighting, about 1.3e-4 below 560/141, which answers T-079's open "the same points may carry the bound further". Not S5: point covers are capped below 3.99.
Novelty: previously-published Present in an identified source
Records: n = 12 register evidence 1 evidence 2 evidence 3 evidence 4 source 1 source 2 review 1 review 2

T-094

Claim: Two rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 5 October 2026, prove $s (67) \geq 212 / 25$ = 8.48 and $s (84) \geq 9411 / 1000$ = 9.411.

Each is a density of uniform rectangles in D4 orbits, 536 at $n = 67$ and 686 at $n = 84$ , and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-082, T-090 and T-091. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Both supersede the source's certificates of T-090 at the same count, mixed_n67_L8475 and mixed_n84_L94075.

Both values are above what the record held at their counts: the verified 1691/200 and 47/5, by 0.025 and 0.011, and T-090's reported 339/40 and 3763/400, by 0.005 and 0.0035. A packing of n + 1 squares contains one of n, but at $n = 68$ and 85 the record already holds more in both lanes (851/100 and 473/50), so neither carries further.

Each certificate was decided here on 5 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants of each 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 not run here.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in two comments of 5 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: Two primary certificates, each on its own reported entry and its own replay entry. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3. The source's checker and axis tables were not run here, so no second method and no reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A replay of the source's own checker, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-2026-10-05 --runners 2 (about 26.5 CPU-hours planned), would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bounds on record at $n = 67$ and 84, above the replayed rectangle and mixed certificates that held them by 0.025 and 0.011. Further sizes from the generator and checker of T-069, T-071, T-072, T-075, T-082, T-090 and T-091, with no new technique, at S3 as those are. The review of 5 October confirmed S3: that they are the source's first certificates verified here by an independently re-implemented replay is a fact about this repository's confirmation, not about the result.
Novelty: previously-published Present in an identified source
Records: n = 67 n = 84 register evidence 1 evidence 2 evidence 3 evidence 4 source review

T-091

Claim: Twelve rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 4 October 2026, after those of T-090, prove $s (53) \geq 3051 / 400$ = 7.6275, $s (54) \geq 1537 / 200$ = 7.685, $s (58) \geq 1587 / 200$ = 7.935, $s (70) \geq 3463 / 400$ = 8.6575, $s (71) \geq 8721 / 1000$ = 8.721, $s (73) \geq 8813 / 1000$ = 8.813, $s (76) \geq 1793 / 200$ = 8.965, $s (87) \geq 479 / 50$ = 9.58, $s (88) \geq 481 / 50$ = 9.62, $s (90) \geq 973 / 100$ = 9.73, $s (91) \geq 781 / 80$ = 9.7625 and $s (94) \geq 199 / 20$ = 9.95. A packing of n + 1 squares contains one of n, so the $n = 71$ value also bounds $n = 72$ , the $n = 88$ value $n = 89$ , the $n = 91$ value $n = 92$ and 93, and the $n = 94$ value $n = 95$ , where each is above the verified bound the record held: 17 counts in all.

Each is a density of 312 to 853 uniform rectangles in D4 orbits and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-082 and T-090. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Ten supersede the source's certificates at the same count: eight of T-082's, and at $n = 88$ and 94 two of T-090's of earlier that day. At $n = 53$ and 54 they are the source's first mixed certificates.

All 12 values were above what the record reported at their counts, by 0.004 to 0.03: over T-082 at eight counts, T-090 at $n = 88$ and 94, and the source's rectangle certificates (T-068, T-074) at $n = 53$ and 54. Over the verified lower bounds they replace, at the twelve counts and the five they carry to, the rise is 0.0125 to 0.1525.

Each certificate was decided here on 5 and 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants of each 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 not run here.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in 12 comments of 4 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: 12 primary certificates, each on its own reported entry and its own replay entry. $n = 72$ , 89, 92, 93 and 95 take the $n = 71$ , 88, 91 and 94 certificates directly, their masses being below those counts, so no step is added. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3. The source's checker and axis tables were not run here, so no second method and no reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need a human oversight record and two adversarial AI reviews of the claim by distinct reviewers: the 5 October review of these certificates is one, and the review of the census route of the same day decided the route, not these certificates' mathematics. Replays of the source's own checker (about 131 CPU-hours planned, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-evening-2026-10-04 --runners 6) would add a reproduction with the producer's code beside the rung, and only they bear on the five bundles that record their proof run on macOS arm64 (OF-2); a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bounds on record at 17 counts from $n = 53$ to 95 when decided, and at 14 since T-090's certificates were decided at $n = 72$ , 93 and 95 on 6 October; the strongest reported at all 12 of its own. Further sizes from the generator and checker of T-069, T-071, T-072, T-075, T-082 and T-090, with no new technique, at S3 as those are; the 5 October review of these certificates confirmed S3.
Novelty: previously-published Present in an identified source
Records: 17 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 evidence 11 evidence 12 evidence 13 evidence 14 evidence 15 evidence 16 evidence 17 evidence 18 evidence 19 evidence 20 evidence 21 evidence 22 evidence 23 evidence 24 source review

T-090

Claim: Sixteen rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 3 and 4 October 2026, prove $s (42) \geq 2739 / 400$ = 6.8475, $s (43) \geq 2763 / 400$ = 6.9075, $s (44) \geq 2789 / 400$ = 6.9725, $s (51) \geq 747 / 100$ = 7.47, $s (56) \geq 3121 / 400$ = 7.8025, $s (57) \geq 3149 / 400$ = 7.8725, $s (67) \geq 339 / 40$ = 8.475, $s (69) \geq 431 / 50$ = 8.62, $s (72) \geq 219 / 25$ = 8.76, $s (75) \geq 447 / 50$ = 8.94, $s (84) \geq 3763 / 400$ = 9.4075, $s (86) \geq 9503 / 1000$ = 9.503, $s (88) \geq 769 / 80$ = 9.6125, $s (93) \geq 247 / 25$ = 9.88, $s (94) \geq 497 / 50$ = 9.94 and $s (95) \geq 1993 / 200$ = 9.965. A packing of n + 1 squares contains one of n, so the $n = 95$ value also bounds $n = 96$ , where it is above the verified bound the record held: 17 counts in all.

Each is a density of 358 to 630 uniform rectangles in D4 orbits and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-082. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Nine supersede the source's certificates at the same count: eight of T-082's, and at $n = 84$ T-071's. At the other seven they are the source's first mixed certificates.

All 16 values were above what the record reported at their counts, by 0.003 to 0.0375: over T-082 at eight counts, T-071 at $n = 84$ , and the source's rectangle certificates (T-046, T-068, T-077) at seven. At $n = 88$ and 94 the source's certificates of later that day (T-091) are higher, and at $n = 67$ and 84 those of 5 October (T-094), each verified here, so the verified lower bound there is theirs. Over the verified lower bounds they replace, at the other twelve counts and at $n = 96$ , the rise is 0.005 to 0.1175.

Each certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants of each 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 not run here.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in 16 comments of 3 and 4 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: 16 primary certificates, each on its own reported entry and its own replay entry. $n = 96$ takes the $n = 95$ certificate directly, its mass being below that count, so no step is added. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3. The source's checker and axis tables were not run here, so no second method and no reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need a human oversight record and two adversarial AI reviews of the claim by distinct reviewers: the 5 October review of these certificates is one, and the review of the census route of the same day decided the route, not these certificates' mathematics. Replays of the source's own checker (about 146 CPU-hours planned, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-2026-10-04 --runners 8) would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bounds on record at 13 counts from $n = 42$ to 96, and the strongest reported at 12 of its own. Further sizes from the generator and checker of T-069, T-071, T-072, T-075 and T-082, with no new technique, at S3 as those are; the 5 October review of these certificates confirmed S3.
Novelty: previously-published Present in an identified source
Records: 17 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 evidence 11 evidence 12 evidence 13 evidence 14 evidence 15 evidence 16 evidence 17 evidence 18 evidence 19 evidence 20 evidence 21 evidence 22 evidence 23 evidence 24 evidence 25 evidence 26 evidence 27 evidence 28 evidence 29 evidence 30 evidence 31 evidence 32 source review

T-092

Claim: For seven counts, $s (n)$ is at most the bound certified here from Francisco Couzo's packing as published on 3 October 2026: $s (208) \leq 14.926534459703512$ , $s (209) \leq 14.953939011860643$ , $s (228) \leq 15.604602454552252$ , $s (263) \leq 16.742270262031792$ , $s (272) \leq 16.968165867864400$ , $s (303) \leq 17.924341009860250$ and $s (306) \leq 17.963438139777141$ .

Each bound is the side the source prints, or where the printed pose's own side rounds up past it, that side plus one unit of its fifteenth decimal at $n = 208$ , 209 and 263 and two units at $n = 306$ . Each printed side is below the one Couzo published at the same count on 26 or 27 September (T-056), by 8.3e-6 at $n = 303$ up to 0.0105 at $n = 208$ , and below the Kingbird catalogue as captured on 30 September 2026.

Each bound is certified here twice, by T-056's two methods, which share no geometry code. The first is an exact rational packing, the source's decimal pose rounded to rationals at centre dilation 1, decided pair by pair and wall by wall over Q by two checkers that share no code. The second is outward-rounded interval arithmetic on the source's own pose as printed, with the true cosine and sine of each angle, which decides every pair and wall at 40 digits.

Francisco Couzo, franciscouzo/square-packing. The author said on jlevy/squares#227 that he found the 102 and 103 packings "with the help of Claude", and the repository itself states no AI assistance.
Composition: Both machine entries decide all seven bounds, each at its case's verified value, by different methods (exact-algebraic on a rational rounding of each pose; interval-certified on the printed pose), so every load-bearing part is C3, with two machine methods shown beside the rung. Their common mode is stated in E-franciscouzo-2026-10-03-interval-replay. The comparisons with Couzo's earlier sides and the retained catalogue are recorded values and set no rung.
Next rung: V4 and C4 need a second adversarial AI review of the complete claim by a reviewer distinct from the one of 2026-10-05, and a human oversight record. At $n = 306$ both routes refute the printed side for the printed pose, so the printed side itself needs a pose refined beyond the source's binary64 digits, or coordinates the source prints at higher precision; that would be a new packing, not a replay.
Significance: Seven substantive case results: each moves both the best known side and the verified ceiling at its count, by 8.3e-6 to 0.0105, below the packings that held them (T-056). S3 by the anchor "a substantive case result", as T-056; the source publishes no method, so nothing here is a reusable technique. Kept by the review of 2026-10-05, beside T-056 (S3, 49 such results) and T-088 and T-089 (S2, which move a best known side and no verified bound).
Novelty: previously-published Present in an identified source
Records: 7 cases register evidence 1 evidence 2 evidence 3

T-082

Claim: Twenty-two rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 3 October 2026, prove $s (51) \geq 373 / 50$ = 7.46, $s (52) \geq 151 / 20$ = 7.55, $s (55) \geq 966 / 125$ = 7.728, $s (58) \geq 1581 / 200$ = 7.905, $s (69) \geq 2153 / 250$ = 8.612, $s (70) \geq 3459 / 400$ = 8.6475, $s (71) \geq 1741 / 200$ = 8.705, $s (73) \geq 8809 / 1000$ = 8.809, $s (74) \geq 3547 / 400$ = 8.8675, $s (75) \geq 223 / 25$ = 8.92, $s (76) \geq 224 / 25$ = 8.96, $s (86) \geq 19 / 2$ = 9.5, $s (87) \geq 191 / 20$ = 9.55, $s (88) \geq 48 / 5$ = 9.6, $s (89) \geq 193 / 20$ = 9.65, $s (90) \geq 389 / 40$ = 9.725, $s (91) \geq 39 / 4$ = 9.75, $s (92) \geq 977 / 100$ = 9.77, $s (93) \geq 493 / 50$ = 9.86, $s (94) \geq 248 / 25$ = 9.92, $s (95) \geq 249 / 25$ = 9.96 and $s (96) \geq 997 / 100$ = 9.97.

Each is a density of 303 to 571 uniform rectangles in D4 orbits and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-069, T-071, T-072 and T-075. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Six supersede certificates of the same source this record holds verified at the same count: at $n = 76$ (T-072), 87, 91, 92 and 96 (T-075), and 90 (T-069).

21 of the values were above what the record reported at their counts, by 0.015 to 0.125: over the source's rectangle certificates (T-046, T-070, T-074, T-077) at 13 counts, and over its earlier mixed certificates directly at five and by monotonicity at $n = 86$ , 88 and 93. At $n = 96$ , 997/100 is below the side 10 that Evan Daniel's s(k^2 - 4) = k reports there (T-081).

The source's later certificates are higher at 16 of the counts, T-090's of 3 and 4 October at $n = 51$ , 69, 75, 86, 93 and 95 and T-091's of 4 October at $n = 58$ , 70, 71, 73, 76, 87, 88, 90, 91 and 94, each verified here, so the verified lower bound there is theirs. At the other six, $n = 52$ , 55, 74, 89, 92 and 96, these are the verified lower bounds, above those they replace by 0.005 to 0.03. A packing of n + 1 squares contains one of n, but at each next count the record already holds more, so none carries further.

Each certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants of each scaled below coverage one were refused: confirmed, independently re-implemented. At $n = 96$ the first control failed closed, its mutant scaled to 99/100 verifying at the one direction it ran, which showed the certificate's slack there and not a fault (think-0uia); the repaired control, which runs that mutant at every net direction, refused it at 187 of the 201, each at an exact capture below one.

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 not run here.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in 22 comments of 3 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: 22 primary certificates, each on its own reported entry and its own replay entry. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3. The source's checker and axis tables were not run here, so no second method and no reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need a human oversight record and two adversarial AI reviews of the claim by distinct reviewers: the 3 October review of these certificates is one, and the review of the census route of 5 October decided the route, not these certificates' mathematics. Replays of the source's own checker (about 162 CPU-hours planned, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-2026-10-03 --runners 8) would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest verified lower bounds on record at six counts, $n = 52$ , 55, 74, 89, 92 and 96, and the strongest reported at five of them (at $n = 96$ T-081 reports 10); at the other 16 the source's later certificates (T-090, T-091) are higher. Further sizes from the generator and checker of T-069, T-071, T-072 and T-075, with no new technique, at S3 as those are; the 3 October review of these certificates proposed S3, which the replays leave standing.
Novelty: previously-published Present in an identified source
Records: 22 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 evidence 11 evidence 12 evidence 13 evidence 14 evidence 15 evidence 16 evidence 17 evidence 18 evidence 19 evidence 20 evidence 21 evidence 22 evidence 23 evidence 24 evidence 25 evidence 26 evidence 27 evidence 28 evidence 29 evidence 30 evidence 31 evidence 32 evidence 33 evidence 34 evidence 35 evidence 36 evidence 37 evidence 38 evidence 39 evidence 40 evidence 41 evidence 42 evidence 43 evidence 44 source review

T-081

Claim: Evan Daniel's theorem s(k^2 - 4) = k for every integer k >= 5, published in his repository on 3 October 2026: the lower half at k = 5, 6 and 7 by his $s (21)$ , $s (32)$ and $s (45)$ certificates, and at every k >= 8 by one family of periodic measures; the upper half by the k x k grid.

This entry's scope is the projection of that theorem onto the cases this record holds, n <= 324, which is k = 5 to 18: $s (21) = 5$ , $s (32) = 6$ , $s (45) = 7$ , $s (60) = 8$ , $s (77) = 9$ , $s (96) = 10$ , $s (117) = 11$ , $s (140) = 12$ , $s (165) = 13$ , $s (192) = 14$ , $s (221) = 15$ , $s (252) = 16$ , $s (285) = 17$ and $s (320) = 18$ . The reported theorem does not end at k = 18. New to this record are k = 10 to 18; $s (21)$ , $s (32)$ and $s (45)$ are T-052, T-051 and T-053, with T-054 a second route at k = 7, $s (60) = 8$ is T-062, and wand125 published $s (77) = 9$ first, on 1 October (T-067).

For each k >= 8 the measure on [0,k]^2 is a corner module on [0,3]^2 in each corner, a profile of period one in a band of width 3 along each wall, and Lebesgue measure on [14/5, k - 14/5]^2, of total mass k^2 - 4D with D = 214770225571/200000000000, so that 4D exceeds 4. The source states that every closed unit square in [0,k]^2 has mass at least one, and reduces that, for every k >= 8, to one finite statement on the 9 x 9 box, Valid9.

Its Lean development kernel-checks, the source reports, that the tilted part, ValidTilt9, implies the theorem for every k >= 8, with the D4 reduction and the axis-parallel face proved; it does not prove ValidTilt9. ValidTilt9 is the claim of the source's exact-rational checker qx2_zm.py, the program that decided T-064's Valid7, which the source reports accepting all 16,200 root boxes in 115,268 leaves. The source reports three adversarial reviews of the certificate by AI agents and none by outside mathematicians.

A second, separately written exact checker reports ValidTilt9 too: wand125's, the checker whose Valid7 run was replayed here for T-064, in valid7-independent-check on 6 October 2026, over centres [0, 9/2]^2 and u in [0, 7/16] with no symmetry used, in 28,350 roots and 9,537,343 leaves, none uncertified. Its read log says qx2_zm.py was not read; both checkers were written with Claude models, by their sources' own statements.

Here the bundle's fast verify.sh passed on the retained bytes; it recomputes no tilted leaf. wand125's verify_tilt9.sh and an audit of its records passed here, 30 of its roots were certified again here, 21 of them with their published leaves, and two lighter covers were refused; that re-decides about 3% of its recorded cost. The reduction was built here from the retained bytes on 3 October, with only the three standard axioms. Nothing here has yet replayed either run of ValidTilt9 in full, and the verified lower bounds are unchanged.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method, at the request of jlevy/squares#316. The source's CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.
Composition: Compound, and the minimum is set by the lower half at k >= 8. The upper half is the k x k grid, E-basic-grid-upper. At k = 5, 6 and 7 the lower half is the source's earlier certificates, which this record holds as T-052, T-051 and T-053 (with T-054 beside it at k = 7), and this entry adds no evidence to them.

At k >= 8 it is two parts in series: the finite premise ValidTilt9, reported by the source's run (E-k2m4-evand-validtilt9-qx2-report) and by wand125's independent run (E-k2m4-wand125-validtilt9-report), and the reduction from it to every k >= 8, the source's Lean (E-k2m4-evand-lean-report), built here with its axiom receipt (E-k2m4-evand-bentz4-lean-build). The reduction is replayed and the finite premise is not: both runs of ValidTilt9 are reports, so the premise sets the minimum, the entry is V0, and the claim-chain review of 3 October reads it, C1.

The scope lists the cases this record holds; the theorem's own domain is every k >= 5, and a scope that can say so is think-kqi1.
Next rung: ValidTilt9 replayed in full by either checker: qx2_zm.py on the box-9 cover, about 226 CPU-hours at the source, in the shards the evand packet's replay plan gives, compared root for root with the published record (think-4uir); or wand125's run, the independent route, priced here from a sample at about 435 CPU-hours as published, or about 118 under its last options (think-hwpr). The Lean reduction is replayed already (E-k2m4-evand-bentz4-lean-build), and the claim chain is reviewed. Then a replayed entry for ValidTilt9, and the verified lower bounds at k = 10 to 18.

A replay of wand125's run lifts the ValidTilt9 part only, and with Daniel's own Lean beside it the result's code relation stays that of the producer's reduction.

The clean-room verifier's Milestone C (docs/project/specs/active/plan-2026-10-03-measure-verifier-milestone-c.md) remains a first-party route; it plans points and segments only, and box 9 also needs a uniform-polygon primitive and an exact argument inside the Lebesgue block, where every square has mass exactly one.
Significance: Scored as a claim: a bound family, the deficit-4 case of Friedman's conjecture, which would settle nine open cases of this record, k = 10 to 18, and every later k if the finite premise and its reduction pass replay and review here. S4 by the anchor "a reusable technique, bound family", as T-064 is; not S5, since no case it settles is central to this project. The score gates nothing.
Novelty: previously-published Present in an identified source
Records: 14 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 source review

T-080

Claim: A measure of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 proves $s (101) \geq 257 / 25$ = 10.28. Its mass is below 102 to 105, and a packing of any of those counts contains one of 101, so it also gives $s (102), s (103), s (104), s (105) \geq 257 / 25$ : five counts in all.

The measure is 333 point, 897 segment and 4 rectangle orbits under D4, with exact rational geometry, of total mass 10099999/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000. It is accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp: an outward-rounded interval branch and bound over centre boxes in one quadrant of the centre domain, which the measure's quarter-turn symmetry makes stand for the whole, counting points in the closed core, segments by the length inside and rectangles by the area inside.

The value exceeds Green's reported 10.2467362... at $n = 101$ to 104, by more than 0.0332, and Nagamochi's 1 + sqrt(86) = 10.2736... at $n = 105$ . It passed a complete replay here on 2 October 2026 of the source's unchanged checker and replay functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replay runs the source's own algorithm and is not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#294. The source says parts of the work were produced with AI assistance under human direction.
Composition: One primary certificate, on its reported entry and its replay entry. $n = 102$ to 105 take it directly, its mass being below them, so no step is added. The replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, read before the replay. A second machine method would be a method-distinct decision of rotated coverage for a linear measure; the replay here runs the source's own checker.
Significance: Raised the verified lower bound at five counts, $n = 101$ to 105, from Nagamochi's closed form past Green's reported values, by a certificate kind new to this record that adds points and segments to the rectangle densities. Further sizes from one generator at S3, as T-073; the 2 October review of the certificates confirmed S3. Nagamochi's closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03); the verified floor it left at these counts, Karakuş's general bound (T-083), is lower still.
Novelty: previously-published Present in an identified source
Records: 5 cases register evidence 1 evidence 2 source review

T-079

Claim: $s (12) \geq 15680000 / 3949423$ = 3.97020020..., by re-weighting Evan Daniel's 1,736 points. It raises the verified lower bound by 33774720/31204391123, about 0.0010824, over squarepacker's $s (12) \geq 31360 / 7901$ (T-078), and leaves a gap of 117692/3949423, about 0.0298, to the grid's 4.

The certificate keeps the points of Daniel's $s (12)$ certificate (T-049), every coordinate scaled by 3951000/3949423, and solves for new weights by linear programming over their 223 D4 orbits, with rows from the near-tight cells Daniel's verifier reports; the total is 14970347/1250000 = 11.9762776 < 12. Every closed unit square in [0, 15680000/3949423]^2, at every angle, captures weight at least one, decided over the rational angle net 2 arctan(k/96000) with Daniel's per-bin shrink.

It was decided here three ways on 2 and 3 October 2026: by Daniel's verifier, built unmodified with overflow checks on, in the producing run and again by a review lane that did not produce it, both VERIFIED over all 39,765 bins, least captured weight 10000045/10^7; and by this repository's parent-core interval branch and bound over all 39,765 rows, which shares no code with the sweep or with the LP and gives the strict $s (12) > 15680000 / 3949423$ . Both checkers refuse two mutated certificates.

This project's re-weighting, after Evan Daniel's certificate and verifier (T-049), whose method is Burns's and Massaccesi's, and after squarepacker's rescaling of that certificate (T-078), which showed a finer net leaves room for a larger container. The same work rescaled Daniel's certificate whole to $s (12) \geq 1568000 / 395039$ , which this bound implies.
Composition: Primary at $n = 12$ : one certificate, no monotonicity. Two methods decide it: Daniel's arrangement sweep (exact-algebraic), run by the producing lane and replayed by the review lane, and this repository's directed-rounding interval coverage of every row (interval-certified); that is C3, with two methods shown beside the rung. The exact audit of the file is a premise check, well-formedness only, and decides nothing. All rest on the certificate, the counting step and the shrink lemma on the net $N = 96000$ , and the native rows are the sweep's bins and sigma_k by design.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. Rung 5 needs a proof-assistant formalization reviewed by human experts; Daniel's Lean pose-box-tree checker, which kernel-checks T-049's certificate, could in principle take this one. The same 1,736 points carried the bound further: the LP total was 0.024 below 12 at this side, and squarepacker's re-weighting of them reached 7943/2000 (T-095). The exact value remains open; the conjecture is $s (12) = 4$ .
Significance: Raises the verified lower bound at $n = 12$ again, by 0.0016 over Daniel's T-049 and 0.0011 over T-078, to within 0.0298 of 4, the best verified bound at $n = 12$ this repository knows of. T-049, which raised it by 0.0086, is S3, and the step here is new weights on Daniel's geometry, not a rescaling alone. Not S4: the method, a weighted cover re-optimised at a finer net, is Daniel's, Burns's and Massaccesi's. Not S5: pure point covers are capped well short of 4, so the central question at $n = 12$ does not move.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 12 register evidence 1 evidence 2 evidence 3 evidence 4 source 1 source 2 review

T-078

Claim: $s (12) \geq 31360 / 7901$ = 3.96911783..., by squarepacker (Ryu Sungjoon) after Evan Daniel, published on 2 October 2026 and reported on jlevy/squares#309. It raises the verified bound by 15680/31216851, about 0.000502, over T-049.

The certificate is Daniel's 1,736-point weighted certificate of T-049 with every coordinate and the container multiplied by 7902/7901 and the weights unchanged, total 11.9738036 < 12. The rescaling and its verification on the finer angle net are squarepacker's; the certificate and the verifier that decides it are Daniel's. Every closed unit square in [0, 31360/7901]^2, at every angle, captures weight at least one, decided over the rational angle net 2 arctan(k/24000) with a per-bin shrink of 1/(cos d + sin d). The nets $N = 6000$ and 12000 refuse it, which a pass at one net does not need: each net's test is a complete sufficient condition.

It was replayed here on 2 October 2026 by Daniel's verifier and by squarepacker's own indep_check.cpp at $N = 24000$ , least captured weight 10000056/10^7 at bin 0 in both, as the source records, and decided completely by this repository's native parent-core interval branch and bound over all 9,942 rows, which shares nothing with the two sweeps and gives the strict $s (12) > 31360 / 7901$ .

squarepacker (Ryu Sungjoon), s12-lower-bound, archived as Zenodo 10.5281/zenodo.23106582, after Evan Daniel's certificate and verifier (T-049). The source's README says the rescaling, the verification runs and its checker were prepared with the help of an AI assistant from Anthropic, which it names.
Composition: Primary at $n = 12$ : one certificate, no monotonicity. Two entries whose methods differ decide it: the producer's two arrangement sweeps, Daniel's verify and squarepacker's indep_check.cpp (exact-algebraic, replayed here, one method in two implementations), and this repository's directed-rounding interval coverage of every row (interval-certified); that is C3, with the two methods shown beside the rung. All three rest on the certificate, the counting theorem and the shrink lemma on the net $N = 24000$ , and the native rows are Daniel's bins and sigma_k by design.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; the one retained review was written by the lane that ran the replays, so a separately prompted read is the next step. Rung 5 needs a proof-assistant formalization reviewed by human experts; the source notes that Daniel's Lean pose-box-tree checker, which kernel-checks T-049's certificate, could in principle take this one. The exact value remains open; the conjecture is $s (12) = 4$ .
Significance: True and replayed in full by three checkers, and it raises the verified lower bound at $n = 12$ , but by 0.000502, moving the gap to the grid's 4 from 0.0314 to 0.0309. It rescales T-049's certificate until a finer angle net stops accepting it, which spends that certificate's slack and adds no technique, as the source says itself; T-061 is the precedent for scoring such a step by what it changes.
Novelty: previously-published Present in an identified source
Records: n = 12 register evidence 1 evidence 2 evidence 3 source 1 source 2 review 1 review 2

T-077

Claim: Three rectangle-density certificates in wand125/square-packing-bounds, raised on 2 October 2026, prove $s (20) \geq 49 / 10$ = 4.9, $s (42) \geq 2731 / 400$ = 6.8275 and $s (70) \geq 3451 / 400$ = 8.6275. Each raises the certificate T-068 reports at its count, which T-074 replayed, by 0.0025 to 0.0125.

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The certificates were built with Tokoharu's solver, their weights scaled by one exact factor each to that mass. The same revision raised four more rungs, at $n = 59$ , 77, 91 and 93, which are below what this record holds there: the exact values $s (59) = 8$ and $s (77) = 9$ , and the same revision's mixed certificates at $n = 91$ and 92, the latter carried to $n = 93$ (T-075).

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, in two cloud batches on 3 October 2026 whose receipts were merged into the packet on 6 October, and each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction: confirmed, reproduced with the producer's code. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises.

At $n = 20$ and 42 the values raised the verified lower bound, from T-074's 1959/400 and 1363/200, and later on 6 October T-104's check2 certificate raised it again at $n = 20$ and T-090's mixed certificate at $n = 42$ ; at $n = 70$ the mixed certificate of T-091 holds more.

Each was also decided by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions at Tokoharu's threshold 10001/10000 in the census of 3 October, and two mutants of each scaled below that threshold were refused on 6 October: confirmed, independently re-implemented as well. It shares no code with the source's checker and runs the same method, a second implementation and not a second method.

wand125 after Tokoharu and Levy, square-packing-bounds. No issue requests these certificates; the import of 2 October took them in from the source's own commits. The source says parts of the work were produced with AI assistance under human direction.
Composition: Three primary certificates at one source revision, each on its own count, with two complete replays each: the source's own checker, reproducing its records, and sqverify-fast on the census route for format T that the review of 6 October set and these certificates meet, an independent re-implementation. One interval-certified method: C3, and no second method.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record: the 2 October review of the afternoon certificates was the retaining lane's own and read them before the replays, and the 6 October review of the format T route decided the route, not these certificates' mathematics. A second machine method would be a method-distinct decision of rotated coverage; the source's C++ checker and sqverify-fast decide it by one method, and the exact preflight does not decide it.
Significance: Raised the verified lower bound at $n = 20$ and 42, by 0.0025 and 0.0125 over T-074's replayed certificates, and the strongest rectangle-density lower bounds on record at all three counts. Raised rungs from one generator, as T-068's were, at S3; the 2 October review of the afternoon certificates proposed S3.
Novelty: previously-published Present in an identified source
Records: n = 20 n = 42 n = 70 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source 1 source 2 review 1 review 2

T-076

Claim: A measure of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 proves $s (82) \geq 233 / 25$ = 9.32. The measure is of the kind and checker of T-073: point masses, segments of uniform density and rectangles of uniform density, in D4 orbits with exact rational geometry, of total mass 82 - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, with 86 point, 222 segment and 774 rectangle orbits. They are those of T-073's $n = 83$ certificate, every coordinate scaled by exactly 932/935, with the masses solved again.

The source reports it accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, byte for byte the verifier of T-073, in 153,579,479 nodes, and says the full replay was run again from the published tarball by the same implementation. The value exceeds Green's reported 2 sqrt(2) + (247 + 12 sqrt(2))/41 = 9.2667335..., the value this record reported at $n = 82$ , by 0.0532664..., and does not rest on Green's argument, which jlevy/squares#308 reports the published material does not establish for k >= 4. Its mass is below 83, where T-075's 937/100 is higher.

It passed a complete replay here on 3 October 2026 of the source's unchanged checker and replay functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replay runs the source's own algorithm and is not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 2 October 2026 on jlevy/squares#294. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim from one certificate, on its reported entry and its replay entry. The replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone, which the 2 October review of the linear certificates read function by function.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; the review of these certificates was the retaining lane's own, and the checker's separately prompted review is T-073's. A second machine method would be a method-distinct decision of rotated coverage for a linear measure; the replay here runs the source's own checker. linear-control n82 would put a negative control on this certificate itself; the checker's controls are $n = 101$ 's.
Significance: Raised the verified lower bound at $n = 82$ from Nagamochi's closed form past Green's reported value, the one count from 82 to 85 where the record still reported Green's value, by a certificate of T-073's kind that does not rest on Green's argument. A further size from one generator and checker, at S3 as T-073 is; the 2 October review of the afternoon certificates proposed S3, which the replay leaves standing. Nagamochi's closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03); on the merged record the verified floor this replay raised at $n = 82$ was Karakuş's general bound (T-083), lower still.
Novelty: previously-published Present in an identified source
Records: n = 82 register evidence 1 evidence 2 source review

T-075

Claim: Six rectangle-density certificates that wand125/square-packing-bounds published on the afternoon of 2 October 2026 prove $s (83) \geq 937 / 100$ = 9.37, $s (85) \geq 473 / 50$ = 9.46, $s (87) \geq 237 / 25$ = 9.48, $s (91) \geq 97 / 10$ = 9.7, $s (92) \geq 39 / 4$ = 9.75 and $s (96) \geq 249 / 25$ = 9.96. A packing of 86 squares contains one of 85, and likewise at 88 and 93, so they also give $s (86) \geq 473 / 50$ , $s (88) \geq 237 / 25$ and $s (93) \geq 39 / 4$ .

Each is a density of 279 to 728 uniform rectangles in D4 orbits and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-069, T-071 and T-072. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Two supersede certificates this record held verified at the same count, at $n = 85$ (T-071) and $n = 92$ (T-069).

Each value is above what the record reported at its count: by 0.02 at $n = 83$ , over the linear certificate of T-073, and by 0.0025 to 0.095 at the others, over the source's rectangle certificates (T-068, T-074) and its earlier mixed ones. At $n = 96$ it is the first certificate in this record above Nagamochi's 1 + sqrt(79) = 9.8882.... At $n = 83$ , 85 and 87 the source measures its improvement from 92667/10000, below Green's reported 9.2667335..., so those figures overstate the margin; the bounds are unaffected.

Each certificate passed a complete replay here on 2 and 3 October 2026 of the source's unchanged checker and per-angle functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replays run the source's own algorithm and are not an independent decision of coverage. sqverify-fast, this repository's clean-room measure verifier, also decided each at all 201 net directions of the retained candidate in its census of 3 October 2026, and on 6 October two mutants of each scaled below coverage one were refused: independently re-implemented, beside the reproduction with the producer's code. It 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.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in four comments of 2 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: Six primary certificates, each on its own reported entry and its own replay entry. $n = 86$ , 88 and 93 take the $n = 85$ , 87 and 92 certificates directly, their masses being below those counts, so no step is added. Each replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. The axis tables are a second implementation for one direction and not a second method. Each certificate also has a third entry, sqverify-fast at all 201 net directions of the retained candidate: an independent implementation of the same method, recorded beside the rung and not a condition of it.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, written by the retaining lane before any replay. A second machine method would be a method-distinct decision of rotated coverage; the replays here, the source's own checker and sqverify-fast, run one method. The source's checker's controls were made on the $n = 37$ certificate of the same kind; mixed-control n96 would put one on a certificate of this release. sqverify-fast's controls are on all six.
Significance: Raised the verified lower bound at nine counts, $n = 83$ to 96, the first above Nagamochi's value at $n = 96$ . Further sizes from the generator and checker of T-069, T-071 and T-072, with no new technique, at S3 as those are; the 2 October review of the afternoon certificates proposed S3, which the replays leave standing.
Novelty: previously-published Present in an identified source
Records: 9 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 evidence 11 evidence 12 evidence 13 evidence 14 evidence 15 evidence 16 evidence 17 evidence 18 source 1 source 2 review

T-073

Claim: Two measures of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 prove $s (101) \geq 257 / 25$ = 10.28 and $s (83) \geq 187 / 20$ = 9.35. A packing of 102 to 105 squares contains one of 101, so the first also gives $s (102), s (103), s (104), s (105) \geq 257 / 25$ .

Each certificate is a measure of a kind this record has not registered before: point masses, segments of uniform density and rectangles of uniform density, given in D4 orbits with exact rational geometry, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000. It has 333 point, 897 segment and 4 rectangle orbits at $n = 101$ , and 86, 222 and 774 at $n = 83$ .

The source reports each accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, the verifier of its superseded $s (50) \geq 147 / 20$ certificate: an outward-rounded interval branch and bound over centre boxes in one quadrant of the centre domain, which the measure's quarter-turn symmetry makes stand for the whole, counting points in the closed core, segments by the length inside and rectangles by the area inside. It says each full replay was run again from the published tarball, by the same implementation and not an independent one.

At $n = 101$ the value exceeds Green's reported 2 sqrt(2) - 1 + (810 + 18 sqrt(5))/101 = 10.2467362..., by more than 0.0332, and so it does at $n = 102$ to 104, where this record held that value by monotonicity; at $n = 105$ it exceeds Nagamochi's 10.2736.... At $n = 83$ it exceeds Green's reported 9.2667335..., held here by monotonicity from $n = 82$ , by 0.083266...; the source's "more than 0.0833" is measured from 9.2667, below Green's value, and is false, though the bound is unaffected.

Here the source was retained, its exact premises were recomputed from the retained bytes, and the replay's checks before its first angle were run on both pinned tarballs, and a negative control at $n = 101$ refused two mutated measures. Each certificate passed a complete replay here of the source's unchanged checker and replay functions on its pinned tarball, $n = 101$ 's on 2 October 2026 and $n = 83$ 's on 3 October: all 201 directions, each returning the certificate's own record. The replays run the source's own algorithm and are not an independent decision of coverage. $n = 101$ 's five counts are also registered as replayed in their own entry (T-080).

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#294 and its comment of 2 October 2026. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim per certificate, each on its own reported entry; $n = 102$ to 105 take the $n = 101$ certificate directly, its mass being below them, so no step is added. Each replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone, which the 2 October review read function by function. $n = 83$ 's replay entry is cited here; $n = 101$ 's is entered once, in T-080, which carries $n = 101$ to 105 in the verified lane, so a case's citation names one result.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, read before the replays. A second machine method would be a method-distinct decision of rotated coverage for a linear measure; the replays here run the source's own checker. linear-control n83 would put a negative control on that certificate itself; the checker's controls are $n = 101$ 's.
Significance: Raised the verified lower bound at five counts, $n = 101$ to 105 (T-080), and confirmed $n = 83$ 's value below the replayed 937/100 there; when registered, the strongest lower bounds on record at six counts, past Green's reported values at $n = 83$ and $n = 101$ and past Nagamochi's at $n = 105$ , by a certificate kind new to this record that adds points and segments to the rectangle densities. Further sizes from one generator at S3. The 2 October review of the certificates confirmed S3: the kind is new here, but the technique is the measure argument with Tokoharu's net and checker, extended by two capture bounds.
Novelty: previously-published Present in an identified source
Records: 6 cases register evidence 1 evidence 2 evidence 3 source review

T-072

Claim: $s (76) \geq 447 / 50$ = 8.94, by a rectangle density of wand125/square-packing-bounds checked at coverage one, published on 2 October 2026.

The certificate is a density of 317 uniform rectangles with no point mass, of total mass 7599999/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, such that the shrunken core of every rotated unit square at each net direction captures mass at least one. The source reports it accepted at every oblique net angle by code/mixed_rotated_verify.cpp, the research copy of Tokoharu's verify.cpp that its $s (50)$ certificate (T-048) and the certificates of T-069 and T-071 use, which accepts coverage at least 1 where Tokoharu's requires 10001/10000, and at angle zero by exact integer tables. The least oblique lower bound it records is 1.00000000051.

The value exceeds the source's rectangle certificate at this count, 357/40 = 8.925 (T-068), which it says it supersedes, and Nagamochi's 1 + sqrt(61) = 8.8102....

Here the exact premises were recomputed from the retained bytes, and the certificate passed a complete replay on 2 October 2026 of the source's unchanged checker and per-angle functions on the pinned tarball: all 201 directions, each returning the certificate's own record. The replay runs the source's own algorithm and is not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 2 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim on one certificate. E-n076-wand125-mixed-894-source-replay replays it in full with the source's own checker: one interval-certified method, replayed here, C3. Coverage is decided by the source's C++ checker alone, and the axis tables are a second implementation for one direction and not a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A second machine method would be a method-distinct decision of rotated coverage; the replay here runs the source's own checker. The checker's controls were made on the $n = 37$ certificate of the same kind; a control on this certificate, devtools.audit_wand125_point_and_mixed mixed-control n76, would bind them to this measure as well.
Significance: Raised the verified lower bound at $n = 76$ by 0.05, over wand125's $n = 75$ rectangle certificate by monotonicity (T-045), by the certificate kind of T-048, T-069 and T-071: a further size from one generator, a substantive case result at S3. The 2 October review of the certificate confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 76 register evidence 1 evidence 2 source review

T-071

Claim: Two rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 2 October 2026, prove $s (84) \geq 47 / 5$ = 9.4 and $s (85) \geq 471 / 50$ = 9.42. The $n = 85$ certificate's mass is below 86 and 87 too, so it gives $s (86) \geq 471 / 50$ and $s (87) \geq 471 / 50$ as well: four counts in all.

Each certificate is a density of uniform rectangles with no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000: 661 rectangles at $n = 84$ and 587 at $n = 85$ . Each is accepted at every oblique net angle by code/mixed_rotated_verify.cpp, the research copy of Tokoharu's verify.cpp that the source's $s (50)$ certificate (T-048) and the five certificates of T-069 use, which accepts coverage at least 1 where Tokoharu's requires 10001/10000, and at angle zero by exact integer tables. The least oblique lower bounds are 1.00000000089 at $n = 84$ and 1.0000000017 at $n = 85$ .

Both values exceed Green's reported 2 sqrt(2) + (247 + 12 sqrt(2))/41 = 9.2667335..., which this record holds at both counts by monotonicity from $n = 82$ , by more than 0.1332 and 0.1532; the source's own improvement figures are measured from 9.2667, below Green's value, and are not lower bounds. At $n = 86$ and 87 the $n = 85$ value is also above the rectangle certificates T-068 reports there, 1873/200 and 941/100.

Each certificate passed a complete replay here on 2 October 2026 of the source's unchanged checker and per-angle functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replays run the source's own algorithm and are not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 2 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: Two primary certificates, each on its own reported entry and its own replay entry. $n = 86$ and 87 take the $n = 85$ certificate directly, its mass being below those counts, so no step is added. Each replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone; the axis tables are a second implementation for one direction and not a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A second machine method would be a method-distinct decision of rotated coverage; the replays here run the source's own checker. The checker's controls were made on the $n = 37$ certificate of the same kind.
Significance: Raised the verified lower bound at four counts, $n = 84$ to 87, past Green's reported value at $n = 84$ and 85 by the widest margins over it on record, by the certificate kind of T-048 and T-069. Further sizes from one generator at S3; the 2 October review of these certificates proposed S3, which the replays leave standing.
Novelty: previously-published Present in an identified source
Records: n = 84 n = 85 n = 86 n = 87 register evidence 1 evidence 2 evidence 3 evidence 4 source review

T-074

Claim: Twenty-nine rectangle-density certificates in wand125/square-packing-bounds, raised or added between 29 September and 1 October 2026, prove $s (19) \geq 1927 / 400$ , $s (20) \geq 1959 / 400$ , $s (26) \geq 2213 / 400$ , $s (27) \geq 1127 / 200$ , $s (28) \geq 2289 / 400$ , $s (29) \geq 2319 / 400$ , $s (30) \geq 47 / 8$ , $s (31) \geq 2381 / 400$ , $s (38) \geq 1309 / 200$ , $s (39) \geq 1327 / 200$ , $s (40) \geq 67 / 10$ , $s (41) \geq 169 / 25$ , $s (42) \geq 1363 / 200$ , $s (43) \geq 551 / 80$ , $s (44) \geq 2777 / 400$ , $s (53) \geq 3043 / 400$ , $s (54) \geq 3069 / 400$ , $s (55) \geq 617 / 80$ , $s (56) \geq 3113 / 400$ , $s (68) \geq 851 / 100$ , $s (69) \geq 1717 / 200$ , $s (70) \geq 69 / 8$ , $s (74) \geq 3539 / 400$ , $s (75) \geq 89 / 10$ , $s (88) \geq 3791 / 400$ , $s (89) \geq 1913 / 200$ , $s (93) \geq 3889 / 400$ , $s (94) \geq 1961 / 200$ and $s (95) \geq 49259 / 5000$ .

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The $n = 56$ certificate's mass is below 57 and 58 too, which gives $s (57), s (58) \geq 3113 / 400$ : 31 counts in all.

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, in sixteen cloud batches on 2 October 2026 whose receipts were merged into the packet the same day, and each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises. A thirtieth certificate replayed in the same batches, rect_n76_L8925, is below the mixed certificate that holds $n = 76$ (T-072) and is not registered here.

wand125, square-packing-bounds, built with and checked by Tokoharu's solver and interval verifier. The source keeps its credit to this project for the method lineage and says parts of the work were produced with AI assistance under human direction.
Composition: Twenty-nine primary certificates. $n = 57$ and 58 take the $n = 56$ certificate directly, its mass being below those counts, so no step is added. One interval-certified method, replayed here: C3. The replay decides coverage with the source's own checker, so it confirms the source's run rather than adding a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, read before the replays. A second machine method would be a method-distinct decision of rotated coverage; global coverage is decided by the source's C++ checker here and, under T-068, by sqverify-fast, one method in two implementations, and the exact preflight does not decide it.
Significance: Raised the verified lower bound at 31 counts from $n = 19$ to 95, over the replayed certificates of T-045 and T-070, wand125's point certificate at $n = 56$ (T-044) and Nagamochi's closed form at $n = 57$ and 58, on Tokoharu's method with larger certificates. Substantive case results, S3, as T-045 and T-070; the 2 October review of T-068 confirmed S3 for these certificates. The closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03); the floors it left at $n = 57$ and 58, T-044's 381/50 and Karakuş's general bound, are lower still.
Novelty: previously-published Present in an identified source
Records: 31 cases register evidence 1 source review

T-069

Claim: Five rectangle densities of wand125/square-packing-bounds, checked at coverage one and published between 29 September and 1 October 2026, prove $s (37) \geq 161 / 25$ = 6.44, $s (65) \geq 167 / 20$ = 8.35, $s (66) \geq 421 / 50$ = 8.42, $s (90) \geq 48 / 5$ = 9.6 and $s (92) \geq 969 / 100$ = 9.69.

Each certificate is a density of uniform rectangles with no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000: 350 rectangles at $n = 37$ , 787 at $n = 65$ , 631 at $n = 66$ , 762 at $n = 90$ and 832 at $n = 92$ . Each is accepted at every oblique net angle by code/mixed_rotated_verify.cpp, the research copy of Tokoharu's verify.cpp that the source's $s (50)$ certificate uses (T-048), which accepts coverage at least 1 where Tokoharu's requires 10001/10000, and at angle zero by exact integer tables. The least oblique lower bounds run from 1.000000000042 at $n = 90$ to 1.0000000054 at $n = 37$ .

At $n = 65$ the value exceeds Green's reported 8.2899656, which is 2 sqrt(2) + 71/13, and at $n = 92$ Nagamochi's 1 + sqrt(75) = 9.6602540. At $n = 37$ , 66 and 90 it exceeds the rectangle certificates of the same source that this record reports.

Each certificate passed a complete replay here on 2 October 2026 of the source's unchanged checker and per-angle functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replays run the source's own algorithm and are not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim per count, each on its own certificate, its own reported entry and its own replay entry. Each replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone; the axis tables are a second implementation for one direction and not a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A second machine method would be a method-distinct decision of rotated coverage; the replays here run the source's own checker. The checker's controls were made on the $n = 37$ certificate.
Significance: Raised the verified lower bound at five counts, by 0.03 to 0.34 over Nagamochi's closed form, two of them past Green's and Nagamochi's reported values, by the certificate kind of T-048. Further sizes from one generator at S3; the 2 October review of these certificates confirmed S3. Since Karakuş's finding (T-085, merged 2026-10-03) the closed form is itself a reported bound, and the margins over the verified floor it left behind, Karakuş's general bound, are larger.
Novelty: previously-published Present in an identified source
Records: 5 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 source review

T-068

Claim: Thirty-four rectangle-density certificates of wand125/square-packing-bounds, one at each of 34 counts from $n = 19$ to $n = 95$ , published between 29 September and 1 October 2026, prove the values below. They raise 32 of the values T-046 reports and add $n = 87$ and $n = 93$ .

Up to $n = 44$ the values are $s (19) \geq 1927 / 400$ , $s (20) \geq 1959 / 400$ , $s (26) \geq 2213 / 400$ , $s (27) \geq 1127 / 200$ , $s (28) \geq 2289 / 400$ , $s (29) \geq 2319 / 400$ , $s (30) \geq 47 / 8$ , $s (31) \geq 2381 / 400$ , $s (38) \geq 1309 / 200$ , $s (39) \geq 1327 / 200$ , $s (40) \geq 67 / 10$ , $s (41) \geq 169 / 25$ , $s (42) \geq 1363 / 200$ , $s (43) \geq 551 / 80$ and $s (44) \geq 2777 / 400$ .

From $n = 53$ they are $s (53) \geq 3043 / 400$ , $s (54) \geq 3069 / 400$ , $s (55) \geq 617 / 80$ , $s (56) \geq 3113 / 400$ , $s (66) \geq 1677 / 200$ , $s (68) \geq 851 / 100$ , $s (69) \geq 1717 / 200$ , $s (70) \geq 69 / 8$ , $s (74) \geq 3539 / 400$ , $s (75) \geq 89 / 10$ , $s (76) \geq 357 / 40$ , $s (86) \geq 1873 / 200$ , $s (87) \geq 941 / 100$ , $s (88) \geq 3791 / 400$ , $s (89) \geq 1913 / 200$ , $s (90) \geq 3831 / 400$ , $s (93) \geq 3889 / 400$ , $s (94) \geq 1961 / 200$ and $s (95) \geq 49259 / 5000$ .

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, at core side 9977/10000 on 201 net half-angles of step 83/40000, built with Tokoharu's solver, its weights scaled to that mass. The source reports each accepted at all 201 net directions by Tokoharu's unchanged interval checker, verify.cpp, and replayed in full from its published files on a second machine before publication.

Each certificate was decided here by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate at Tokoharu's threshold 10001/10000, in the census of 3 October 2026, and on 6 October two mutants of each scaled below that threshold 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.

Thirty of the certificates, all but $n = 66$ , 86, 87 and 90, were also replayed here in full with the source's own checker on 2 October, each reproducing the upstream accepting run at every direction; 29 of them are registered again in T-074, and $n = 76$ 's in none. The case records decide which certificate is current at each count.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#281. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim per count, each on its own certificate at one source revision, the packet's reported entry and its own replay entry. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3, on the census route for format T that the review of 6 October set and these certificates meet. The source's checker was also replayed here on 30 of them, a reproduction with the producer's code beside the rung, recorded in the packet's receipts and, for 29, under T-074; no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record: the 2 October review of these certificates is one, and the 6 October review of the format T census route decided the route, not these certificates' mathematics. A complete replay of the source's own checker on the four certificates it has not run on, rect_n66_L8385, rect_n86_L9365, rect_n87_L941 and rect_n90_L95775 (10.7 CPU-hours by the upstream per-angle times), would add a reproduction with the producer's code beside the rung for those counts; a second machine method would be a method-distinct decision of rotated coverage.
Significance: The strongest rectangle-density lower bounds on record at 34 counts when published, each above the certificate T-046 reports there; through T-074's replays, 29 of them raised the verified lower bound. Further sizes and raised rungs from one generator at S3. The 2 October review of these certificates confirmed S3.
Novelty: previously-published Present in an identified source
Records: 34 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 evidence 7 evidence 8 evidence 9 evidence 10 evidence 11 evidence 12 evidence 13 evidence 14 evidence 15 evidence 16 evidence 17 evidence 18 evidence 19 evidence 20 evidence 21 evidence 22 evidence 23 evidence 24 evidence 25 evidence 26 evidence 27 evidence 28 evidence 29 evidence 30 evidence 31 evidence 32 evidence 33 evidence 34 evidence 35 source review

T-067

Claim: $s (77) = 9$ : the lower half by wand125's mixed cover of 1 October 2026, the upper half by the $9 \times 9$ grid. The cover's total is below 78 too, so it gives $s (78) = 9$ as well, a second route to the k = 9 case of the family Evan Daniel reports (T-064).

The cover is 28,273 weighted points of [0,9]^2 plus mass spread uniformly along 6,420 segments, total 43347137744028965/2^49 = 76.999984600031... < 77, exactly D4-invariant, in Evan Daniel's mixed format. It is Daniel's $s (60) = 8$ cover cut at 4 in both coordinates with a central band inserted, plus a symmetric band measure of mass about 16.43 found by a linear program, the whole scaled to just below 77. Eighty-four points that lay on loaded grid lines were replaced by segments of length 2/1000 on the same lines, which the source says its exact checker needs. Every closed unit square in [0,9]^2 captures mass at least one, so a packing of 77 at side below 9, scaled up to side 9, would capture at least 77.

The source reports the cover accepted by Daniel's checkers: zmx2 over 8,100 symmetry-reduced roots and over 64,800 unreduced ones, and zm_mixed.py in exact rational arithmetic over 129,600 roots, none uncertified. It has no Lean statement.

zmx2 was replayed here in full on 2 October 2026, with --pair-points as the source ran it: all 8,100 D4 roots in one run and all 64,800 unreduced roots in two region runs read as one record, none uncertified, with the box totals the source states. Its run records are not published, so no root-for-root comparison was possible. The source's own verification script cannot pass as published, because its checksum file lists run logs that are not in the repository, so its two zmx2 steps were run here as this repository's own sequence.

wand125, square-packing-bounds, building directly on Evan Daniel's $s (60)$ cover, format and checkers. Registration was requested in jlevy/squares#279. The source says parts of the work were produced with AI assistance under human direction.
Composition: Compound, and the minimum is set by the lower half. The lower half is E-n077-wand125-mixed-cover-zmx2-replay (interval-certified, zmx2 replayed here), the upper half the grid (E-basic-grid-upper, exact-algebraic); the equality is C3, as T-066 reads its own. The checker is Daniel's, which also decides T-062 and T-066. The zm_mixed.py sweep the source reports is not replayed here, and sets nothing.

$n = 78$ takes the same certificate directly, its total being below 78, so no step is added. At registration this entry's scope was $n = 77$ alone, since T-064 registers $s (78) = 9$ (result-import.md, register table); it took in $n = 78$ at stage 4, on 2 October 2026, because the verified bound there rested on this replay alone, T-064's own route being unreplayed until its Valid7 replay and Lean build passed here on 2 and 3 October. The verified bound at $n = 78$ still cites this replay alone. T-064 keeps its claim, which names this route.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. The source's zm_mixed.py sweep, about 94 CPU-hours at the source, recorded as a second, exact-algebraic entry, would give the lower half a second machine method (think-mx3k). V5 would need a Lean statement for this cover and a human expert's review of the formalization.
Significance: An exact value for a case that was open, and a second route to $s (78) = 9$ : a substantive case result at S3, set beside T-062, whose cover it extends. The 2 October review of the $s (59)$ and $s (77)$ covers confirmed the draft: the band insertion that carries a side-8 cover to side 9 is the one new idea, and at one instance it is a case result, not yet a technique.
Novelty: previously-published Present in an identified source
Records: n = 77 n = 78 register evidence 1 evidence 2 evidence 3 source review

T-066

Claim: $s (59) = 8$ : the lower half by wand125's mixed cover of 1 October 2026, the upper half by the $8 \times 8$ grid. The cover's total is below 60 and 61 too, so it gives $s (60) = 8$ and $s (61) = 8$ as well, a second route to T-062 and T-063.

The cover is 26,308 weighted points of [0,8]^2 plus mass spread uniformly along 5,240 segments of length 1/50 on the interior grid lines, total 1474762899/25000000 = 58.99051596 < 59, exactly D4-invariant, in Evan Daniel's mixed format. Every closed unit square in [0,8]^2 captures mass at least one, so a packing of 59 at side below 8, scaled up to side 8, would capture at least 59. It was made with Daniel's line-cover linear program from his $s (60) = 8$ cover, and scaled to a margin the source puts at about 0.09% over the exact threshold.

The source reports the cover accepted by Daniel's zmx2, with outward-rounded binary64 enclosures, over 6,400 symmetry-reduced roots and 51,200 unreduced ones, none uncertified. Its exact-rational evidence is two zm_mixed.py runs whose joint coverage cannot be checked from what is published, so it is not recorded here. It has no Lean statement.

zmx2 was replayed here in full on 2 October 2026, over all 6,400 D4 roots and all 51,200 unreduced roots, none uncertified, with the box totals the source states; its run records are not published, so no root-for-root comparison was possible.

wand125, square-packing-bounds, building directly on Evan Daniel's method, format, checkers and $s (60)$ cover. Registration was requested in jlevy/squares#280. The source says parts of the work were produced with AI assistance under human direction.
Composition: Compound, and the minimum is set by the lower half. The lower half is E-n059-wand125-mixed-cover-zmx2-replay (interval-certified, zmx2 replayed here), the upper half the grid (E-basic-grid-upper, exact-algebraic); the equality is C3, as T-052 and T-053 read theirs. The checker is Daniel's, which also decides T-062, so the two routes to $s (60)$ and $s (61)$ share one checker. The source's two-run zm_mixed.py composite is not recorded as evidence (review finding F1).
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A complete zm_mixed.py sweep, about 127 CPU-hours at the source, recorded as a second, exact-algebraic entry, would give the lower half a second machine method; the per-root records the source published on 4 October in answer to finding F1 can be checked against it (think-mx3k). V5 would need a Lean statement for this cover and a human expert's review of the formalization.
Significance: An exact value for a case that was open, by Daniel's recipe applied one count lower, and a second route to T-062 and T-063: a substantive case result, not a new technique or a family, so S3 beside T-062 and below T-052, which introduced line mass. The 2 October review of the $s (59)$ and $s (77)$ covers confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 59 register evidence 1 evidence 2 evidence 3 source review

T-093

Claim: $s (17) > 18641771 / 4000000$ = 4.66044275, by Guzhou0806 / N17 project's R071 release of 30 September 2026, on the charge of R068 (T-043). It raised the verified bound by exactly 11/4000000 over T-043's 116511/25000 and leaves 0.0150873 to Bidwell's 4.67553009.

The certificate keeps every point orbit, rule orbit and weight of R068, and its budget of 17000448944 units of 10^-9, and rebuilds every strict core for the smaller parent side 18452000/18641771 in the container 4613/1000, over 5,114 parent-angle intervals that refine R068's 4,991. The source reports a least core charge of 1000026844 over all 5,114 intervals, so seventeen cores exceed the budget by 7404. It refines R070 of the day before, $s (17) > 46604427 / 10000000$ = 4.6604427 over 5,107 intervals, which it supersedes.

It was replayed here in full on 5 October 2026, the first replay of it outside its source: the source retains a completion summary of its own run and says the row records are missing, and its GitHub Actions replay passed unobserved by its publisher. The source's run_public.js ran Guzhou0806's C++ checker and Kleddamag's Node BigInt checker over every interval, and devtools.audit_guzhou_r071 found the two fresh ledgers identical row for row, every row at 1000026844; the two checkers are T-043's byte for byte and one event-cell method. Both refuse two mutated certificates.

Guzhou0806 / N17 project, n17-square-packing, on R068's continuation of Kleddamag's 4.66001 charge. The package credits Kleddamag for the 4.66001 framework, proof and BigInt checker (Kleddamag building on Squares Project (Joshua Levy), Mira and Guzhou0806), and discloses AI assistance.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record, as T-043's next_rung says; a second machine method needs the native parent-core route to read the weighted and winning-subset features, which it still cannot. The r071-bound artifact of the source's CI run, which expires on 2026-12-29 and could not be fetched here, holds a third ledger that could be compared row by row with the fresh ones; it would add a comparison, not a rung. R070's obstruction puts the reach of this charge below 186417711/40000000 (review RF-7), so a higher bound needs a new charge or an argument across parents. Rung 5 needs a proof-assistant formalization reviewed by human experts.
Significance: The case's verified lower bound, 2.75 x 10^-6 above T-043 from T-043's charge unchanged, a finer catalogue and rebuilt cores: the construction of T-042, which is S2, adding no orbit, rule, weight or method. T-043's step was 156 times this one and T-039's, the S3 precedent the draft weighed, 83 times; R070's obstruction shows the charge cannot reach 1/40000000 further, so this is the end of a recipe rather than an opening. Holding the verified field is a fact about the frontier, not the claim; a reader who weighs it above the size of the step would score S3 by T-039, and the score gates nothing.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-064

Claim: Evan Daniel's theorem s(k^2 - 3) = k for every integer k >= 6, dated 29 September 2026 by the source and public in its repository on 30 September: the lower half by one family of periodic measures, the upper half by the k x k grid.

This entry's scope is the projection of that theorem onto the cases this record holds, n <= 324, which is k = 6 to 18: $s (33) = 6$ , $s (46) = 7$ , $s (61) = 8$ , $s (78) = 9$ , $s (97) = 10$ , $s (118) = 11$ , $s (141) = 12$ , $s (166) = 13$ , $s (193) = 14$ , $s (222) = 15$ , $s (253) = 16$ , $s (286) = 17$ and $s (321) = 18$ . The reported theorem does not end at k = 18.

For each k the measure on [0,k]^2 is a corner module in each corner, a profile of period one along each wall and Lebesgue measure on the middle square, of total mass k^2 - 4D with D = 423621306389/500000000000, so that 4D exceeds 3. The source states that every closed unit square in [0,k]^2 has mass at least one, and reduces that, for every k >= 6, to one finite statement it calls Valid7: every closed unit square in [0,7]^2 has mass at least one under the k = 7 measure.

The source's Lean development takes Valid7 as a hypothesis and, the source reports, kernel-checks that it implies the theorem for every k >= 6; it does not prove Valid7. Valid7 is the claim of the source's exact-rational Python checker, qx2_zm.py, which the source reports accepting all 9,800 root boxes in 32,079 leaves, with the axis-parallel poses decided by a separate exact enumeration; it reports six adversarial reviews of the certificate by AI agents and none by outside mathematicians.

On 2 October 2026 wand125 published a second, separately written exact-rational checker of Valid7 on the same cover, which reports all 156,800 root boxes of the unreduced pose space certified in 9,640,060 leaves, about 626 core-hours, and says Daniel's checker and lemma write-ups were not read. The two share the statement, the cover and its format.

Here both parts of the lower half were rebuilt. Daniel's checker was replayed in full on the retained cover on 2 and 3 October 2026, in two shards: all 9,800 root boxes and 32,079 leaves, none uncertified, each root's leaves equal to the published run's, which discharges Valid7 for this cover. The Lean reduction was built from retained bytes with the pinned toolchain, and bentz_of_valid7 depends only on the standard axioms. Together they prove the theorem here, for every k >= 6.

wand125's second checker was reviewed on 2 October and found sound but for one defect in its exact positivity test (D-1), which cannot make a true Valid7 fail. It was replayed here in full on 2 and 3 October with that test's accepting case made a refusal, in fourteen shards: all 156,800 root boxes certified in 9,808,968 leaves, none uncertified, and each root the source ran under its published code with the published leaves. So Valid7 is decided here by two separately written exact checkers, Daniel's reproduced with the producer's code and wand125's independently re-implemented.

So $s (97) = 10$ , $s (118) = 11$ , $s (141) = 12$ , $s (166) = 13$ , $s (193) = 14$ , $s (222) = 15$ , $s (253) = 16$ , $s (286) = 17$ and $s (321) = 18$ are proved here, nine cases closed. $s (33) = 6$ and $s (46) = 7$ were proved already on Bentz's proofs, $s (61) = 8$ on Daniel's $s (60)$ cover (T-063), and $s (78) = 9$ on wand125's $s (77)$ cover (T-067); this family proves each again by another route.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method. The source's CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.
Composition: Compound, and the minimum is set by the lower half. The upper half is the k x k grid, E-basic-grid-upper, replayed exactly here. The lower half is two parts in series: the finite premise Valid7 for the k = 7 cover, decided by the full replay of Daniel's exact checker (E-k2m3-evand-valid7-qx2-replay, exact-algebraic, same-implementation), and the reduction from Valid7 to every k >= 6, kernel-checked here (E-k2m3-evand-bentz-lean-build, proof-assistant-checked, with its axiom receipt). Each is V3 and C3, so the result is.

Valid7 has a second replayed checker, wand125's separately written one, replayed here in full with D-1 guarded (E-k2m3-wand125-valid7-independent, exact-algebraic, independent-implementation). It is the same method as Daniel's, so it adds no distinct method and moves no rung. No human formalization review is retained, so the Lean build does not reach rung 5. The scope lists the cases this record holds, since a scope is a set of cases; the theorem's own domain is every k >= 6, and a scope that can say so is think-kqi1.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the claim and a human oversight record; the one retained review read the second checker's method and the reduction's statement. Valid7's two replayed checkers are both exact-algebraic; a checker of another kind, interval-certified or a Lean proof of Valid7, would add a method beside the rung. Rung 5 needs human formalization reviews of the Lean statement and a Lean proof of Valid7 itself.

The source has begun that proof: by 4 October its Lean reduces Valid7 to the tilted part ValidTilt7 (bentz_of_validTilt7, with the D4 reduction and the axis face proved) and kernel-checks the 689 LEB and 374 CAP leaves of the run's 32,079 with Lemmas U and K, as reported and not built here; the tree that joins the leaves and the 26,829 PIECE and EXACT leaves remain. think-kqi1 tracks a scope that can state an infinite family.

The source also reports an interval-certified checker: its zmx2 with area density decides Valid7 by a sweep of the whole pose space, about 400 CPU-seconds at the source, retained with its run records in the 4 October evand packet and not replayed here (E-k2m3-evand-family-report, think-83zc). The source still waits for a second reader of its newest lemmas.
Significance: Scored as a claim: a bound family, which would settle eleven open cases of this record and every later k if the finite premise and its reduction pass replay and review here. S4 by the anchor "a reusable technique, bound family"; the score gates nothing.
Novelty: previously-published Present in an identified source
Records: 13 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source 1 source 2 review 1 review 2

T-084

Claim: s(k^2 - 1) = k for every integer k >= 3: Karakuş 2026, Corollary 1.2. The lower half is his rectangle bound, Theorem 1.1, at a = b = k, through Corollary 6.1, and is the value of Corollary 6.2 at N = k^2 - 1; the upper half is the k x k grid with one square removed. k = 2, $s (3) = 2$ , is classical and is not this entry's.

This entry's scope is the cases this record holds, k = 3 to 18: $s (8) = 3$ , $s (15) = 4$ , $s (24) = 5$ , $s (35) = 6$ , $s (48) = 7$ , $s (63) = 8$ , $s (80) = 9$ , $s (99) = 10$ , $s (120) = 11$ , $s (143) = 12$ , $s (168) = 13$ , $s (195) = 14$ , $s (224) = 15$ , $s (255) = 16$ , $s (288) = 17$ and $s (323) = 18$ . Nagamochi 2005 stated the family first (T-007); Nagamochi's Lemma 1, on which that proof rests, is false (T-085), and Karakuş's proof uses neither that lemma nor the paper's technical lemmas.

The proof was read in full and re-derived here. Its Proposition 5.1 is machine-checked here, independently re-implemented (devtools.check_karakus_strip_measure), by its own decomposition rather than the paper's Lemma 5.2. Hakan Karakuş, arXiv:2609.37410, 29 September 2026.
Composition: Compound, an equality. The upper half is the grid, E-basic-grid-upper, replayed exactly here. The lower half is Corollary 6.1 at a = b = k >= 3, where Delta(k) = 1, and its load-bearing step, Proposition 5.1, is decided by E-karakus-strip-measure-interval; E-karakus-strip-lower is the read of the whole proof. Read and not machine-checked are the chord profile (5.3), the reduction to it, the case split with its reflection and central symmetry, the soundness of the coded rules (the review of 2026-10-06), Theorem 1.1's scale-and-sum and Corollary 6.1's step to s(k^2 - 1) >= k, each of which estimates nothing over the pose space. So the lower half is at the machine check's rung, and the equality takes it.
Next rung: V4 and C4 need a second adversarial AI review of the machine check by a distinct reviewer and a human oversight record; the review of 2026-10-06 is the first. chelokot's Lean development proves the same family by an augmented measure (Records.NearSquare.squareMinusOne_isMinimumSide); the build of 2 October 2026 retained for T-086 compiled it and its Audit module printed standard axioms for it, but no statement-fidelity reading of it is retained, so it is not yet evidence here. Registered with that reading, it would be a second machine method beside the rung.
Significance: An infinite family of exact values, proved again after the published proof was found incomplete; every value was already held as proved. S3 by the anchor "a substantive case result or machine audit".
Novelty: previously-published Present in an identified source
Records: 16 cases register evidence 1 evidence 2 evidence 3 source

T-083

Claim: For every nonsquare integer $8 \leq N \leq 324$ , Karakuş 2026, Corollary 6.2 gives $s (N)$ >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4), which is strictly above sqrt(N).

The bound is his rectangle bound, Theorem 1.1, at a square container whose side t solves t^2 - (t - floor(sqrt(N))) = N. The rectangle bound is proved by a strip measure that uses neither the scoring lemma nor the technical lemmas of Nagamochi 2005. It equals ceil(sqrt(N)) exactly at N = m^2 - 1, which is T-084, and is strictly below Nagamochi's Theorem 2, T-007, at every other nonsquare N.

The proof was read in full and re-derived here. Its Proposition 5.1, the strip measure's per-square bound on which the rectangle bound rests, is machine-checked here, independently re-implemented: an exact branch and bound over the square's orientation, side and cut height (devtools.check_karakus_strip_measure), by its own decomposition rather than the paper's Lemma 5.2. Hakan Karakuş, arXiv:2609.37410, 29 September 2026.
Composition: Derived. The load-bearing step, Proposition 5.1, is decided by E-karakus-strip-measure-interval over the whole pose space; E-karakus-strip-lower is the read of the whole proof. Read and not machine-checked: the chord profile (5.3), the reduction of the measure of an open square to it, the case split by which strip lines meet the square with its reflection and central symmetry, the soundness of the coded rules (the review of 2026-10-06), Theorem 1.1's scale-and-sum, and Corollary 6.2's step to $s (N)$ . Each is a fact of plane geometry, measure additivity or arithmetic that estimates nothing over the pose space, so the derived rung is the machine check's. Lemma 5.2 and the paper's three strip cases are not used and stay read.
Next rung: V4 and C4 need a second adversarial AI review of the machine check by a distinct reviewer, and a human oversight record that checks its trust boundary, the meaning of the decided inequalities against Proposition 5.1 and the reviews' findings; the review of 2026-10-06 is the first. A second machine method, such as a formalization of Proposition 5.1, would be recorded beside the rung. A stronger floor at any n is a new result, not a rung change.
Significance: Since 2 October 2026 the verified floor of the open cases that rested on T-007 where no registered bound is stronger, replacing its proof at a weaker value. S3 by the anchor "a substantive case result or machine audit"; the score is the theorem's, not ours.
Novelty: previously-published Present in an identified source
Records: 301 cases register evidence 1 evidence 2 source

T-061

Claim: $s (11) > 3875000000 / 999999999$ = 3.875000003875000003875..., by Ke Wang and Can Li's Zenodo record of 29 September 2026, reported on jlevy/squares#247. The step over 31/8 is exactly 31/7999999992, about 3.9e-9.

The certificate is Kleddamag's v1.0.2 global-certificate.json (T-037) with thirteen threshold-charge orbits raised, which puts the minimum core charge at 1000047559 on every one of the 12,028 parent-angle intervals, and with the parent and every core side scaled by 999999999/1000000000, so that the container over the parent is (31/8) times 1000000000/999999999. Eleven disjoint cores exceed the budget 11000095024 by 428125 integer units.

One more step of the scale factor, 1 - 2*10^-9, already fails at row 615, so these data admit at most about twice the step. Scaling Kleddamag's certificate alone already passes a full sweep, so the reweighting enlarges the surplus but is not needed for the bound.

It was replayed here in full by the authors' two verifiers and by Kleddamag's own Python and JavaScript event-cell sweeps, and decided a second time by this repository's native parent-core interval branch and bound, which accepted all 12,028 rows with no stalled box.

Ke Wang and Can Li, Zenodo record 23038546, building on Kleddamag and this project. The record makes no statement about AI assistance.
Composition: Primary at $n = 11$ : one certificate and the bound L/A', with no monotonicity step. Two evidence entries whose methods differ both passed on the retained certificate, which is C3 with the two methods shown beside the rung: the event-cell sweeps (exact-algebraic: the authors' primary, Kleddamag's Python and JavaScript, and the authors' re-implementation are one method) and this repository's directed-rounding interval coverage of every row (interval-certified). Both rest on the certificate and on T-037's counting and transfer theorem, whose hypotheses were re-checked at the scaled sides.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the complete claim and a human oversight record; the 2026-09-30 review is a same-project read and is not recorded in this entry's reviews. Rung 5 needs a proof-assistant formalization of the transfer theorem and the counting argument, reviewed by human experts. The reweighting is not needed for the bound: Kleddamag's own weights at the scaled sides pass Kleddamag's full event-cell sweep with the same minimum 999962528 and surplus 107864 (the packet's scale-only receipt), so the scaling alone carries the step and the thirteen raised orbits only enlarge the surplus to 428125.
Significance: It strictly raises the verified lower bound at a central case, which the S5 anchor names, but by 31/7999999992, about 3.9e-9, spending the slack of T-037's certificate after a reweighting; the release itself calls it a proof-of-slack result, and these data admit at most about twice the step. The case's gap to Trump's packing, 0.0020836, does not change at any printed digit, so it is scored as a citable detail rather than movement; T-042 is the precedent for scoring a true, fully replayed step by what it changes.
Novelty: previously-published Present in an identified source
Records: n = 11 register evidence 1 evidence 2 evidence 3 source 1 source 2 review

T-060

Claim: $s (11) = T = (6 u + 4) / (1 + 2 u - u^{2}) = 3.8770835900228141773078970601 \dots$ , where $u$ is the unique root in $(9 / 25, 37 / 100)$ of $5 u^{8} - 10 u^{7} - 2 u^{6} + 14 u^{5} + 12 u^{4} - 6 u^{3} + 2 u^{2} + 2 u - 1 = 0$ . Independent rotations and boundary contact are allowed.

The proof, published by Queuingtheorydotcom on 29 September 2026, is confirmed here by a re-implementation sharing components with the source -- complete repository geometric executions on kernels the source also bundles -- and a mapped mathematical audit.

Queuingtheorydotcom, 11SquaresOptimal, Astra-assisted work building on Squares Project and Kleddamag. The matching construction is Trump's.
Composition: The complete 2184-pattern cover and 2180 exclusions leave four symmetric survivors. The D4 bridge reduces these to case 438; the 14-round root induction and complete 10-node capture graph eliminate all far branches and force the near state into the reviewed local region. Exact local isolation and the endpoint embedding argument exclude every side below $T$ . The exact Trump witness attains $T$ . All premises have completed exact executions, which is V3/C3. One adversarial AI review is retained and mapped, by GPT-6 Astra at max reasoning; it is not a human oversight record. No second machine method is claimed.

The Lean 4 formalization in 11SquaresFormalized states the whole claim as one theorem, ElevenSquare.optimality, which the 6 October statement audit reads as this claim with the same exact $T$ . Its source reports that its full verification run, resumed from earlier validated receipts, passed with an axiom audit that trusts Lean's compiler for 13,308 native_decide axioms. That run is private and recorded as reported, so it sets no rung; the build here covers the statement closure and the upper half only.
Next rung: V5 needs two records the register lacks. One is a complete build of the 11SquaresFormalized proof that the record can rest on, with its axiom receipt: run here from the retained pin at its toolchain, or a third party's build retained here. The source's own run is private and is recorded as reported (E-n011-lean-formalization-report). The other is a retained formalization review by a named human expert who is not its author and states competence in Lean and in the mathematics, checking statement fidelity, the definitions, the axioms (the 13,308 native_decide axioms the theorem trusts to Lean's compiler among them) and that build.

V4 and C4 need the owner's oversight record and a second adversarial AI review by a reviewer distinct from Astra. C5 needs that build made here at the pinned toolchain, an open_review pointer and two such expert reviews. Engineering follow-up think-e2ot will orchestrate fresh whole-ensemble replay through reviewed state-equivalence bindings. Simplification was to precede the n11 explainer, tracked separately, which is now the eleven-square optimality paper.
Significance: Resolves global optimality for $n = 11$ , a central case, by closing the gap to Trump's exact construction, beyond the previous strict T-037 lower bound.
Novelty: previously-published Present in an identified source
Records: n = 11 register evidence 1 evidence 2 evidence 3 source 1 source 2 review 1 review 2

T-059

Claim: wand125/square-packing-tools reports that its exact general_pose_tree checker reproduces all 12028 per-row core minima of Kleddamag's n11 certificate, release v1.0.2, with global row minimum 999962528 units and all witnesses replayed. This is the source's row-scan equality claim only, not a new proof of the global counting theorem or independent rectangle-density coverage.

wand125, square-packing-tools, 29 September 2026, building on Tokoharu's rectangle method, solver and verifier and on Evan Daniel's point verifier. The source discloses Codex and Claude assistance.
Next rung: V4 and C4 need a second, independent row-minimum method (the checker and the claim share one implementation, so no method diversity is claimed) and the review record the ladder requires. A separate branch-and-bound over the 12,028 rows is the natural second method; the existing native parent-core branch and bound already decides the same rows for the counting premise (T-037).
Significance: A reusable independent row-checking implementation worth replaying; the existing n11 bound does not change.
Novelty: previously-published Present in an identified source
Records: n = 11 register evidence 1 evidence 2

T-058

Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows

Claim: A rectangle-density certificate with core side B = 9977/10000 and the 201-direction net of step D = 83/40000 cannot have mass below n at any side L >= α U, where U is the side of any packing of n unit squares and α = B(1 + D) = 399908091/400000000; if every orientation of that packing lies in the net's D4 orbit, L >= B U suffices. For each $n = 1$ ..100 an exact certificate of n disjoint net cores proves the ceiling at a side no larger than the corrected value of the source's update, B UB(n) at the 64 integer-grid rows of its table and α UB(n) at the other 36, with UB(n) the table's rounded value. The source's original B UB(n) is proved at the 64 grid rows only; at the 36 others it is neither proved nor refuted.

wand125, square-packing-tools, 29 September 2026, building on Tokoharu's rectangle method, solver and verifier and on Evan Daniel's point verifier. The source discloses Codex and Claude assistance.
Next rung: Rung 4 needs two retained adversarial reviews of the ceiling lemma and the exact verifier, by distinct reviewers, and a human oversight record. Separately, and moving no rung: decide whether a certificate with mass below n exists between B UB(n) and the proved ceiling at the 36 non-grid rows.
Significance: A tooling limit that could prematurely stop searches; resolving its premises affects safe integration but establishes no new packing bound.
Novelty: previously-published Present in an identified source
Records: n = 1–100 register evidence 1 evidence 2

T-070

Claim: Sixteen rectangle-density certificates in wand125/square-packing-bounds, added or raised on 27 and 28 September 2026, prove $s (29) \geq 579 / 100$ , $s (38) \geq 327 / 50$ , $s (39) \geq 663 / 100$ , $s (41) \geq 1351 / 200$ , $s (51) \geq 2977 / 400$ , $s (52) \geq 1507 / 200$ , $s (53) \geq 1519 / 200$ , $s (57) \geq 1567 / 200$ , $s (59) \geq 198 / 25$ , $s (67) \geq 1691 / 200$ , $s (69) \geq 343 / 40$ , $s (71) \geq 1737 / 200$ , $s (72) \geq 437 / 50$ , $s (73) \geq 439 / 50$ , $s (86) \geq 1871 / 200$ and $s (95) \geq 49209 / 5000$ . They are rect_n29_L579, rect_n38_L654, rect_n39_L663, rect_n41_L6755, rect_n51_L74425, rect_n52_L7535, rect_n53_L7595, rect_n57_L7835, rect_n59_L792, rect_n67_L8455, rect_n69_L8575, rect_n71_L8685, rect_n72_L874, rect_n73_L878, rect_n86_L9355 and rect_n95_L98418.

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. A certificate's mass is below every larger count too, which gives nine more, 25 counts in all: $s (42), s (43), s (44) \geq 1351 / 200$ from $n = 41$ , $s (54), s (55) \geq 1519 / 200$ from $n = 53$ , $s (60) \geq 198 / 25$ from $n = 59$ , $s (68) \geq 1691 / 200$ from $n = 67$ , $s (70) \geq 343 / 40$ from $n = 69$ , and $s (74) \geq 439 / 50$ from $n = 73$ .

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, twelve in cloud batches on 29 September 2026 whose receipts were merged into the packet on 2 October, and rect_n51_L74425, rect_n57_L7835, rect_n72_L874 and rect_n73_L878 in batches on 3 October merged on 6 October; each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises.

Two more certificates replayed in the 3 October batches, rect_n58_L789 and rect_n91_L9645, are below the mixed certificates that hold $n = 58$ and 91 (T-091) and are not registered here.

wand125, square-packing-bounds, built with and checked by Tokoharu's solver and interval verifier. The source keeps its credit to this project for the method lineage and says parts of the work were produced with AI assistance under human direction.
Composition: Sixteen primary certificates. The nine other counts take a smaller count's certificate directly, its mass being below them, so no step is added. One interval-certified method, replayed here: C3. The replay decides coverage with the source's own checker, so it confirms the source's run rather than adding a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method would be a method-distinct decision of rotated coverage; global coverage is still decided by the source's C++ checker alone, and the exact preflight does not decide it.

At $n = 59$ and 60 superseded on 2026-10-02 by the exact values $s (59) = 8$ (T-066) and $s (60) = 8$ (T-062), when their covers' replays landed, at $n = 86$ the same day by $s (86) \geq 471 / 50$ from the replayed $n = 85$ mixed certificate (T-071), and at $n = 29$ , 38, 39, 41 to 44, 53 to 55, 68 to 70, 74 and 95 by wand125's raised certificates of 1 October (T-074), when their replays were merged; stays true as stated. The counts it took on the morning of 6 October, $n = 51$ , 57 and 72, went later that day to T-090's mixed certificates when sqverify-fast decided them, and $n = 52$ , the last count it held, to T-082's the same day.
Significance: Raised the verified lower bound at 23 counts from $n = 29$ to 95, by 0.004 to 0.34, over wand125's weighted point certificates (T-044), Tokoharu's $n = 29$ certificate (T-047) and Nagamochi's closed form, on Tokoharu's method with larger certificates, and on 6 October at $n = 51$ , 57 and 72, by 0.0425, 0.0525 and 0.019 over the mixed certificates of T-048 and T-091 and T-074's $n = 56$ certificate. Substantive case results, S3, as T-045. The closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03).
Novelty: previously-published Present in an identified source
Records: 25 cases register evidence 1 source review

T-063

Claim: $s (61) = 8$ , as a corollary of $s (60) = 8$ (T-062), which Evan Daniel published with it on 28 September 2026.

Removing a square from a packing leaves a packing, so $s (61)$ is at least $s (60)$ , and the $8 \times 8$ grid holds 61 unit squares. The $s (60)$ cover's total, 59.86, is below 61 too, so its certificate decides both counts.

T-062's lower half was replayed here in full on 2 October 2026, and this value with it. wand125's $s (59) = 8$ (T-066), replayed here the same day, gives it a second route by the same deduction, and the same source's family s(k^2 - 3) = k (T-064) states it again at k = 8, by a separate certificate, replayed here on 3 October with the reduction built. wand125's point-only cover for $s (61)$ itself, which Evan Daniel also reports certifying, was replayed here on 2 October by Daniel's zmx2, VERIFIED-D4 over all 6,400 roots: a route that uses points only, on a cover built by someone else.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method. The source's CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.
Composition: Derived, and the minimum is set by its premise. The inputs are the lower half of T-062, E-n060-evand-mixed-cover-zmx2-replay (interval-certified, replayed here), whose scope takes in $n = 61$ because the cover's total is below 61; monotonicity by deletion, E-n061-evand-derived-report, read and found sound; and the $8 \times 8$ grid, E-basic-grid-upper. It stands at T-062's V3/C3 and never above it. wand125's $s (59) = 8$ (T-066) is a second route under the same checker.

So is wand125's separate point-only cover for $s (61) = 8$ , replayed here by zmx2 (E-n061-wand125-point-cover-zmx2-replay): a second route by the source's pinned checker, the same checker as T-062's, so no second method. Evan Daniel's reports of certifying that cover with zeromargin.py and zmcheck (E-n061-wand125-point-cover-evand-replay-report) are not replayed here.
Next rung: Nothing of its own: this entry rises with T-062. wand125's point-only cover was replayed here by zmx2 (think-hxrz); replaying Daniel's zeromargin.py or zmcheck on it would give $s (61) = 8$ a route that rests neither on zmx2 nor on T-062's cover.
Significance: A routine consequence of T-062, immediate once its premise is proved, though it closes a second open case. The 2 October review of the geometric premises confirmed the draft.
Novelty: previously-published Present in an identified source
Records: n = 61 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 source 1 source 2 review 1 review 2

T-062

Claim: $s (60) = 8$ : the lower half by Evan Daniel's mixed cover of 28 September 2026, the upper half by the $8 \times 8$ grid.

The cover is 23,744 weighted points of [0,8]^2 plus mass spread uniformly along 5,216 segments of length 1/50 on the interior grid lines, total 748233441/12500000 = 59.85867528 < 60, exactly D4-invariant. Every closed unit square in [0,8]^2 captures mass at least one, a boundary point and a segment along an edge counting in full, so a packing of 60 at side s below 8, its centres scaled by 8/s, would give 60 pairwise disjoint closed unit squares in [0,8]^2 capturing at least 60.

The source certifies that at margin zero with two separately written checkers: zm_mixed.py in exact rational arithmetic over 102,400 root boxes, and the Rust zmx2 with outward-rounded binary64 enclosures over 6,400 symmetry-reduced roots and 51,200 unreduced ones, none uncertified. It has no Lean theorem for this cover.

zmx2 was replayed here in full on 2 October 2026, over all 6,400 D4 roots and all 51,200 unreduced roots, every root's census equal to the source's; zm_mixed.py was not run. wand125's $s (59) = 8$ (T-066), replayed here the same day with the same checker, gives this value a second route by monotonicity.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method. Registration was requested in jlevy/squares#256. The source's CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.
Composition: Compound, and the minimum is set by the lower half. The lower half is E-n060-evand-mixed-cover-zmx2-replay (interval-certified, zmx2 replayed here), the upper half the grid (E-basic-grid-upper, exact-algebraic). As for T-052 and T-053, the two machine entries certify different halves; the lower half stands on one replayed method, and the equality is C3. The second route, T-066's $s (59)$ cover, was made from this cover and decided by the same zmx2, so it adds a certificate and not a method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review, of the geometric premises, is retained. A complete zm_mixed.py sweep, about 20 CPU-hours at the source, recorded as a second, exact-algebraic entry, would give the lower half a second machine method (think-mx3k); the shared author, agent and pose-space architecture stay the residual common-mode risk. V5 would need a Lean data file and top theorem for this cover, which the source has not attempted, and a human expert's review of the formalization.
Significance: An exact value for a case that was open, by the line-cover recipe of T-052 and T-053 carried to side 8, adding no technique: a substantive case result, S3. The 2 October review of the geometric premises confirmed the draft, and noted that the family argument that lifted T-053 to S4 applies here exactly as there, so the two should carry one score; which one is the rescoring pass's (think-qh3s).
Novelty: previously-published Present in an identified source
Records: n = 60 register evidence 1 evidence 2 evidence 3 source review

T-055

Claim: $s (21) = 5$ by a second, point-only route: the lower half by wand125's point-only measure, completed on 28 September 2026 with a Lean 4 reduction, the upper half by the $5 \times 5$ grid.

The lower half is Evan Daniel's s21_lower_4.9950.txt support (T-050) scaled by 1001/1000 and re-weighted, 4,604 D4-invariant entries of total 2624862500021/125000000000, with capture threshold q = 249987/250000 over every closed unit square in [0,5]^2, so that 21q exceeds the total by 999979/125000000000.

The capture statement is decided by the source's own exact rational replay of a retained certificate tree over 5,000 root boxes, and Lean proves minSide 21 = 5 from that one hypothesis.

That replay was run here in full on 29 September 2026, the source's runner unchanged: every stage passed, and its records match the source's M1 run, 45,436 of the 45,446 output files byte for byte and the rest in every certified quantity. The Lean overlay was not built here.

wand125 after Evan Daniel, square-packing-bounds. The source names Daniel's mixed-cover proof (T-052) as first, claims no priority, and says its code was developed with Codex.
Composition: Compound, and the minimum is set by the lower half. The lower half is E-n021-wand125-point-endpoint-source-replay (exact-algebraic, the source's runner replayed here), the upper half the grid (E-basic-grid-upper); the equality is C3. The runner is the source's, so the replay confirms the source's run rather than adding a second method, and the Lean reduction, not built here, adds nothing to the rung.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A build of the Lean overlay with its axiom receipt, and the runner's lemmas formalised (review F10), would be the road to rung 5. The controls end at the first root box that holds a pose, so neither exercises the sieve or frontier stages; a control refused deeper in the tree would test more of the runner.
Significance: A second, point-only route to a value T-052 already holds, with a positive-margin threshold rather than a zero-margin cover and a Lean reduction of its own. A citable detail that changes no theorem.
Novelty: previously-published Present in an identified source
Records: n = 21 register evidence 1 evidence 2 evidence 3 source review

T-054

Claim: $s (45) = 7$ by a second, point-only route: the lower half by wand125's point-only measure of 28 September 2026, the upper half by the $7 \times 7$ grid.

The measure is 12,645 D4-invariant points, total 12666371418707823/2^48 = 44.99999100001 < 45. Its capture condition, every closed unit square in [0,7]^2 having mass at least one with a boundary point counting, is decided by Evan Daniel's unmodified zmx2 with --d4 --pair-points.

The source's verify.sh was replayed here in full on 28 and 29 September 2026: VERIFIED-D4 on all 4,900 roots, 1,295,460 boxes, the source's reference census.

wand125 after Evan Daniel, square-packing-bounds. The source uses Daniel's checker, states that Daniel published $s (45) = 7$ first (T-053) and claims no priority, and says parts of the work were produced with AI assistance under human direction.
Composition: Compound: the lower half is E-n045-wand125-point-cover-source-replay (interval-certified, on the source-chosen zmx2, the same implementation as T-053's replay), the upper half the grid (E-basic-grid-upper). The two machine methods certify different halves; the lower half sets the minimum at one method, and the equality is C3. The unreduced --full sweep was not run here.
Next rung: The unreduced zmx2 --full sweep would remove the reliance on the D4 reduction (review F2). A second machine method would need a checker other than zmx2 to decide this point cover; the exact zeromargin.py route reads point covers and is the reachable one. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.
Significance: A second certificate for a value T-053 already holds, decided by the same checker. It shows the value yields to points alone, with a uniform pointwise margin of 8.35e-4, a citable detail that changes no theorem.
Novelty: previously-published Present in an identified source
Records: n = 45 register evidence 1 evidence 2 evidence 3 source review

T-048

Claim: $s (50) \geq 37 / 5$ = 7.4, by wand125's mixed rectangle-density certificate of 28 September 2026. It exceeds Green's reported 2 sqrt(2) + 101/25 + 3 sqrt(14)/25 = 7.3174260 by more than 0.0825739888. The certificate's mass is below 51 too, so it gives $s (51) \geq 37 / 5$ as well.

The certificate is a D4-expanded rectangle density of 553 orbit representatives, total mass 4999999/100000 < 50, core side 9977/10000 and 201 net half-angles of step 83/40000. It is accepted at every oblique net angle by code/mixed_rotated_verify.cpp, a research copy of Tokoharu's verify.cpp that accepts coverage at least 1 instead of 10001/10000 and checks each net node's unit-square centre domain, and at angle zero by exact integer tables.

The source's complete replay was run here on 29 September 2026 on the pinned bundle, with the source's driver and checker unchanged: all 201 directions, each equal to the shipped record. This repository's exact audit checked the premises and the integrity of all 621 bundle files and all 200 oblique inputs. The replay runs the source's own algorithm and is not an independent decision of coverage.

wand125 after Evan Daniel and Tokoharu, square-packing-bounds, the $n = 50$ measure resting on Tokoharu's format and checker. The source says parts of the work were produced with AI assistance under human direction.
Composition: One certificate. E-n050-wand125-mixed-740-source-replay replays it in full with the source's own driver and checker: one interval-certified method, replayed here, C3. Coverage is decided by the source's C++ checker alone, and the axis tables are a second implementation for one direction and not a second method. $n = 51$ takes the same certificate directly, its mass being below 51, so no step is added.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained. A second machine method would be a method-distinct decision of rotated coverage; the replay here runs the source's own checker. The checker's controls were made on the $n = 37$ certificate of the same kind; devtools.audit_wand125_point_and_mixed mixed-control on this certificate would bind them to this measure as well.
Significance: Raised the verified lower bound at $n = 50$ by 0.317 over Nagamochi's closed form, and at $n = 51$ by 0.236, and stands 0.0826 above Green's reported value at $n = 50$ . The two changes to Tokoharu's checker were read and found legitimate on 2026-09-28. A substantive case result at S3. The closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03), so the margins over the verified floor it left, Karakuş's general bound, are larger.
Novelty: previously-published Present in an identified source
Records: n = 50 n = 51 register evidence 1 evidence 2 source review

T-043

Claim: $s (17) > 116511 / 25000$ = 4.66044, by Guzhou0806 / N17 project's R068 release of 28 September 2026, continuing Kleddamag's public 4.66001 charge (T-041). It raised the verified bound by exactly 43/100000 over T-041 and leaves 0.0150901 to Bidwell's 4.67553009.

The 889 rule orbits and their weights are 4.66001's byte for byte; one zero-weight site orbit moves by 1/12500 and one weighted four-site point orbit is added, for a budget of 17000448944 units of 10^-9, and the strict cores are rebuilt over 4,991 parent-angle intervals with parents of side 115325/116511 in the container 4613/1000. Seventeen cores at the minimum core charge 1000026844 exceed the budget by 7404.

It was replayed here in full on 28 and 29 September 2026, the first local validation on record: the package says its publisher ran no local validation and did not observe its CI. Guzhou0806's own C++ checker and Kleddamag's Node BigInt checker ran over every interval, and devtools.audit_guzhou_r068 found the fresh and published ledgers identical row for row; the two checkers are one event-cell method.

Guzhou0806 / N17 project, n17-square-packing. The package credits Kleddamag for the charge, the proof and the Node BigInt checker (Kleddamag building on Squares Project (Joshua Levy), Mira and Guzhou0806), and discloses AI assistance.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; the mapped 2026-09-28 review is not recorded in this entry's reviews, and whether it counts is the owner's decision. The native parent-core route models only k-of-m threshold atoms and cannot read the weighted and winning-subset features, so extending it to them is the reachable second machine method. The review's code-independent exact sweep reproduced nine sampled rows; completing and retaining it would make the exact leg this repository's own but is the same method. Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-10-05 by Guzhou0806's R071 (T-093), which keeps this charge; stays true as stated.
Significance: The case's verified lower bound, +0.00043 over T-041 from one added point orbit on an unchanged charge and a finer catalogue. A substantive case result at S3 by the T-015 and T-032 precedent; the movement is small, and this repository's replay is the first local validation of it on record.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-042

Claim: $s (17) > 233009 / 50000$ = 4.66018, by Guzhou0806 / N17 project's R067 release of 28 September 2026. R068 (T-043) superseded it the same day, so it never held the case's verified field.

The certificate is Kleddamag's 4.66001 charge (T-041) unchanged, budget 17000402008 units of 10^-9, at the larger parent side 32950/33287, with strict cores rebuilt over 2,808 parent-angle intervals whose endpoints contain every 4.66001 endpoint. Seventeen cores at the minimum core charge 1000026844 exceed the budget by 54340.

It was replayed here in full on 29 September 2026 by the source's paired launcher, Guzhou0806's C++ checker with Kleddamag's Node BigInt checker, and devtools.audit_guzhou_r068 found the fresh and published ledgers identical on all 2,808 rows; the two checkers are one event-cell method.

Guzhou0806 / N17 project, n17-square-packing, on Kleddamag's charge. The release names itself as made "with AI assistance".
Next rung: No rung change is sought: the result never held a field, and a method-distinct decision would be spent better on R068, which runs the same charge further. Superseded on its publication day, 2026-09-28, by Guzhou0806's R068 (T-043), before it held the verified field; stays true as stated.
Significance: True and replayed in full, but superseded on its publication day by R068 before it held any field: a citable intermediate, +0.00017 over T-041, that changes no standing bound. Registered because it was replayed and reviewed here.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 source review

T-056

Claim: For each of 49 counts n from 68 to 307, $s (n)$ is at most the verified upper bound its case record carries, from Francisco Couzo's packings as published on 27 September 2026. The counts are 68, 102, 103, 105, 106, 110, 123, 130, 131, 132, 152, 154, 155, 156, 172, 177, 180, 181, 182, 199, 206 to 210, 228, 236 to 241, 259, 263, 268 to 273, 292, 297 and 301 to 307.

Each bound is the side the source prints, from 8.798795237222592 at $n = 68$ to 17.981030548643712 at $n = 307$ , or, where the printed pose's own side rounds up past it, that side plus one unit of its fifteenth decimal at 26 counts and 2, 3 and 2 units at $n = 206$ , 259 and 305. Every printed side is below the best known side this project recorded before, by 3.0e-8 at $n = 172$ up to 0.022 at $n = 301$ , and below the live catalogue of 29 September 2026.

Each bound is certified here twice, by methods that share no geometry code. The first is an exact rational packing, the source's decimal pose rounded to rationals at centre dilation 1, decided pair by pair and wall by wall over Q by two checkers that share no code. The second is outward-rounded interval arithmetic on the source's own pose as printed, with the true cosine and sine of each angle, which decides every pair and wall at 40 digits and encloses the pose's extent to within 1e-39.

Francisco Couzo, franciscouzo/square-packing. The author said on jlevy/squares#227 that he found the 102 and 103 packings "with the help of Claude", and the repository itself states no AI assistance.
Composition: Both machine entries decide all 49 bounds, each at its case's verified value, by different methods (exact-algebraic on a rational rounding of each pose; interval-certified on the printed pose), so every load-bearing part is C3, with two machine methods shown beside the rung. Their common mode is stated in E-franciscouzo-2026-09-27-interval-replay. The comparisons with earlier sides and the live catalogue are recorded values and set no rung.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the complete claim and a human oversight record; the two reviews so far, of 2026-09-29 and 2026-09-30, are same-project reviews of each route and are not recorded in this entry's reviews. At $n = 206$ , 259 and 305 both routes refute the printed side for the printed pose, so the printed side itself needs a different pose, refined beyond the source's binary64 digits, or coordinates the source prints at higher precision; that would be a new packing, not a replay.
Significance: Forty-nine substantive case results at once: each moves the best known side at its count, 28 of them by more than 0.001, and each moves this repository's verified ceiling off the integer grid onto a certified packing, so 46 cases stop trailing their report. S3 by the anchor "a substantive case result"; the source publishes no method, so nothing here is a reusable technique.
Novelty: previously-published Present in an identified source
Records: 49 cases register evidence 1 evidence 2 evidence 3

T-053

Claim: $s (45) = 7$ : the lower half by Evan Daniel's mixed cover of 27 September 2026, the upper half by the $7 \times 7$ grid.

The cover is 19,989 weighted points of [0,7]^2 plus mass spread uniformly along 3,912 segments of length 1/50 on the interior grid lines, total 2238676387/(5*10^7) = 44.77352774 < 45, exactly D4-invariant. Every closed unit square in [0,7]^2 captures mass at least one, with boundary points and edge segments counting in full.

The source certifies it at margin zero with the same two checkers as T-052, zm_mixed.py over 78,400 roots of the D4 region and zmx2 over 4,900 D4 roots and 39,200 unreduced roots. Nothing about this cover is in Lean beyond lemmas proved for every side.

zmx2 was replayed here in full on 28 and 29 September 2026, over all 4,900 D4 roots and all 39,200 unreduced roots, every root's census equal to the source's; zm_mixed.py was only sampled.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.
Composition: Compound: the lower half is E-n045-evand-mixed-cover-zmx2-replay (interval-certified), the upper half the grid (E-basic-grid-upper, exact-algebraic). As for T-052, the two machine methods certify different halves; the lower half stands on one replayed method and sets the minimum, and the equality is C3. wand125's point-only lower half (T-054) is a second certificate under the same checker and is registered separately.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A complete zm_mixed.py re-sweep, about 16.5 CPU-hours at the source, recorded as a second, exact-algebraic entry, would give the lower half a second machine method; the source's own zm_mixed.py record was imported from a working-name run (review finding F1), so a fresh complete run also retires that finding. V5 would need a top theorem and the checker programs formalised, and a human expert's review of the formalization.
Significance: An exact value for a case that was open, 0.169 above Nagamochi's closed form, and the third of the family s(k^2 - 4) = k for k = 5, 6, 7 with T-051 and T-052. The technique is T-052's, which alone would argue for S3 by the calibration T-020 wrote, but that calibration exempts a case that changes character, and this one goes from open to solved; with its two siblings it is a bound family, so S4.
Novelty: previously-published Present in an identified source
Records: n = 45 register evidence 1 evidence 2 evidence 3 source review

T-052

Claim: $s (21) = 5$ : the lower half by Evan Daniel's mixed cover of 27 September 2026, the upper half by the $5 \times 5$ grid.

The cover is 7,536 weighted points of [0,5]^2 plus mass spread uniformly along 1,872 segments of length 1/50 on the interior grid lines, total 522368729933/(25*10^9) = 20.894749197 < 21, exactly D4-invariant. Every closed unit square in [0,5]^2 captures mass at least one, a boundary point and a segment along an edge counting in full.

The source certifies that at margin zero with two separately written checkers, zm_mixed.py in exact rational arithmetic and the Rust zmx2 with outward-widened binary64 enclosures, and Lean proves minSide 21 = 5 from the one checker statement.

zmx2 was replayed here in full on 28 and 29 September 2026, D4-reduced over 2,500 roots and unreduced over 20,000, every root's census equal to the source's; zm_mixed.py was only sampled.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.
Composition: Compound: the lower half is E-n021-evand-mixed-cover-zmx2-replay (interval-certified, zmx2 replayed here), the upper half the grid (E-basic-grid-upper, exact-algebraic). The two machine entries differ in method, but they certify different halves; the lower half, which sets the minimum, stands on one replayed method, and the equality is C3, as T-008 reads its own halves.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A complete zm_mixed.py re-sweep, about 12.9 CPU-hours at the source, recorded as a second, exact-algebraic entry, would give the lower half a second machine method; the shared pose-space architecture stays the residual common-mode risk. V5 would need the checker statement S21CheckerCover proved in Lean and a human expert's review of the formalization; the reduction s21_eq_five_of_checker, from that statement to $s (21) = 5$ , was kernel-checked here on 2026-09-30 with the standard axioms only (the 2026-09-28 packet's receipts/lean/).
Significance: An exact value for a case that was open, the second of the s(k^2 - 4) family after T-051, and the release that introduced line mass on the grid lines, which is what makes a zero-margin cover below the count possible at an integer side. S4 by the anchor "a reusable technique"; T-053 reuses it at $n = 45$ .
Novelty: previously-published Present in an identified source
Records: n = 21 register evidence 1 evidence 2 evidence 3 source review

T-046

Claim: wand125/square-packing-bounds reports one standing rectangle-density certificate for each of 48 counts from $n = 18$ to $n = 95$ , added or raised between 26 and 28 September 2026.

Up to $n = 44$ the standing values are $s (18) \geq 939 / 200$ , $s (19) \geq 963 / 200$ , $s (20) \geq 979 / 200$ , $s (26) \geq 553 / 100$ , $s (27) \geq 28 / 5$ , $s (28) \geq 143 / 25$ , $s (29) \geq 579 / 100$ , $s (30) \geq 1173 / 200$ , $s (31) \geq 1187 / 200$ , $s (37) \geq 257 / 40$ , $s (38) \geq 327 / 50$ , $s (39) \geq 663 / 100$ , $s (40) \geq 1339 / 200$ , $s (41) \geq 1351 / 200$ , $s (42) \geq 679 / 100$ , $s (43) \geq 1373 / 200$ and $s (44) \geq 1387 / 200$ .

From $n = 51$ they are $s (51) \geq 2977 / 400$ , $s (52) \geq 1507 / 200$ , $s (53) \geq 1519 / 200$ , $s (54) \geq 3067 / 400$ , $s (55) \geq 771 / 100$ , $s (56) \geq 777 / 100$ , $s (57) \geq 1567 / 200$ , $s (58) \geq 789 / 100$ , $s (59) \geq 198 / 25$ , $s (60) \geq 397 / 50$ , $s (61) \geq 199 / 25$ , $s (66) \geq 67 / 8$ , $s (67) \geq 1691 / 200$ , $s (68) \geq 1699 / 200$ , $s (69) \geq 343 / 40$ , $s (70) \geq 431 / 50$ , $s (71) \geq 1737 / 200$ , $s (72) \geq 437 / 50$ , $s (73) \geq 439 / 50$ , $s (74) \geq 221 / 25$ , $s (75) \geq 889 / 100$ , $s (76) \geq 223 / 25$ , $s (77) \geq 223 / 25$ , $s (78) \geq 1791 / 200$ , $s (86) \geq 1871 / 200$ , $s (88) \geq 189 / 20$ , $s (89) \geq 191 / 20$ , $s (90) \geq 191 / 20$ , $s (91) \geq 1929 / 200$ , $s (94) \geq 1959 / 200$ and $s (95) \geq 49209 / 5000$ . Here $n = 77$ and $n = 90$ take the $n = 76$ and $n = 89$ certificates, whose masses lie below 77 and 90.

The certificates were built with Tokoharu's solver, their weights scaled by one exact rational factor to mass n - 1/100 (n - 1/1000 at $n = 27$ ) before the recorded run, and each was accepted there at every net direction by Tokoharu's unchanged interval checker, verify.cpp.

Here this repository's exact preflight checked that every checker input is the one the upstream run recorded, and checked the mass, net and partition premises. The values stand here as reported; those whose certificates have since been replayed in full are registered again as replayed results, in T-045 and T-070.

wand125 after Tokoharu and Levy, square-packing-bounds as of 27 September and 28 September 2026. The source says parts of the work were produced with AI assistance under human direction.
Composition: One claim per count, from two source revisions: the counts whose case cites E-wand125-rectangle-report keep the 27 September certificate, and those citing the 28 September entries the raised or new one. $n = 77$ and $n = 90$ are reported-lane transfers by the mass condition, which add no premise to the source claim they rest on. Every part stands at V0/C0.
Next rung: The complete 201-direction replay of the 18 standing certificates no replay here has carried. Twenty-one passed and were entered on 2026-10-02, and each count they carry is registered again as replayed while staying in this scope, as $n = 27$ did: $n = 18$ , 19, 20, 26, 30, 40, 61, 75 and 78, with 76 and 77 by mass, in T-045, and $n = 29$ , 38, 39, 41, 52, 53, 59, 67, 69, 71, 86 and 95, with 42 to 44, 54, 55, 60, 68, 70 and 72 to 74 by mass, in T-070. Six more passed on 3 October and were entered on 6 October: $n = 51$ , 57, 72 and 73 in T-070, and $n = 58$ and 91, which are below the mixed certificates that hold those counts.

Of the 18 left, the 1 October revision raised 16 (T-068), whose own certificates carry those counts, and $n = 37$ and 60 are held above these certificates by replayed bounds (T-069's $s (37) \geq 161 / 25$ and $s (60) = 8$ , T-062), so no replay of the source's checker is planned for them. sqverify-fast's census of 3 October verified all 48 standing certificates at all 201 directions; with control receipts and a review of the census route that reads them, it would carry every count of this entry. C1 before that, by recording the 2026-09-27 scaling review as an external_review on the report entries.
Significance: Scored as a claim: if the replays pass, these are the strongest lower bounds on record at 48 counts, above Green's DS7 reports and every certificate the register holds at those sizes. The score is of what the reports would establish; V0/C0 says how far they stand, and the score gates nothing.
Novelty: previously-published Present in an identified source
Records: 48 cases register evidence 1 evidence 2 evidence 3

T-045

Claim: Twelve rectangle-density certificates in wand125/square-packing-bounds, added on 26 and 27 September 2026, prove $s (18) \geq 939 / 200$ , $s (19) \geq 963 / 200$ , $s (20) \geq 979 / 200$ , $s (26) \geq 553 / 100$ , $s (27) \geq 28 / 5$ , $s (30) \geq 1173 / 200$ , $s (31) \geq 148 / 25$ , $s (32) \geq 119 / 20$ , $s (40) \geq 1339 / 200$ , $s (61) \geq 199 / 25$ , $s (75) \geq 889 / 100$ and $s (78) \geq 1791 / 200$ . They are rect_n18_L4695, rect_n19_L4815, rect_n20_L4895, rect_n26_L553, rect_n27_L56, rect_n30_L5865, rect_n31_L592, rect_n32_L595, rect_n40_L6695, rect_n61_L796, rect_n75_L889 and rect_n78_L8955.

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100 (n - 1/1000 at $n = 27$ ), such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The $n = 27$ certificate's mass 26.999 is also below 28, and the $n = 75$ certificate's 74.99 below 76 and 77, so they give $s (28) \geq 28 / 5$ and $s (76), s (77) \geq 889 / 100$ too: 15 counts in all.

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp: $n = 27$ , 31 and 32 on 27 September 2026, and the other nine in cloud batches on 29 September, whose receipts were merged into the packet on 2 October. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises.

wand125, square-packing-bounds, built with and checked by Tokoharu's solver and interval verifier. The source keeps its credit to this project for the method lineage and says parts of the work were produced with AI assistance under human direction.
Composition: Twelve primary certificates. $n = 28$ takes the $n = 27$ certificate and $n = 76$ and 77 the $n = 75$ one directly, their masses being below those counts, so no step is added. The $n = 32$ part rests on the complete replay receipt retained beside the others (receipts/replay/rect_n32_L595), which E-wand125-rectangle-source-replay's limitations name; that entry's scope lists only the counts whose verified lower bound it carries, and $n = 32$ never held one. One interval-certified method, replayed here: C3.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method would be a method-distinct decision of rotated coverage; global coverage is still decided by the source's C++ checker alone, and the exact preflight does not decide it.

Every certificate of this revision that T-046 reports is now replayed and registered here, and only $n = 18$ still holds this entry's value. At $n = 32$ superseded on intake, 2026-09-27, by Evan Daniel's $s (32) = 6$ (T-051), before it held the verified field; at $n = 61$ on 2026-10-02 by $s (61) = 8$ (T-063), when the $s (60)$ cover's replay landed; on the same day at $n = 76$ by $s (76) \geq 447 / 50$ (T-072) and at $n = 77$ and 78 by $s (77) = 9$ and $s (78) = 9$ (T-067), when their replays landed; and at $n = 19$ , 20, 26 to 28, 30, 31, 40 and 75 by wand125's raised certificates of 1 October (T-074), when their replays were merged. Stays true as stated.
Significance: Raised the verified lower bound at $n = 27$ and 28 by 0.092 and at $n = 31$ by 0.21 over Tokoharu's values (T-047), on Tokoharu's method with larger certificates. Substantive case results, S3.
Novelty: previously-published Present in an identified source
Records: 15 cases register evidence 1 source review

T-041

Claim: $s (17) > 466001 / 100000$ = 4.66001, by the bounds/4.66001/ package of Kleddamag's 17-squares-certified-bound, published untagged on 27 September 2026. It raised the verified bound by exactly 1999/100000 over T-040.

It keeps v1.1.0's checkers byte for byte and combines two completed charge candidates: 224 point, 225 two-of-three, 30 three-of-five, two four-of-seven, 154 weighted-threshold and 254 pairwise-intersecting winning-subset orbits, 7,048 feature images over 20,856 sites, all 86 distinct rules of capacity one, over 2,168 parent-angle intervals with parents of side 461300/466001 in the container 4613/1000. Seventeen cores at the minimum core charge 1000026844 exceed the budget 17000402008 (units of 10^-9) by 54340.

It was replayed here in full on 27 September 2026 by the source's Python and JavaScript BigInt checkers, which agree on every interval and with both published ledger pairs; they are one event-cell method.

Kleddamag, 17-squares-certified-bound, building on Squares Project (Joshua Levy), Mira and Guzhou0806, as its ATTRIBUTION.md states, produced with AI agents (OpenAI Codex) under Kleddamag's direction.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method is not at hand: the native parent-core route cannot read the weighted and winning-subset features, and the review's independent exact sweep, which shares no code with the source and reproduced all 2,168 row minima, was review scratch and is the same method. Retaining that sweep as a repository instrument would make the exact leg this repository's own. Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-09-29 by Guzhou0806's R068 (T-043), which continues this charge; stays true as stated.
Significance: Raised the verified lower bound at $n = 17$ by 0.01999 on T-040's checkers and budget rules, with a larger dictionary of the same features. A substantive case result, S3 by the T-015 and T-032 precedent.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-051

Claim: $s (32) = 6$ : the lower half by Evan Daniel's weighted closed cover of 26 September 2026, the upper half by the $6 \times 6$ grid.

The cover is 13,085 D4-invariant rational points of [0,6]^2, total weight 3171350535386/10^11 = 31.713505354 < 32, such that every closed unit square in [0,6]^2 captures weight at least one, a boundary point counting. A packing at side below 6 would scale to 32 disjoint closed unit squares capturing at least 32.

The margin is zero at the grid, so the source decides the cover by exhaustive exact pose-space subdivision of its D4 fundamental region (zeromargin.py, 7,200 root boxes), and a Lean theorem derives minSide 32 = 6 from that one checker hypothesis.

The complete zeromargin.py sweep was replayed here on 27 September 2026 with the source's runner, checker and settings: all 7,200 roots certified, 164,130 boxes, every root's census equal to the source's.

The source's third checker, zmx2, was run here on the same cover on 29 September 2026 with --pair-points: all 3,600 roots of the D4 region certified, 1,405,342 boxes, none uncertified. zmx2 is a second implementation of the same subdivision by the same author and agent. Its point test derives from zeromargin.py's, but it closes germs by its own pair lemma and decides by exact integer tests on outward-rounded binary64 enclosures of chord ends. On 2 October 2026 zmx2 was run here again with --sym-atoms, without the D4 fold, over all 28,800 roots of the whole pose space: none uncertified, and every root's census equal to the source's run.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.
Composition: Compound. The upper half is the $6 \times 6$ grid replayed exactly by the basic-bounds checker (E-basic-grid-upper), a construction its replay establishes, so under epistemics.md's Scope and Composition the equality takes the C of its lower half, which sets the rung.

The lower half rests on complete replays here of the source's zero-margin pose-space subdivision by two checkers whose method values differ: zeromargin.py, exact-algebraic, over the 7,200 roots of the D4 fundamental region (E-n032-evand-closed-cover-source-run), and zmx2, interval-certified, over the 3,600 roots of the D4 region (E-n032-evand-closed-cover-zmx2-replay) and, with --sym-atoms and no fold, over all 28,800 roots of the whole pose space (E-n032-evand-zmx2-full-sym-replay). That is C3, with the two methods shown beside the rung.

The two checkers differ in how germs are closed (zmx2's pair lemma Z and pair dynamic programme over lines one unit apart, against zeromargin.py's monotone chains), in arithmetic and in their parsers. Taken as zeromargin.py's run and zmx2's run without the fold, they no longer share the D4 fold, the region or any symmetry premise: the zmx2 side rests on its own Lemma A (search/ZMX2.md section 4.9, proved, and checked against the code by the 2 October review) where zeromargin.py's rests on the fold, which the source's Lean development kernel-checks, and nothing in zeromargin.py corresponds to Lemma A.

The common mode is what remains shared: the author and AI agent and the statement decided; the point-in-square formulation, G0-G3 with the corner-and-vertex rule, one design in two arithmetics, which for a point cover is the primitive that decides most leaves; the pose parametrisation; and the branch-and-bound architecture. The source's Lean development kernel-checks the formulation on zeromargin.py's side, which bounds the common mode and does not remove it. The two checkers are separately written, not independent.

The point test is shared by derivation: on 1 October 2026 the source stated that zmx2 was written to be independent of zm_mixed.py only and that its author was permitted to read zeromargin.py, so Lemma P derives from zeromargin.py's G0-G3 (jlevy/squares#238, retained as packing/resources/web/evand-square-packing-2026-10-01/issue-238-author-2026-10-01.json). The brief that governed that reading was not public when this entry was reviewed. The source published it on 3 October, and the 4 October evand packet retains it (s12/tasks/s21-finish/xcheck.md, think-83zc): it forbids zm_mixed.py and its write-up and permits zeromargin.py, zmcheck and their write-ups, as the source said.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; whether a mapped same-project review of another author's result counts toward them is the owner's decision. V5 needs the source's hypothesis-free Lean theorem s32_eq_6 built, which decides a generated box tree in the kernel with zeromargin.py only as the tree generator's untrusted oracle, and a human expert's review of the formalization. Mathlib's build cache became reachable on 2026-09-30, and the conditional s32_eq_six_of_checker and s13_eq_4 were built here (the 2026-09-28 packet's receipts/lean/). The cost that remains is compute: a probe puts s32_eq_6 at 14 to 28 CPU-hours emitted in about 320 parts of about 4 GB each, with 8 to 10 GB for the generator (think-angg).

Beside the rung, the source's zmx2 --full --pair-points --sym-atoms run was replayed here on 2 October (think-48e1, E-n032-evand-zmx2-full-sym-replay), which took the D4 fold out of what the two checkers share. It changes neither rung: it is evidence of the kind already held, a machine replay of the source's own checker with a mapped review, as section 8 of the 2 October review says, and what it changes is the composition that rung 4's reviewers and oversight record will read.
Significance: An exact value for a case that was open, the first this record holds for s(k^2 - 4) with k >= 4, 0.204 above Nagamochi's closed form. S4 by the anchor "a reusable technique": the zero-margin closed cover decided by exhaustive pose-space subdivision, which the same author carried to $n = 21$ and 45 (T-052, T-053) and wand125 to $n = 45$ (T-054). Not S5: $n = 32$ is not a central open case of this project.
Novelty: previously-published Present in an identified source
Records: n = 32 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 evidence 6 source 1 source 2 review 1 review 2 review 3

T-040

Claim: $s (17) > 232001 / 50000$ = 4.64002, by Kleddamag's 17-squares-certified-bound v1.1.0 release of 26 September 2026. It raised the verified bound by exactly 1/50 over R052 (T-039).

It widens v1.0.0's charge with weighted-threshold features (coefficients 2, 1, 1, 1 at threshold 3) and pairwise-intersecting winning-subset features, every distinct rule of capacity one: 280 point, 155 two-of-three, 26 three-of-five, one four-of-seven, 54 weighted and 30 winning-subset orbits over 2,048 parent-angle intervals, with parents of side 32950/33143 in the container 4613/1000. Seventeen cores at the minimum core charge 998727933 exceed the budget 16978369232 (units of 10^-9) by 5629.

It was replayed here in full on 27 September 2026 by the source's launcher, running its Python and JavaScript BigInt checkers over every interval; they agree with each other and with all three published ledger pairs, and they are one event-cell method.

Kleddamag, 17-squares-certified-bound, building on Squares Project (Joshua Levy), Mira and Guzhou0806, as its ATTRIBUTION.md states. Its AUTHORS.md says the work was produced with AI agents (OpenAI Codex) under Kleddamag's direction.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. This repository's native parent-core route models only k-of-m threshold atoms, so it cannot yet read the weighted and winning-subset features; extending it to them is the reachable second machine method. The only packing/tests control is the retention check of the packet's compressed files; the adversarial control is the source's controls.py, replayed here (eight corrupted certificates rejected by both checkers). Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-09-27 by Kleddamag's 466001/100000 (T-041); stays true as stated.
Significance: Raised the verified lower bound at $n = 17$ by 0.02, with T-041's 0.01999 the largest step of the ladder after R012's 0.02305, and it is the release that introduced the weighted-threshold and winning-subset charge features that 4.66001, R067 and R068 reuse. Those features could argue for S4's reusable technique, but they are others' extension of the weighted method this project banked, so it is held at S3 as T-032 was.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-039

Claim: $s (17) > 231001 / 50000$ = 4.62002, by Guzhou0806 / N17 project's R052 release of 25 September 2026. It raised the verified bound by 1142853/4992650000, about 0.000229, over T-038.

The certificate has 2,354 point orbits and 514 two-of-three and 54 three-of-five threshold orbits (18,585 sites) over 15,721 parent-angle intervals, parents of side 230650/231001 in the container 4613/1000. Seventeen cores at the minimum core charge 999426274093 exceed the budget 16990246659579 (units of 10^-12) by 2 units, no slack beyond integer rounding.

It was replayed here in full on 25 September 2026 by the source's Python/Numba and Node/BigInt sweeps, each reproducing the source's own row ledger byte for byte; they are one event-cell method. This repository's native parent-core route verifies every exact premise but refuses the coverage at its engine ceilings.

Guzhou0806 / N17 project, n17-square-packing, on Kleddamag's v1.0.0 mixed point/threshold parent-core architecture (T-038). The release names itself as produced "with AI assistance" and credits Kleddamag, Mira-acc and this repository's threshold lineage.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method needs the native parent-core engine's static ceilings raised past this certificate's 8,937 atoms, 9,261 sites and 44,685 member slots, so that its complete coverage decision can run; its four-row sizing run is not a decision. Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-09-27 by Kleddamag's v1.1.0 (T-040); stays true as stated.
Significance: A substantive case result that held the verified field at $n = 17$ for two days. Its movement, 0.000229, is small, and it adds three-of-five groups under the k-of-m rule already reviewed at $n = 11$ to T-038's architecture. S3 by the T-015 and T-032 precedent, at the low end of it.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-089

Claim: $s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , by two packings by Allen Chang, the first optimized by David Ellsworth, certified here exactly. The Kingbird catalogue prints their sides as 9.63475764863108, of degree 672, and 9.83881526994826, cut short from 9.838815269948262260..., the root near it of the degree-41 polynomial it prints; it leaves the degree-672 polynomial to the picture's source, whose side a third party's parse reads as 9.634757648631082029.... None of those is certified here.

In the catalogue's words, $s (83)$ was "Improved by Allen Chang in September 2026, with GPT-5.6 Sol and GPT-6 Astra", "Optimized by David Ellsworth in September 2026", and its "Polynomial root solution found by Allen Chang in September 2026", improving the Hajba and Cantrell packing by about 6.8e-5; $s (87)$ was "Improved and optimized by Allen Chang in September 2026, working with GPT-6 Astra, with help from 'TheMagicAnimals'", improving Ellsworth's packing of February 2026 by about 2.2e-6.

The catalogue states no checker, and its pictures, which hold the poses, could not be fetched. Each certificate is a third party's binary64 parse of the picture, rounded to rationals and dilated about its centre by 1 + 1e-13: exact rational packings of sides 9.6347576486319454... and 9.8388152699491448..., 8.65e-13 and 8.85e-13 above the printed sides, each decided pair by pair and wall by wall over Q by two checkers of this repository that share no geometry or verification code. Each bound above is that side rounded up at the printed precision.

Allen Chang and David Ellsworth, September 2026, in the Squares in Squares catalogue.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the claim and a human oversight record; the review of 2026-10-05 is one. The verified bounds sat 8.7e-13 and 8.9e-13 above the printed sides because the certificates are binary64 parses dilated by 1 + 1e-13; the poses refined on their active contacts, Evan Daniel's exact certificates (T-101), have since brought both cases' verified upper bounds within one unit of the printed sides' last place. A method-distinct second route is what C4 needs.
Significance: Lowers the best known side at two counts, by about 6.8e-5 and 2.2e-6. S2 by the anchor "a citable detail that changes no theorem".
Novelty: previously-published Present in an identified source
Records: n = 83 n = 87 register evidence 1 evidence 2 evidence 3 evidence 4

T-088

Claim: $s (69) \leq 8.82719465572975$ , by David Ellsworth's packing of 69 unit squares, certified here exactly. The Kingbird catalogue prints the packing's side as 8.82719465572973, cut short from 8.827194655729738914..., the root near it of the degree-38 polynomial the catalogue prints; neither of those two is certified here.

In the catalogue's words, the packing was "Refound by David Ellsworth in September 2026, using his modified version of Thomas Schadt's simulated annealing program, starting from $s (69)$ found by Maurizio Morandi in June 2010", and "Optimized by David Ellsworth in September 2026". It is the packing whose side hmbelvedere's UnitSquare release reported in July 2026 as 8.8272055078..., which the catalogue lists as "Improved by hmbelvedere in July 2026, using an undisclosed iterative method (probably with AI)", after David W. Cantrell's improvement of August 2023; the optimization lowers that side by about 1.1e-5.

The catalogue states no checker, and its picture, which holds the pose, could not be fetched. The certificate is a third party's binary64 parse of that picture, rounded to rationals and dilated about its centre by 1 + 1e-15: an exact rational packing of side 8.8271946557297478..., 1.8e-14 above the printed side, decided pair by pair and wall by wall over Q by two checkers of this repository that share no geometry or verification code, and refused by both when its side is cut by 1e-15 or one square is moved by 1e-6. The bound above is that side rounded up at the printed precision.

David Ellsworth, September 2026, in the Squares in Squares catalogue.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the claim and a human oversight record; the review of 2026-10-05 is one. The verified bound sat 2e-14 above the printed side because the certificate is a binary64 parse; the pose refined on its active contacts, Evan Daniel's exact certificate (T-101), has since brought the case's verified upper bound within one unit of the printed side's last place. A method-distinct second route is what C4 needs.
Significance: Lowers the best known side at one count by about 1.1e-5, by optimizing a packing already reported. S2 by the anchor "a citable detail that changes no theorem".
Novelty: previously-published Present in an identified source
Records: n = 69 register evidence 1 evidence 2

T-036

Claim: Every packing of eleven unit squares in a square container, six at orientation 0 and five sharing one orientation modulo pi/2 with half-tangent in [91442076901/250000000000, 73154061521/200000000000] (an interval containing [t* - 10^-6, t* + 10^-6] for Trump's exact half-tangent t*), has container side at least U = 3.877083590022814..., the exact side of Trump's packing, and a packing in the family has side exactly U only if it is a quarter-turn image of Trump's pose with the squares relabelled within the two classes.

Reflections leave the family, since they send the tilt theta* to pi/2 - theta*, whose half-tangent 0.464 is outside the box, so that Z/4 x S6 x S5 orbit is the whole equality set. The statement is T-035's reduction composed with the first clause of the BC-240 local theorem at Trump's pose. It says nothing about any other tilt, about packings whose orientation classes are not six plus five, or about $s (11)$ .
Composition: Compound, and the minimum is set by BC-240's first clause. The reduction, T-035 on E-n011-h236-rung0-reduction, is V3/C3. The first clause, on E-n011-trump-local-theorem-first-clause, is an audited proof whose steps are prose and whose radius computation alone is machine-checked, so method proof-audited with a proof block supports V3 and nothing higher, which fixes the composed theorem at V3. The checker derives C3 from the reduction's entry; this note is why the declared confirmation is C2.

The composing step is the one the closed-tree review derived again: undoing the quarter turn carries a Trump-degenerate leaf's packing at side s' <= U to a translate by (0, U - s') inside [0,U]^2, a sup-norm isometry on centre differences, so the first clause forces Trump's labelled pose and the four wall contacts force s' = U. The angle window, 2.0e-6 radians, is below rho_row.

Confirmation is C2 by the C table: the clause's replay command, the source-distinct BC-241 checker, passes at this head, and a proof-audited entry does not derive above C2.

2026-10-02: the radius generator gained a replay mode, so rho_row = 808514697/200000000000 now carries an exact-algebraic entry of its own, E-n011-trump-isolation-radius (replayed-here, same-implementation, passed), and C3 for the radius is derived. It does not set the composed theorem's minimum, because the closed-tree review's question, whether BC-240's audited prose steps are load-bearing for confirmation, is answered yes.

The replay and the BC-241 checker confirm the quantities the proof consumes (the moduli, the curvature constants, the stresses, the caps and their aggregate arithmetic) and not the steps that use them. Those steps are prose, and the first clause rests on each: that the second-order remainder of every active function is at most K/2 times the squared norm, so that the modulus forces the norm to be at least 2 kappa/K; that every packing in the ball selects one of the 128 branches, which needs the inactive-feature gap cap, itself a single-source computation; and the composing step through the quarter turn.

A mutation or a perturbed radius is refused by the replay, but a wrong prose step would not be. So C2 stands until those steps are mechanised or reviewed again by a distinct reviewer with the oversight record the ladder asks for.
Next rung: The replay mode the closed-tree review asked for exists, and the radius has its own exact-algebraic entry, but the composition note judges BC-240's audited prose steps load-bearing, so C3 needs those steps mechanised or reviewed again by a distinct reviewer, not more replay of the radius. V4 would need BC-240's closing steps mechanised rather than only the quantities they consume, and with C4 a second adversarial AI review by a distinct reviewer and a human oversight record. Retaining the per-face dual witnesses and recomputing the gap cap from distinct source are the closure review's two bounded residual tasks.
Significance: A substantive case result and machine audit: the first optimality statement with an equality case for a family containing Trump's packing. Stromquist 2003's bound for squares at 0 and 45 degrees is an earlier restricted-orientation statement, and it is not attained and its family does not contain Trump's packing. It moves no bound on $s (11)$ , and its cost, about 1.7e8 nodes for one box, argues against S4's reusable technique.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3

T-035

Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within rho of his pose

Claim: Every packing of eleven unit squares in a square container, six at orientation 0 and five sharing one orientation modulo pi/2 with half-tangent in [91442076901/250000000000, 73154061521/200000000000] (an interval containing [t* - 10^-6, t* + 10^-6] for Trump's exact half-tangent t* = 0.365769307604677...), whose side is at most the rational U_hi of the certificate header (U_hi - U = 2.03e-45), lies, after the quarter turn that puts its tilted centroid in the closed upper-right quadrant and the relabelling that orders each class by x + y/4, strictly within rho = 808514697/200000000000 of Trump's labelled image (rotation 1, labels [3,4,2,5,0,1,8,10,6,9,7]) in every centre coordinate, with every tilted orientation within 2.0e-6 radians of Trump's; every other packing in the family has side greater than U_hi.

The statement is a machine-verified reduction, not optimality: a closed exact cell tree of 139,441,005 records, replayed in full by an independent reader, puts every small packing in the family near Trump's pose. It does not invoke the BC-240 local theorem. T-036 composes it with that theorem's first clause.
Composition: Not compound. One certificate tree and one reader verdict carry the whole statement, with no local theorem, monotonicity or composition step between them. The reader closes a Trump-degenerate leaf by exact arithmetic, bounding every centre coordinate of the cell's packings within rho of the labelled image, which is why the statement stops at the rho-ball rather than at Trump's pose. The entry is C3 (repository-origin, exact-algebraic, a certificate, a replay and a passing status). One adversarial AI review is retained, the mapped 2026-09-24 closed-tree review: a Fable max W2 review by a different agent than the lanes that produced the tree, inside the same project. It is not an external review and not a human oversight record.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would need a proof-assistant formalization of the cell-tree contract and its reader, reviewed by human experts. Three steps would strengthen the record without moving a rung. The certificate could be retained where a gate can reach it; until then the manifest is the record's only hold on it. The reader could compare the declared box with t* +- 10^-6 itself, a comparison only the review has made. A reader that accepts a non-root cell would make the $n = 11$ negative control reader-checked. The mathematics beyond this result is rung 1 of the ladder, H-112, which H-242's pilot (BC-383) prices: a per-node bound that closes this box in far fewer than 1.7e8 nodes comes before more boxes.
Significance: A substantive case result and machine audit at the smallest open case: one certificate tree, closed with no unresolved leaf, reduces a whole restricted family to a neighbourhood of Trump's pose. It moves no bound on $s (11)$ , its family is one box of half-tangents 2e-6 wide, and its cost, about 1.7e8 nodes for that one box, argues against S4's reusable technique. The 2026-09-24 closed-tree review scored the composed theorem S3 on the same grounds.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1

T-050

Claim: $s (21) \geq 5000 / 1001$ = 4.995004995..., by Evan Daniel's weighted point certificate of 23 September 2026. It raised the verified bound by 2878/25025, about 0.115, over T-034.

The certificate is 4,604 D4-invariant rational points in 580 orbits, total weight 260057/12500 = 20.80456 < 21, decided over the rational angle net at $N = 6000$ by the method of T-049.

It was replayed here on 27 September 2026 by the source's Rust verifier at $N = 6000$ , least captured weight 10000083/10^7 at bin 684 as the source records, and by its exact Python re-check on four bins including the binding one; the two are one angle-net method.

Evan Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method would be the native parent-core decision, which certified rows 0 to 2,378 of 2,486 before it was stopped and projects to about 2.8 CPU-hours complete; its journal was not retained, so it must run again. Superseded as the verified lower bound on 2026-09-29 by Evan Daniel's $s (21) = 5$ (T-052); stays true as stated.
Significance: Moved $n = 21$ by 0.115 over T-034 to within 5/1001 of the grid's 5, past this repository's own fixed-shrink ceiling of 4.9885 for the format. A substantive case result, S3; superseded within two days by the same author's exact value.
Novelty: previously-published Present in an identified source
Records: n = 21 register evidence 1 evidence 2 source review

T-034

Claim: $s (21) \geq 122 / 25$ , from this project's weighted fractional unavoidable-set certificate at container side 122/25 = 4.88. The case held 97/20 = 4.85 from T-021 in this register's verified field, and the movement is +0.03. The certificate does not reach $n = 20$ : its total mass is 5036431/250000 = 20.145724, above twenty, so Condition 2 gives it $n = 21$ and upward, and T-021's 97/20 rung, retained beside it, still carries $n = 20$ .

The size takes no monotonicity step, for the reason T-020 gives: only Condition 2 mentions n among the five conditions, so an atom set of mass M certifies its side for every integer strictly above M. From $n = 22$ on the register already holds 5, so the certificate is true there and weaker.

The site set is the plainest the generator offers: auto grids (34, 46, 56) at inset 1/2 with no seed and no windows, converged below twenty-one in 36 LP rounds. At the same side a T-021-seeded set ran out its deadline unconverged, and a five-window set converged but stalled the interval route on a seam where atom rows sit exactly B apart; neither run decides anything about the side.
Composition: Primary at $n = 21$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim.

Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes, as for T-021; that is C3, with the two methods shown beside the rung. The standard-library verifier the review also ran is a second implementation of the sweep's method, so it adds nothing to that count.

One adversarial AI review is retained, the mapped 2026-09-23 review by Fable at max thinking, which covers the complete claim: it re-decided the frozen bytes by three routes, checked each of the five conditions and the $n = 21$ -only scope, and found nothing that refutes them. A different agent than the lane that produced the certificate wrote it, inside the same session; it is not an external review, and it is not a human oversight record.
Next rung: The margin is 0.854276 below twenty-one, more than five times T-021's 0.151277 below twenty, so the side was not pushed to where the covering value stops it. H-240 froze it below X-047's estimated additive crossing near 4.886, which is an estimate and not a bound, and this run did not test it.

A certificate for $n = 21$ cannot exist above ceil(sqrt(21)) * B = 4.9885, leaving 0.1085 of runway above 122/25, and the best known packing is the trivial 5, so the ceiling binds first: the method could in principle close the case to within 0.0115 of the upper bound and can never reach it.

The next rung is a pre-registered side above 122/25 on the same unseeded construction, read from the restricted optimum a converged run reaches. Set B's refusal is the instrument note to carry with it: a window lattice at pitch B puts atom rows exactly B apart, and the interval route stalls on the seam where an upright square has both closed edges on such a pair. A stall there is not a counterexample, and it decides nothing.

V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would need a proof-assistant formalization of the five-condition theorem, reviewed by human experts.
Significance: Scored at S3 by the calibration note T-020 wrote and T-021 applied: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at one of the same sizes, one rung higher, and $n = 21$ does not change character at 4.88. What it adds beyond the rung is a reading about the instrument rather than a theorem: the stock generator with no seed and no windows converged where the seeded and windowed site sets did not decide. The side also sits above 39/8 = 4.875, where both of H-062's constructions crossed twenty at $n = 20$ ; a mass above twenty is consistent with that wall and says nothing about $n = 20$ .
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 21 register evidence 1 evidence 2

T-047

Claim: $s (11) \geq 381 / 100$ , $s (26) \geq 1377 / 250$ and $s (29) \geq 571 / 100$ , by Tokoharu's rectangle-density certificates of 22 September 2026. The $n = 26$ and $n = 29$ certificates also give $s (27), s (28) \geq 1377 / 250$ and $s (30), s (31) \geq 571 / 100$ by monotonicity.

Each certificate is an exact nonnegative rational density on rectangles, D4-expanded, of total mass below its count, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one.

Each of the three passed a complete 201-direction replay here on 22 September 2026 of the source's interval checker, behind this repository's exact preflight of the candidate-to-input enclosures, orbit normalization, total mass, net and shrink premises and the axis-event partition.

Tokoharu, square-packing-density-bounds, after Levy and wand125: the author's announcement says it was inspired by @ojoshe and @wand125, and the README credits jlevy/squares, through wand125's attribution, for row and column generation and branch-and-bound. The rectangle-density method and its interval verifier are Tokoharu's.
Composition: Three primary certificates and four monotone transfers (27 and 28 from $n = 26$ , 30 and 31 from $n = 29$ ), each one deletion step that adds no premise, so every count keeps the replay's V3/C3. One interval-certified method, replayed here: C3.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method would be a method-distinct decision of global rotated coverage; the exact rational polygon and derivative probes exercise the interval kernel but decide no coverage. Superseded as the verified lower bound by wand125's replayed rectangle certificates at $n = 27$ , 28 and 31 on 2026-09-27 (T-045), and at $n = 26$ and 30 (T-045) and $n = 29$ (T-070) on 2026-10-02; at $n = 11$ it never held the field, being below T-033 and Kleddamag's 31/8 (T-037). Stays true as stated.
Significance: Scored at S4 by the anchor "a reusable technique": the rectangle-density certificate and its interval verifier are Tokoharu's, and wand125's certificates at 48 counts (T-045, T-046) and its $n = 50$ checker (T-048) are built on them. Its own bounds moved $n = 26$ to 31 by 0.02 to 0.38 over Nagamochi's closed form; at $n = 11$ it is below the standing bound and never held a field.
Novelty: previously-published Present in an identified source
Records: 7 cases register evidence 1 evidence 2 evidence 3 source review

T-044

Claim: Ten exact weighted point certificates in wand125/square-packing-bounds, of 22 September 2026, prove $s (26) \geq 109 / 20$ , $s (29) \geq 557 / 100$ , $s (39) \geq 13 / 2$ , $s (40) \geq 13 / 2$ , $s (53) \geq 369 / 50$ , $s (55) \geq 377 / 50$ , $s (56) \geq 381 / 50$ , $s (69) \geq 841 / 100$ , $s (70) \geq 171 / 20$ and $s (72) \geq 861 / 100$ .

Four more counts follow from the same files: $s (41) \geq 13 / 2$ from the $n = 39$ file and $s (71) \geq 171 / 20$ from the $n = 70$ file by monotonicity, and $s (52) \geq 369 / 50$ and $s (68) \geq 841 / 100$ because the $n = 53$ and $n = 69$ files have exact mass below 52 and 68, and only the mass condition mentions the count.

Three more follow by monotonicity since 2 October 2026: $s (54) \geq 369 / 50$ from the $n = 53$ file, $s (57) \geq 381 / 50$ from the $n = 56$ file and $s (73) \geq 861 / 100$ from the $n = 72$ file. Nagamochi's closed form, stronger at those counts, had held their verified fields until its published proof was found incomplete (T-085).

Every file was replayed here in full on 22 September 2026 by this repository's exact verifier, sqpack.fractional.certificate, over the complete 201-direction net through the source's adapter check_with_sqpack.py.

wand125, square-packing-bounds, building on this project: the source says "The method is not ours", credits jlevy/squares for the generator, and checks with this project's verifier.
Composition: Ten primary certificates and seven derived counts. At $n = 41$ , 54, 57, 71 and 73 the derivation is monotonicity from the $n = 39$ , 53, 56, 70 and 72 files; at $n = 52$ and 68 it is the exact mass inequality on the $n = 53$ and $n = 69$ files, recorded as their own audited entries. Each step is one exact comparison and adds no premise, so every count keeps the replay's V3/C3. All five entries are exact-algebraic, which is one machine method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. An interval-certified decision of the same ten files with this repository's fractional interval route, which decides weighted point certificates of this schema, would be a second machine method. No packing/tests control reads this packet; the control named is the verifier's own adversarial suite.

wand125's replayed rectangle certificates superseded these as the verified lower bound on 2026-10-02 at $n = 40$ (T-045), at $n = 39$ , 41, 52, 53, 55 and 68 to 72 (T-070), and at $n = 56$ (T-074), the last count they held then. The three counts this entry took the same day when Karakuş's finding (T-085) moved Nagamochi's closed form out of the verified lane, $n = 54$ , 57 and 73, were superseded too, at $n = 54$ and 57 by T-074 and at $n = 73$ by T-070, as the two lines were merged on 2026-10-03; it holds none, and stays true as stated. At $n = 26$ and $n = 29$ superseded on intake, 2026-09-22, by Tokoharu's density certificates (T-047), before either held the verified field; stays true as stated.
Significance: The first certificate-backed lower bounds at twelve counts from 39 to 72, above the Green values DS7 has reported without proof since 2009 and above Nagamochi's closed form by 0.02 to 0.21. A substantive family of case results at S3; the generator and verifier are this project's, run by someone else, so it is not S4's new technique.
Novelty: previously-published Present in an identified source
Records: 17 cases register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source review

T-037

Claim: $s (11) > 31 / 8$ = 3.875, by Kleddamag's 11-squares-certified-bound v1.0.2 release of 22 September 2026. It raised the case's verified lower bound by 0.048002 over T-033 and leaves 0.0020836 to Trump's 3.8770835900.

The certificate is an exact nonnegative D4-invariant system of point and k-of-m threshold charges over 12,028 contiguous parent-angle intervals, each with a closed core strictly inside every parent of the interval, parents of side 764/775 in a container of side 191/50. Eleven disjoint cores would carry at least 11 x 999962528 against a budget of 10999479944, which they exceed by 107864 integer units, and compactness makes the bound strict.

It was replayed here in full by the source's two exact event-cell sweeps, in Python and JavaScript, and decided a second time by this repository's native parent-core interval branch and bound (devtools.verify_kleddamag_n11_native), which accepted all 12,028 rows with no stalled box.

Kleddamag, 11-squares-certified-bound, building on Squares Project (Joshua Levy): the release's ATTRIBUTION.md says the work builds on this project and that global-certificate.json was developed from T-026's certificate.
Composition: Primary at $n = 11$ : one certificate and the bound L/A, with no monotonicity step. Two evidence entries whose methods differ both passed on the retained certificate, which is C3 with the two methods shown beside the rung: the source's exact event-cell sweeps (exact-algebraic, replayed here; the Python and JavaScript scanners are one method) and this repository's directed-rounding interval coverage of every row (interval-certified). Both rest on the source's certificate and on the same counting and transfer theorem, which the two 2026-09-22 reviews read.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the complete claim and a human oversight record. The two 2026-09-22 reviews are mapped and each reads one route; neither is recorded in this entry's reviews, and whether a same-project review of another author's certificate qualifies is the owner's decision rather than missing work. Rung 5 needs a proof-assistant formalization of the transfer theorem and the counting argument, reviewed by human experts. Superseded on 2026-09-29 by Wang and Li's $s (11) > 3875000000 / 999999999$ (T-061), which is this certificate reweighted and scaled, 3.9e-9 higher, and by T-060's exact $s (11)$ = T; stays true as stated.
Significance: Scored at S5 by the rubric's anchor, movement on a central open case. $n = 11$ is a central open case of this project, and T-019 gave not being one as its first reason for staying below S5. The bound closes about 96 per cent of the gap the register held, from 0.0501 to 0.0021 below Trump's packing, and it is this project's T-026 certificate adopted and carried further outside it. The mathematics is Kleddamag's; this repository's part is the replay and the method-distinct second decision.
Novelty: previously-published Present in an identified source
Records: n = 11 register evidence 1 evidence 2 evidence 3 source 1 source 2 review 1 review 2

T-033

Claim: $s (11) \geq 955000$ *sqrt(2073600042893309449)/359341754646249 = 3.826997548829543624, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on the 2880-step direction net.

The 584 point atoms and 320 threshold atoms of the retained 191/50 threshold certificate, on the same D4-symmetric sites and with every weight multiplied by 500000000/498684619, form a threshold certificate of the retained form at side 191/50 on the 2880-step net with shrunken side 249507/250000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell charge exactly 1. The rescaled total budget is 5483661432/498684619 = 10.996251384, still strictly below eleven.

The shrink is T-026's and does not move under refinement; what this net changes is the largest half-gap tangent, 207107/1440000000 over the 2881 directions of the net, exactly half the 1440-step value, and the bound rises with it.

For every positive rational q with q^2 below 129600002680831840562500000000/129126496632245014779129770001, common scaling of the point atoms, the threshold atoms' points, the container side and the shrunken side preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 191/50. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.

What that conclusion does not carry is stated with it. No individual certificate is supplied at the supremum, because the sharpened containment test is equality there; no strict inequality is established there; and 191/50 remains the largest side at which the retained T-025/T-026/T-033 fixed-core family holds an individual-side certificate, which is T-025's and not this result's.
Composition: Derived from a full replay of the 2880-step threshold source's Conditions 1, 1', 2', 3, 4 and 5' by the threshold theorem's own exact sweep, scaling equivariance for Conditions 1 and 1', invariance of Conditions 2' and 3, inverse-dilation preservation of Condition 5', and the sharpened containment lemma T-022 proved by an exact rational comparison of squares; rational density and upward embedding supply the limit step.

The source certificate's own acceptance rests on two machine decisions of one frozen file that fail differently, the exact event-cell sweep of sqpack.fractional.threshold, whose dense-grid and slab evaluations must agree at every direction, and the interval branch and bound of sqpack.fractional.threshold_interval over the doubled net, which never invokes the D4 argument.

The derived bound does not inherit that. The dilation step on top of them is decided by one exact-algebraic entry and by nothing else, and epistemics.md takes a derived claim to the minimum over its inputs and the derivation itself, and the derivation is machine-replayed here like its inputs, so the rung is C3. T-022 read its own case the same way. This reconciliation applies the same rule to T-024, whose earlier declaration counted only the source certificate's two routes.

A method-distinct decision of the corollary itself, an interval enclosure of the strict factor test and of c squared recorded as its own interval-certified entry, not the source's two routes, would give the derivation a second machine method. No review has been mapped, so rung 4, which needs two adversarial AI reviews by distinct reviewers and a human oversight record, is not in reach.

The step is T-022's argument reused unchanged, at the same B and a halved D, whose proof note is cited here rather than rewritten. T-026's 1440-step rung is the control at the coarser net -- the same atoms, the same weights and the same crossing shrink, the two records differing in direction_steps, in their id and provenance, and in no other field -- and the refinement run reproduced its registered surd exactly, so the 0.000550138 between the two rungs is the halving of D alone.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers, of this corollary and the frozen 2880-step certificate, each a document under docs/project/reviews/ entered in docs/project/document-map.yaml with role review, and a human oversight record. A method-distinct decision of the corollary itself would be shown beside the rung. T-026's mapped review is scoped to the 1440-step claim and reviewing that one is not reviewing this one, so it is not cited here.

No self-contained verifiable-claim document is rendered for this rung either: devtools.render_verifiable_claim carries one branch per retained threshold claim and its T-026 branch is bound to that rung's witness direction, surd and proof note, so a third rung needs the branch parameterised and a proof note of its own.

The rescaled budget is 5483661432/498684619, leaving 1869377/498684619 below eleven, and the side this rung reaches is within about 0.000550 of 955000/249507, which is all that any further refinement of the net can buy while the shrink stands where it does. Refinement is therefore near exhausted on these atoms, and re-optimization at a finer net, changed site generation and richer charge atoms remain distinct hypotheses rather than a continuation of this one. A further assurance step would require proof-assistant formalization, external review, or an independently reproduced full decision.
Significance: This is a substantive controlled case result and machine audit of this project's retained threshold family. It moves T-026 from 3.826447410572939744 to 3.826997548829543624 by halving D inside T-022's sharpened containment factor sqrt(1 + D^2)/(B(1 + D)), at T-026's own crossing shrink, which does not rise under refinement; Condition 4's own coarse test B(1 + D) < 1 has the lower ceiling 152800000000000/39926861627361 = 3.826997509248, which this value exceeds. The sharpened factor is what reaches; no atom was re-optimised and no new theorem was needed, since Conditions 1, 1', 2' and 3 are untouched by common scaling and Condition 5' is preserved by inverse dilation of placements exactly as Condition 5 is.

The certificate is the same measure T-026 registered: the rescaled budget is unchanged at 5483661432/498684619 = 10.996251384, still strictly below the eleven a certificate may not reach, and only the net and the half-gap tangent differ.

The unchanged family cannot reach the stronger current verified bound 31/8: its refinement ceiling 955000/249507 is about 3.82755, already below 3.875. The result therefore records controlled method and calibration evidence, rather than a current public-bound advance. Changed weights, sites, parent domains, and richer charge atoms remain unresolved.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source

T-065

Claim: $s (17) \leq 4.6755300936045509516342148538535054$ : seventeen unit squares fit in a square of at most that side. The figure is a rational outward ceiling, not the exact side of the packing.

Kleddamag's release of 21 September 2026 carries John Bidwell's 1998 construction forward as an exact rational witness, $s (17) \leq 4675530093604551 / 10$ ^15, which two exact implementations here and the source's own checker accept: 17 unit squares, 68 vertices contained, 136 pairs separated.

From that seed this project certified the packing at the unique root of its contact chart: the root exists and is unique in a frozen rational box, and all 68 wall and 136 pair obligations hold there, 36 as exact identities and 168 by rational interval bounds, and an independent audit reconstructs every interval record. This lowers Kleddamag's rational ceiling in the seventeenth decimal.

The ceiling agrees with Bidwell's record in the Kingbird catalogue at the catalogue's printed precision. No identity with the catalogue's degree-18 polynomial is proved, no new packing is claimed, and optimality is not claimed.

Kleddamag, Kleddamag/17-squares-certified-bound, for the rational witness of Bidwell's packing; its AUTHORS.md says the work was produced with AI agents under Kleddamag's direction.
Composition: Two parts, both V3/C3. Kleddamag's rational witness is E-n017-kleddamag-rational-upper, replayed here exactly. The registered ceiling is E-n017-certified-endpoint, derived here from that witness and audited here; its accepted root proof and its reviewed symbolic identities are stated premises of the audit. The second part sets the bound.
Next rung: Rung 4 on either axis needs two adversarial AI reviews by distinct reviewers and a retained human oversight record. With exp-245 the packing's side is the catalogue's degree-18 root, so whether to restate this bound at Bidwell's exact algebraic value is a registration decision. First-order stationarity is certified (exp-242), and a local minimum modulo sliders holds in the capture-target form (exp-244, exp-248); global optimality is unproved.
Significance: The first verified upper bound at $n = 17$ below the grid's 5: the case's verified bracket becomes 4.66044 < $s (17) \leq 4.67553009 \dots$ , both ends machine-checked, where the upper end was a reported catalogue value. S3 by the anchor "a substantive case result or machine audit"; it moves no reported bound.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-038

Claim: $s (17) > 461300 / 99853$ = 4.6197910929..., by Kleddamag's 17-squares-certified-bound v1.0.0 release of 21 September 2026. It raised the verified bound by 0.0067450 over Guzhou0806's R012 (T-032).

The certificate is a mixed point and two-of-three threshold charge, exact, nonnegative and D4-invariant, of budget 16998427356 integer units, over 7,853 contiguous parent-angle intervals, each with a closed core strictly inside every parent, in the container 4613/1000 with parents of side 99853/100000. The minimum core charge 1000020517 gives seventeen cores a surplus of 1921433 over the budget, and compactness makes the bound strict.

It was replayed here in full on 21 September 2026 by the source's two complete checkers, a Python and a JavaScript event-cell sweep, which agreed on all 509 histogram bins; they are one method.

Kleddamag, 17-squares-certified-bound, in the lineage the release names itself: Mira, Guzhou0806 / N17 project, and this repository's threshold-atom work. Its AUTHORS.md says the project was directed by Kleddamag with the research, implementation and verification done by OpenAI Codex.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. Unlike the later releases, this certificate carries only point and two-of-three atoms, which this repository's native parent-core route models, so a complete native run of it is the reachable second machine method, which would be shown beside the rung. No test under packing/tests exercises this packet; the control named is the release's own independent_controls.py, which the 2026-09-21 review used to close the containment and centre-domain premises. Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-09-25 by Guzhou0806's R052 (T-039); stays true as stated.
Significance: Raised the verified lower bound at $n = 17$ by 0.0067 over R012 and brought into the record the mixed point/threshold parent-core architecture every later $n = 17$ rung is built on. Scored as T-015 and T-032 were: a result by others replayed here, on this project's weighted method extended by others.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 source review

T-032

Claim: $s (17) \geq 461300 / 99999$ = 4.61304613..., by Guzhou0806's R012 parent-angle certificate of 20 September 2026. Beneath it, $s (17) \geq 4613 / 1000$ by Mira's 1620-atom weighted certificate of 7 September 2026. The case previously held 459/100 from T-019, and the movement is +0.02305 at the R012 value and +0.023 at Mira's side.

The R012 argument is this repository's weighted method with three changes. Parents have side A = 99999/100000 inside [0, L]^2 at L = 4613/1000, and the bound is L/A. A catalogue of 2925 closed parent-angle intervals covers [0, pi/4], and each interval picks its own concentric closed core, by direction and side, strictly inside every parent of the interval. Coverage by the measure is required only over the legal parent-centre square for the interval.

The measure is Mira's, rounded to 10^-5 with weights rounded up, 1616 atoms of mass 424969/25000; every core captures at least gamma = 250023/250000, and 17 gamma exceeds the mass by 701/250000. One size: the measure's mass is just below 17, so the certificate also says $s (18)$ and $s (19)$ are at least this value, and the register already holds 4679/1000 and 24/5 there.

R012 was replayed here by the source's exact checker and decided again by this repository's interval branch and bound. Mira's certificate is in this repository's certificate schema, and both stock verifiers accept it unchanged.

Guzhou0806, n17-square-packing, and Mira, 17squares. Both certificates descend from T-019 and credit it.
Composition: Primary at $n = 17$ on the R012 certificate: the bound is L/A from one catalogue and one measure, with no monotonicity step. That certificate is decided by two evidence entries whose methods differ, the source's exact event-cell replay and this repository's interval decision of all 2925 entries, both passing on the archived bytes; that is C3, with the two methods shown beside the rung.

The exact route is the source's own checker, whose sweep descends from this repository's, so its independence is of method from the interval route and not of authorship from the generator; the proof review also decided every entry with this repository's sweep kernel and agreed on all 2925 minima, as review scratch.

Mira's certificate carries its own pair of entries, the stock exact verifier and the stock interval verifier, and supports the weaker statement $s (17) \geq 4613 / 1000$ independently of R012. The reduction from a packing to R012's finite obligations is a read argument, closed in the review artifact, and is what a formal port would address.
Next rung: Mira's dilation endpoint, 4.61302863588611..., needs a T-022-style proof note for this certificate and is superseded by the R012 value in any case. A retained exact-route instrument for R012 that does not share the source's sweep lineage would make the exact leg this repository's own; the review's scratch pass shows the stock kernel decides every entry in about six minutes once its centre domain is a parameter.

V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record beside the 2026-09-20 proof review. Rung 5 needs a proof-assistant formalization of the reduction, reviewed by human experts. The method's open question is what the selector and the parent-centre restriction gain when the measure is optimised against them; Mira reports a failed attempt at 4.615 on the unrestricted test.
Significance: Raises the verified lower bound at $n = 17$ by 0.02305 over T-019 and closes the gap to Bidwell's 4.67553009 to 0.0625. Scored as T-015 was, and for the same reason: a result by others that this repository replayed, decided by a second method and read closely. Two things keep it at S3. The technique is the one T-017 and T-019 banked, extended by others; and almost all of the movement is Mira's support and weights, R012's selector and parent-centre restriction adding 1.7e-5 on a fixed measure. What those two extensions are worth on a measure optimised against them is open and is not this result.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2 evidence 3 evidence 4

T-031

Claim: At L = 96/25 and B = 9977/10000 on the 181-direction net (half-tangents k*207107/90000000, k = 0..180), the D4-symmetric point measure of total mass 10868617/1000000 = 10.868617 in cases/n11_corner_class_certificate/certificate.json, the retained exp-220 covering, charges at least 2000013/2000000 to every closed B-square at a net direction whose minimum of x + y is at least 1/2 in each of the four corner frames, decided by the exact event-cell sweep and by the interval branch and bound, which agree at that value.

Consequently every packing of eleven unit squares in [0, 96/25]^2 contains a unit square whose minimum of x + y in some corner frame is strictly below 1/2, that is, one that meets the open corner triangle x + y < 1/2 at that corner; equivalently, the all-free (octagon) class of lane-a Theorem B at threshold 1/2 contains no eleven-square packing at side 96/25.

This is a conditional exclusion of one corner-bin class and changes no bound on $s (11)$ ; the all-deep class is separately known to be outside this language's reach (BC-366), so the corner tree cannot close at 96/25 by clipping alone.
Composition: The certificate is a weighted fractional unavoidable-set certificate of the retained form whose row domain is the D4-symmetric convex clip of the admissible core domain by the four corner triangles at depth 1/2, admitted as a domain predicate on the row generator and threaded through both gate routes and the ceiling readers in Session 145 with a residual-7 control on the transported 88-family.

The declared C3 is what the two cited atoms derive. The exact event-cell sweep and the interval branch and bound are the two internal routes of one decide_certificate --corner-clip 1/2 invocation, which refuses the file unless both accept: they share the Certificate parser and the closed-form conditions, carry a byte-identical replay string, and the exact leg records relationship_to_generator: same-implementation, so the repository counts the pair as one confirmation and not as two method-distinct derivations. The Session 146 registration review says exactly that; a genuinely separate derivation of this clipped decision would be a second machine method, and none exists.

The verification rests on the two machine decisions of the frozen bytes, which fail differently: the exact event-cell sweep with the clip as extra half-planes on the per-slab centre range, and the interval branch and bound whose boxes are dropped only when provably inside the excluded set; the two decide the same set only because the clip is D4-invariant and folded, which the instrument's tests check. Conditions 1 to 4 are decided exactly before either route runs.

The conclusion transfers from the shrunken cores to the unit squares without loss in the direction that matters, because the kept domain is closed and a core lies strictly inside its square by Condition 4. The retained bytes declare variant class and the clip, and the gate refuses them without the flag, so the file cannot be read as a bound.
Next rung: The complementary class, in which some core meets a corner triangle, is the hard one: the all-deep class is a Lemma-D theorem for every site set (the transported 88-family keeps residual 7 against a requirement of 7, BC-366), so no point covering excludes it and the corner tree cannot make 96/25 unconditional by clipping. The four mixed D4 classes (fourteen bin vectors) are a bounded follow-up under BC-367 and need a box cut on the row domain. The instrument that could matter is a 2-of-3 threshold atom on ring-centre overlaps inside the all-deep class, which needs the clip wired into the threshold routes.

V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained: the Session 146 registration review is not mapped under docs/project/reviews. The case directory and the gate control test in the T-022 pattern are retained.
Significance: The first decided item of the corner-conditioned point language at a central open size: one D4 class of the corner tree is excluded at 3.84 by a certificate on a clipped domain, with the structural consequence that every eleven-square packing in a square of side 96/25 has a square meeting an open corner triangle of depth 1/2. It moves no bound, and the registration review shows the tree it belongs to cannot close at this side in the point language alone, since the all-deep class is a Lemma-D theorem for every site set (BC-366); the mixed classes are a bounded follow-up.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2

T-030

Claim: $s (18) \geq 4679 / 1000$ = 4.679, from this project's weighted fractional unavoidable-set certificate at container side 4679/1000. The case previously held 1871/400 = 4.6775 from T-029; the movement is +0.0015. The selected external report 9141/2000 = 4.5705 (GitHub; not replayed here) lies 0.1085 below this result.

One size, and not by monotonicity. Only Condition 2 mentions n among the five conditions, so an atom set of mass 71573611/4000000 = 17.89340275 certifies the side for every integer strictly above that mass, which is 18 and upward. It does not reach $n = 17$ , where T-019 still holds 459/100. From $n = 19$ on the register already holds 24/5 = 4.80, so the certificate is true there and weaker.

One rung is retained rather than a ladder. The side was reached by seeding the generator's site set with T-029's own atoms scaled from 1871/400 and adding a windows-5 lattice. Auto at 4679/1000 resolved to (32, 43, 54), the same triple T-029 used. $n = 18$ is off the H-218 sweep, so this retain confirms H-221, not H-218.
Composition: Primary at $n = 18$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-029 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: The retained certificate's total is 71573611/4000000 = 17.89340275, leaving 426389/4000000 = 0.10659725 below eighteen, and that margin is what a further rung has to be found inside. The side above it, 117/25 = 4.68, locked at 18.000000 on the T-019-seeded construction, on T-027's own atoms, on T-028's own atoms, and on three earlier site sets; adding sites can still lower a restricted optimum, so 117/25 is not barred. The remaining interval to that plateau is 0.001. The method's ceiling is 5B = 4.9885, and the best known packing is 4.82287566, so the real runway is 0.1439 to the packing. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

Superseded as the verified lower bound on 2026-10-02 by wand125's replayed $s (18) \geq 939 / 200$ (T-045), so a further rung of this method moves the case only above 4.695; stays true as stated.
Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at $n = 18$ , one size-specific covering rather than T-019's Condition 2 carry, and $n = 18$ does not change character at 4.679. The movement is +0.0015 against T-029's verified field and 0.1085 against the strongest named public candidate. Held below S4 because the technique is the one T-017 already banked, the displacement is of this project's own prior rung rather than a published closed form, and $n = 18$ is not a central open case.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 18 register evidence 1 evidence 2

T-029

Claim: $s (18) \geq 1871 / 400$ = 4.6775, from this project's weighted fractional unavoidable-set certificate at container side 1871/400. The case previously held 187/40 = 4.675 from T-028; the movement is +0.0025. The selected external report 9141/2000 = 4.5705 (GitHub; not replayed here) lies 0.107 below this result.

One size, and not by monotonicity. Only Condition 2 mentions n among the five conditions, so an atom set of mass 17889361/1000000 = 17.889361 certifies the side for every integer strictly above that mass, which is 18 and upward. It does not reach $n = 17$ , where T-019 still holds 459/100. From $n = 19$ on the register already holds 24/5 = 4.80, so the certificate is true there and weaker.

One rung is retained rather than a ladder. The side was reached by seeding the generator's site set with T-028's own atoms scaled from 187/40 and adding a windows-5 lattice. Auto at 1871/400 resolved to (32, 43, 54), not T-028's (32, 43, 53). $n = 18$ is off the H-218 sweep, so this retain confirms H-219, not H-218.
Composition: Primary at $n = 18$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-028 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: T-030 took the next rung at 4679/1000 on 2026-09-19. This result stays true as stated and keeps its bytes at certificate-1871-400.json. The remaining runway to the packing is now T-030's business. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.
Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at $n = 18$ , one size-specific covering rather than T-019's Condition 2 carry, and $n = 18$ does not change character at 4.6775. The movement is +0.0025 against T-028's verified field and 0.107 against the strongest named public candidate. Held below S4 because the technique is the one T-017 already banked, the displacement is of this project's own prior rung rather than a published closed form, and $n = 18$ is not a central open case.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 18 register evidence 1 evidence 2

T-028

Claim: $s (18) \geq 187 / 40$ = 4.675, from this project's weighted fractional unavoidable-set certificate at container side 187/40. The case previously held 467/100 = 4.67 from T-027; the movement is +0.005. The selected external report 9141/2000 = 4.5705 (GitHub; not replayed here) lies 0.1045 below this result.

One size, and not by monotonicity. Only Condition 2 mentions n among the five conditions, so an atom set of mass 35758287/2000000 = 17.8791435 certifies the side for every integer strictly above that mass, which is 18 and upward. It does not reach $n = 17$ , where T-019 still holds 459/100. From $n = 19$ on the register already holds 24/5 = 4.80, so the certificate is true there and weaker.

One rung is retained rather than a ladder. The side was reached by seeding the generator's site set with T-027's own atoms scaled from 467/100 and adding a windows-5 lattice after 117/25 locked at 18.000000. $n = 18$ is off the H-218 sweep, so this retain does not confirm that claim.
Composition: Primary at $n = 18$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-027 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: T-029 took the next rung at 1871/400 on 2026-09-19. This result stays true as stated and keeps its bytes at certificate-187-40.json. The remaining runway to the packing is now T-029's business. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.
Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at $n = 18$ , one size-specific covering rather than T-019's Condition 2 carry, and $n = 18$ does not change character at 4.675. The movement is +0.005 against T-027's verified field and 0.1045 against the strongest named public candidate. Held below S4 because the technique is the one T-017 already banked, the displacement is of this project's own prior rung rather than a published closed form, and $n = 18$ is not a central open case.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 18 register evidence 1 evidence 2

T-027

Claim: $s (18) \geq 467 / 100$ = 4.67, from this project's weighted fractional unavoidable-set certificate at container side 467/100. The case previously held 459/100 = 4.59 from T-019; the movement is +0.08. The selected external report 9141/2000 = 4.5705 (GitHub; not replayed here) lies 0.0995 below this result.

One size, and not by monotonicity. Only Condition 2 mentions n among the five conditions, so an atom set of mass 8937839/500000 = 17.875678 certifies the side for every integer strictly above that mass, which is 18 and upward. It does not reach $n = 17$ , where T-019 still holds 459/100. From $n = 19$ on the register already holds 24/5 = 4.80, so the certificate is true there and weaker.

One rung is retained rather than a ladder. The side was reached by seeding the generator's site set with T-019's own atoms scaled from 459/100 after the uniform grid locked at 18.000000.
Composition: Primary at $n = 18$ : one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-020 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: T-028 took the next rung at 187/40 on 2026-09-19. This result stays true as stated and keeps its bytes at certificate-467-100.json. The remaining runway to the packing is now T-028's business. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.
Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at $n = 18$ , one size-specific covering rather than T-019's Condition 2 carry, and $n = 18$ does not change character at 4.67. The movement is +0.08 against T-019's verified field and 0.0995 against the strongest named public candidate. Held below S4 because the technique is the one T-017 already banked, the displacement is of this project's own prior rung rather than a published closed form, and $n = 18$ is not a central open case.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 18 register evidence 1 evidence 2

T-057

Claim: $s (211) \leq 14.99796070496771500150$ < 15, by Joost de Winter's packing of 16 September 2026: 211 unit squares in a square of that side. It is the first packing of 211 squares below the $15 \times 15$ grid on record, so s(k^2 - k + 1) < k, shown in the Kingbird catalogue for k = 16, 17 and 18 ( $n = 241$ , 273 and 307), holds for k = 15 as well.

The source gives 21-digit poses and the author's report of an outward interval check at 80 digits; it publishes no interval boxes or checker.

The bound is certified here twice, by methods that share no geometry code. The first is an exact rational packing, the source's pose rounded to rationals at centre dilation 1, of side 74989803524838470007/5000000000000000000, 2.1e-14 below the printed side, decided pair by pair and wall by wall over Q by two checkers that share no code. The second is outward-rounded interval arithmetic on the source's own pose as printed, which lies in the printed square as placed with least wall clearance 1.000005e-14 and least pair gap 2.10001e-14, the values the source reports.

Joost de Winter, JoostdeWinter/square-packing-211. The source says nothing about AI assistance.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers of the complete claim and a human oversight record; the same-project reviews of each route (2026-09-29 and 2026-09-30) are not recorded in this entry's reviews. The source's own 80-digit interval run stays unreplayed until it publishes its boxes or checker; this record no longer depends on it.
Significance: A count whose best known packing was the trivial grid now has one below it. On the Kingbird catalogue and this record, which show s(k^2 - k + 1) < k at $n = 241$ , 273 and 307 (k = 16 to 18) and hold the grid at $n = 31$ to 183 (k = 6 to 14), the smallest k shown to satisfy it drops from 16 to 15. S3 by the anchor "a substantive case result"; the margin, about 0.002, changes nothing beyond this count.
Novelty: previously-published Present in an identified source
Records: n = 211 register evidence 1 evidence 2 evidence 3

T-026

Claim: $s (11) \geq 955000$ *sqrt(518400042893309449)/179696714646249 = 3.826447410572939, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on a finer direction net.

The 584 point atoms and 320 threshold atoms of the retained 191/50 threshold certificate, on the same D4-symmetric sites and with every weight multiplied by 500000000/498684619, form a threshold certificate of the retained form at side 191/50 on the 1440-step net with shrunken side 249507/250000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell charge exactly 1. The rescaled total budget is 5483661432/498684619 = 10.996251384, still strictly below eleven.

For every positive rational q with q^2 below 32400002680831840562500000000/32290909254655439869209770001, common scaling of the point atoms, the threshold atoms' points, the container side and the shrunken side preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 191/50. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.
Composition: Derived from a full replay of the 1440-step threshold source's Conditions 1, 1', 2', 3, 4 and 5' by the threshold theorem's own exact sweep, scaling equivariance for Conditions 1 and 1', invariance of Conditions 2' and 3, inverse-dilation preservation of Condition 5', and the sharpened containment lemma T-022 proved by an exact rational-square comparison; rational density and upward embedding supply the limit step.

The confirmation rests on the two machine decisions of the source certificate, which fail differently: the exact event-cell sweep of sqpack.fractional.threshold, whose dense-grid and slab evaluations must agree at every direction, and the interval branch and bound of sqpack.fractional.threshold_interval over the doubled net, which never invokes the D4 argument.

One adversarial AI review is retained and mapped, of the self-contained claim, by a separate agent from the implementation authors; it covers the threshold count, exact sweep, dilation, and endpoint inference. It is not a human oversight record, so the result is C3. T-025's earlier review covers the threshold theorem and T-022's covers the dilation argument; the new review checks their composition with the T-026 values and embedded data.

The 720-step certificate retained beside it crosses at the same shrink and reaches 955000*sqrt(129600042893309449)/89874194646249 = 3.825347845913112, which isolates the remaining 0.0010996 as the effect of D halving alone.
Next rung: The rescaled 1440-step budget is 5483661432/498684619, leaving 1869377/498684619 below eleven. The measured 720- and 1440-step certificates have the same crossing shrink and least unscaled charge. These finite measurements do not decide another net or prove that frozen-atom refinement cannot improve the limit. Re-certification on another net, re-optimization on a finer net, changed site generation, and richer charge atoms remain distinct hypotheses. The retained plateau duals with depth above one are useful separation inputs but not globally feasible obstruction witnesses.

The self-contained T-026 claim embeds the standard-library threshold verifier, the 1440-step certificate, and the exact dilation record; its mapped source-distinct review is the one adversarial AI review retained. V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would require a proof-assistant formalization reviewed by human experts.
Significance: The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0064474 beyond T-025's 191/50 and by about 0.0375930 beyond Stromquist's 2 + 4/sqrt(5), leaving the interval [3.826447410572939, 3.877084] and a bound gap of about 0.0506362.

The movement is the shrink a finer net admits under Condition 4, measured on the frozen T-025 atoms; no atom was re-optimised and no new theorem was needed, since Conditions 1, 1', 2' and 3 are untouched by common scaling and Condition 5' is preserved by inverse dilation of placements exactly as Condition 5 is.

For this fixed certificate, the rescaled budget is 5483661432/498684619 = 10.996251384 against the eleven a certificate may not reach, leaving about 0.0037 of budget. The measured 720- and 1440-step nets share the same crossing shrink and least unscaled charge. These finite measurements leave the effect of another net, a different core side, re-optimized weights, changed sites, and richer charge atoms unresolved.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source

T-025

Claim: $s (11) \geq 191 / 50$ = 3.82, by a threshold certificate: 584 point atoms of mass 271052551/31250000 and 320 threshold atoms, every one 2-of-3, of budget 143352577/62500000, on the D4-symmetric site set at shrunken side 9977/10000 and the 181-direction net.

A threshold atom (S, k, w) charges w to every admissible core holding at least k points of S and costs w floor(|S| / k) of the budget, so the family's total budget is 685457679/62500000 = 10.967322864, strictly below 11, while every event cell at every net direction carries charge at least 100000203/100000000. The counting argument then rules out eleven unit squares with pairwise disjoint interiors in a container of side 191/50.
Composition: One certificate, decided twice. The certificate is instantiated at its named side, 191/50; no dilation family is part of T-025.

Conditions 1, 1', 2', 3 and 4 are closed-form consequences of the frozen bytes and were recomputed without sqpack in the review; Condition 5' is decided by the exact event-cell sweep of sqpack.fractional.threshold, whose dense-grid and slab evaluations agree at every direction and whose witness charge is re-evaluated by membership counting, and independently by the interval branch and bound of sqpack.fractional.threshold_interval over the doubled net, which never invokes the D4 argument.

The two share the loader, the conditions and the net, and no part of how Condition 5' is decided; the gate refuses the record unless both accept and agree on the least charge exactly.

The result is C3 on those two machine routes. One adversarial AI review of the theorem is retained and mapped, by an agent of this project; it found no soundness defect, and its open items -- the second route, the controls, and a stranger-facing statement in a case directory -- this registration closes. It is not a human oversight record.
Next rung: Push the side with the same loop: the LP that produced these atoms is run again at a larger side, and the ceiling family says the answer cannot come from point atoms alone on the stated core domain. The finer-net and dilation continuation has now been measured and registered as T-026, proving $s (11) \geq 3.826447410572939744$ by an exact dilation-limit argument. That is a separate corollary. T-025 itself proves the ordinary lower bound $s (11) \geq 191 / 50$ at V3/C3, and its certificate is instantiated at that named side; V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. The self-contained T-025 claim embeds its standard-library verifier and certificate bytes. T-025 needs no limit argument, so no dilation record belongs at this rung.
Significance: The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by 0.0033905 beyond T-024 and by 0.0311456 beyond Stromquist, leaving the interval [3.82, 3.877084] and a gap of 0.057084. What earns the score beyond the arithmetic is the architecture: the exact depth-one ceiling family retained at this same side proves that no D4-symmetric point-atom measure of mass below eleven exists here, so no certificate of the earlier one-body form could have reached it, and the threshold atoms carry 2.293641 of this budget that the point method cannot have. It is the first certificate of a new kind in this project, decided from frozen bytes by two routes and reviewed adversarially before registration.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 source

T-024

Claim: $s (11) \geq 3175000$ *sqrt(518400042893309449)/598960960743657 = 3.816609502788862, proved by an exact dilation-limit corollary of T-018's retained atoms re-certified on a finer direction net.

The 1121 atoms of the retained 381/100 certificate, on the same D4-symmetric sites and with every weight multiplied by 200000/198931, form a certificate of the retained form at side 381/100 on the 1440-step net with shrunken side 2494953/2500000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell mass exactly 1.

For every positive rational q with q^2 below 3240000268083184056250000000000/3228788092454681596330191602841, common scaling preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 381/100. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.
Composition: Derived from a full replay of the 1440-step source certificate's five conditions, scaling equivariance for Condition 1, invariance of Conditions 2 and 3, inverse-dilation preservation of Condition 5, and the sharpened containment lemma T-022 proved by an exact rational-square comparison; rational density and upward embedding supply the limit step.

The source certificate is decided by two distinct machine methods, the exact event-cell sweep and the interval branch and bound on its frozen bytes. The load-bearing dilation-limit step has one exact-algebraic decision. Both are machine-replayed here, so the whole result is C3 under the derived-claim minimum rule. No review has been mapped for this result.

The 720-step certificate retained beside it is the rung the standalone reader also decides; its own limit, 38100000*sqrt(129600042893309449)/3594594251080001, is registered as evidence and is weaker.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers, of the corollary and the two frozen certificates, and a human oversight record. A method-distinct decision of the corollary itself, rather than a second decision of only the source certificate, would be shown beside the rung; T-022's review covered the argument at the 181-step net and this result reuses it unchanged with new B and D.

The standalone reader verify_claim.py decides the 720-step rung and refuses the 1440-step rung by its declared 1000-direction ceiling; a new reader release that lifts the ceiling would let a stranger decide the registered rung without trusting this project, and that is a change to a retained verifier rather than to the mathematics. The shrink is a 10^-7 grid crossing, not a minimum, so a smaller passing shrink off the grid would move the endpoint by less than the grid step.

Two limits are measured on the frozen atoms: at the original shrink the same atoms fail every finer net at an intermediate direction, so the gain is entirely the shrink the finer net admits, and the exact ceiling family at 191/50 caps every one-body point certificate at unit side 3.8288 on any net containing its six directions.

The next movement is therefore a re-optimised certificate at a finer net (H-153), which the covering LP on the frozen sites already shows to be below eleven at 720 steps, and past the cap the threshold atoms of H-152. The displayed lower bound is proved by a dilation-limit argument rather than one individual-side certificate at the algebraic value.
Significance: The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0065838 beyond T-022 and by about 0.0066095 beyond the T-018 certificate side, for about 0.0277551 beyond Stromquist in total. The movement comes from the shrink that a finer net admits under Condition 4, measured on the frozen T-018 atoms; it is not a new certificate architecture, no atom was re-optimised, and the gap to Trump's packing is still about 0.0604741. The same measurement shows where this route ends: the exact depth-one ceiling family at 191/50 caps every one-body point certificate at unit side 3.8288, and these certificates sit about 0.011 below it.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source

T-023

Claim: At q = 96/25, if four distinct unit squares have selected strict cores of side B = 9977/10000 containing, respectively, the four closed rational patches in arms.endpoint.footprint_union of the retained exp143 receipt, at most five further unit squares fit. Thus the specified four-owner branch contains no eleven-square packing. The five saved rational dots meet every admissible remaining core on the complete 361-direction net; audited containment and boundary arguments transfer this exact check to arbitrary physical angles. This is a conditional exclusion and changes no global bound for $s (11)$ .
Composition: Exact full-net coverage and frozen-data identities are machine-checked and replayed here. Universal owner footprint containment, extension to event boundaries, and strict-core selection for physical squares are audited analytic proof steps, so the composed theorem is V3. Exp145 adds an independent inclusion-exclusion computation of the same finite cover. The analytic owner and physical-angle transfer premises remain shared, and the composed theorem is C3; C4 would need two adversarial AI reviews by distinct reviewers and a human oversight record.
Next rung: Exp145 completes the independent finite-union check. Assemble a review packet for the shared analytic premises before revisiting confirmation. V4 for the whole theorem requires mechanizing the analytic transfer, and with C4 two adversarial AI reviews by distinct reviewers and a human oversight record.

Exp146 enlarged twelve wall-aware footprints, but exp147 transferred no additional tuple. Exp149 refuted the selected five-dot extension and exp150 showed its escape survives individual owner-core constraints. Exp151 also refuted a fixed sixth-site extension. Exp152 found a nonempty two-core site region, but exp153 then refuted the entire fixed-D-plus-one-site family with 188 exact support directions.

Next audit a short actual-core obstruction and determine whether it is a disjoint pair or only a higher-order obstruction before drawing weighted-mass conclusions. Wider owner-class coverage remains a new research result, not a rung change.
Significance: A substantive case exclusion at a central open size, demonstrating that forced occupied area can make a five-dot counting argument sufficient. It excludes one owner combination and does not move the global bound.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3 source

T-022

Claim: $s (11) \geq 38100$ *sqrt(8100042893309449)/899996306539 = 3.810025723614703, proved by an exact dilation-limit corollary of T-018's retained certificate. For every positive rational q with q^2 below 810004289330944900000000/809993351783841654158521, common scaling preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side qL. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.
Composition: Derived from a full replay of T-018's accepted five-condition source, scaling equivariance for Condition 1, invariance of Conditions 2 and 3, inverse-dilation preservation of Condition 5, and a separate sharpened containment lemma proved by an exact rational-square comparison. Rational density and upward embedding then supply the limit step. The mapped, non-superseded source-distinct review independently re-derived the theorem and attacked the endpoint, frozen-Condition-4, compactness, scale, and byte-trust boundaries. It is one adversarial AI review, by an agent of this project, and not a human oversight record, so the result is C3; nor is the review a method-distinct second derivation.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record beside the mapped 2026-09-06 source-distinct review. A second, method-distinct derivation would be shown beside the rung and would not improve the bound. This dilation limit is only the uniform fixed-B, single-core strict-containment supremum. Direction-specific cores, two-core endpoint averaging, or exact fixed-atom core shrinkage might extract more from the same data; the present argument does not establish $s (11)$ greater than the displayed bound.
Significance: The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0000257236147034 beyond T-018. The movement derives from an exact support identity and unused strictness margin in an existing certificate; it is not a new certificate architecture and leaves a gap of about 0.0670578664 to Trump's packing.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2 evidence 3 source

T-021

Claim: $s (20) \geq 97 / 20$ and $s (21) \geq 97 / 20$ , from this project's weighted fractional unavoidable-set certificate at container side 97/20 = 4.85. Both sizes held 24/5 = 4.80 from T-020 earlier the same week in this register's verified fields; the movement is +0.05 at each. The certificate does not reach $n = 19$ : its total mass is 19848723/1000000 = 19.848723, above nineteen, so Condition 2 gives it $n = 20$ and upward and T-020's 24/5 rung, retained beside it, still carries $n = 19$ .

The two sizes take no monotonicity step, for the reason T-020 gives: only Condition 2 mentions n among the five conditions, so an atom set of mass M certifies its side for every integer strictly above M. From $n = 22$ on the register already holds 5, so the certificate is true there and weaker.

Where the rung came from is worth the record. A pre-registered bisection of [24/5, 9977/2000] ran four rungs unattended. The uniform grid walled at this side -- its restricted optimum crossed twenty at round 31 of 32 and stood at 20.000439 with placements still violated -- while a second site set, the 24/5 certificate's own 2260 atoms scaled by 97/96 and unioned with the grids, converged below twenty at the same side. The certificate is the second construction's; the first construction's crossing is why H-062 asks for two.
Composition: Primary at both sizes: one certificate, one accepted verdict, the bound being the container side directly, with no monotonicity or composition step between the artifact and the claim. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository on the frozen bytes; that is C3, with the two methods shown beside the rung. As with T-020 there is no independent evaluator and no review record, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: The margin is 0.151277 below twenty, wider than T-020's 0.077380 was, so the side was not pushed to where the covering value stops it either. What is new is that the stopping side is now bracketed rather than guessed: 39/8 = 4.875 walls on both constructions and 97/20 certifies, so the m = 5 covering wall lies in an interval of width 0.025, and 979/200 = 4.895 and 997/200 = 4.985 wall as well. H-062 asked for 0.02 and the four decided rungs give 0.025, so the hypothesis is unresolved rather than accepted; the round is exp-061 and the remaining rung is the midpoint by the same rule.

One reading of the 4.985 wall does not survive inspection and is recorded here so nobody repeats it. Just below the ceiling the twenty-five axis-parallel B-squares of a 5 x 5 arrangement overlap only in strips of width 5B - L = 0.0035, and the restricted dual is checked at sites only, so a site set with no site in those strips makes twenty-five unit weights dual-feasible and the restricted optimum exactly 25.000000 whatever the covering value is. No uniform inset-1/2 grid at any count the lane could afford puts a site in every strip. The exactly round value this register has learned to distrust has, at m = 5, a mechanism.

The $n = 21$ half of the margin was taken on 2026-09-23: T-034 certifies 122/25 = 4.88 at $n = 21$ from an unseeded site set and supersedes this result there. What this result still held alone was $n = 20$ , whose bound stayed 97/20 because T-034's atoms are too heavy for it, until wand125's replayed rectangle certificate, $s (20) \geq 979 / 200$ (T-045), superseded it there on 2026-10-02. This result stays true as stated at both of its sizes.
Significance: Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at the same two sizes, one rung higher, and neither size changes character at 4.85. What it adds beyond a rung is a measurement rather than a theorem, and the measurement is registered as such: the m = 5 covering wall is bracketed to [97/20, 39/8] -- a certificate at 4.85, crossings above twenty on both constructions at 4.875 -- which is the first bracket this project has put around a covering wall, and it sits far below the method's ceiling of 4.9885 rather than at it. That bears on planning, not on the strength of this claim.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 20 n = 21 register evidence 1 evidence 2

T-086

Claim: s(k^2 - 2) = k for every integer k >= 2: chelokot's Lean theorem Records.NearSquare.squareMinusTwo_isMinimumSide, kernel-checked here. The lower half compensates each square of score at most one under Nagamochi's measure from other squares of the packing, so it does not use Nagamochi's Lemma 1, which is false (T-085); the upper half is the k x k grid with two squares removed.

This entry's scope is the cases this record holds, k = 3 to 18: $s (7) = 3$ , $s (14) = 4$ , $s (23) = 5$ , $s (34) = 6$ , $s (47) = 7$ , $s (62) = 8$ , $s (79) = 9$ , $s (98) = 10$ , $s (119) = 11$ , $s (142) = 12$ , $s (167) = 13$ , $s (194) = 14$ , $s (223) = 15$ , $s (254) = 16$ , $s (287) = 17$ and $s (322) = 18$ . Nagamochi 2005 stated the family first (T-007).

The development was built here with its pinned toolchain and Mathlib's cache, and its axioms are exactly propext, Classical.choice and Quot.sound. chelokot, square-packing-archive, September 2026.
Composition: Compound. The upper half is the grid, E-basic-grid-upper, replayed exactly here; the lower half is E-chelokot-square-minus-two-lean, the source's Lean development rebuilt here with its axiom receipt. A kernel check is machine evidence at rung 3: rung 5 also needs a human expert's review of the formalization, and the statement-fidelity reading retained here is the review lane's.
Next rung: Rung 4 needs two adversarial AI reviews by distinct reviewers and a human oversight record; rung 5 needs a named human expert's review of the formalization, and C5 two, with the open-review pointer the public repository already supports.
Significance: An infinite family of exact values, proved again after the published proof was found incomplete; every value was already held as proved. S3 by the anchor "a substantive case result or machine audit".
Novelty: previously-published Present in an identified source
Records: 16 cases register evidence 1 evidence 2 source

T-085

Claim: Lemma 1 of Nagamochi 2005 -- every square of side in (1, 1.01] inside [0,a] x [0,b] scores more than one against the paper's unavoidable set -- is false for every container with a > 3 and b > 2. Karakuş's square K_t at the lower-left corner, 0 < t <= 1/50, scores 19/20 + t + t^2/2 + t^3/2 < 1 after a concentric shrink, and chelokot's square of side 10001/10000 in [0,4]^2 scores exactly 25009470849041/25584000000000 < 1. The gap is in Section 5.5, Case 6, where Lemma 6 is applied to a contact configuration its hypothesis does not cover.

What holds in its place is Karakuş's strip measure, of total ab - (a + 1 - ceil a), which gives every such square more than one for a >= 2 and b >= 3 (his Proposition 5.1), and the rectangle bound it yields (T-083, T-084). Nagamochi's Theorem 1 at full strength, and so T-007, is unproved and not disproved.

Every printed quantity of both counterexamples was recomputed here in exact rational arithmetic. Hakan Karakuş, arXiv:2609.37410, 29 September 2026; chelokot, square-packing-archive, 4 September 2026, the earlier public record, which Karakuş's paper does not cite. wand125 first reported both, and the defect, to this project in jlevy/squares#295, 2 October 2026.
Next rung: Rung 4 on either axis needs two adversarial AI reviews by distinct reviewers and a retained human oversight record. Whether Lemma 1 holds for a = 3 or for a, b in [2, 3) is undecided; the family does not reach those containers.
Significance: A finding about the published record that removes the proof of the register's most widely cited lower bound at every n >= 10, and with it the verified status of 287 case floors. S3 by the anchor "a substantive case result or machine audit".
Novelty: previously-published Present in an identified source
Records: n = 10–324 register evidence 1

T-020

Claim: $s (19) \geq 24 / 5$ , $s (20) \geq 24 / 5$ and $s (21) \geq 24 / 5$ , from this project's weighted fractional unavoidable-set certificate at container side 24/5 = 4.80.

Two of the three cases previously held Nagamochi's 2005 general closed form in this register's independently verified lower fields -- min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1), which gives 1 + sqrt(13) = 4.6055... at $n = 20$ and 1 + sqrt(14) = 4.7416... at $n = 21$ -- and the third held the 459/100 = 4.59 this project certified earlier the same day as T-019. The movement is +0.21 at $n = 19$ , +0.194449 at $n = 20$ and +0.058343 at $n = 21$ , relative to those verified fields.

The September 2026 DS7 audit records stronger external reports below 4.80 separately, with missing source proofs and table discrepancies explicit; no claim of historical priority follows from the earlier source omissions.

The three sizes take no monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 946131/50000 certifies the side for every integer strictly above that mass, which is 19 and upward. From $n = 22$ on the register already holds 5, so the certificate is true there and weaker; these three are where it moves anything.

One rung is retained rather than a ladder. The side was reached in a single resumed column-generation run, not climbed to, so there is no weaker certificate at this size to compare it against.
Composition: Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. There is no monotonicity or composition step anywhere between the artifact and the claim -- the three sizes come out of Condition 2, which is the only condition that mentions n. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository; that is C3, with the two methods shown beside the rung. Unlike T-017 and T-019 no independent evaluator reviewed the scoring, and no review record is retained, so rung 4 waits on two adversarial AI reviews and a human oversight record.
Next rung: This result's 24/5 rung has total 18.922620, leaving 0.077380 below nineteen, and that margin is where a further rung had to be found. It is a wide margin by this register's standards -- $n = 17$ 's top rung has 0.066920 and $n = 12$ 's has 0.001040 -- which says the side was not pushed to where the covering value stops it, only to where the run was stopped. That is the honest reading of how this one was obtained.

The column generation was halted at round 9 with a restricted optimum of 18.916941, because four more rounds would have cost about 3.75 h to buy margin nothing needed. The side above it was never attempted.

Where the ceiling sits differs across the three sizes and it matters. A certificate for n cannot exist above ceil(sqrt(n)) * B, which for all three is 5B = 4.9885, so 0.1885 of structural runway remains. But at $n = 19$ the best known packing is 4.885618, and no certificate can exceed a side a packing achieves, so the real runway there is 0.085618. At $n = 20$ and $n = 21$ the best known packing is the trivial 5, so the ceiling binds first: the method could in principle close either to within 0.0115 of the upper bound and can never reach it. That asymmetry is the targeting instruction.

A run above 4.885618 that succeeded would contradict the retained $n = 19$ packing and is a refutation to look for rather than a rung to expect; a run between 4.80 and 4.9885 that succeeds moves $n = 20$ and $n = 21$ and stops at $n = 19$ . The quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate, and the cost was the exhaustive tier: the 24/5 rung's exact sweep took 5378 s at its own 2260-atom set, against 173 s for the interval route on the same bytes. The same evening the sweep was rewritten to decide in integers on the weights' common scale, in parallel over directions, and returns the same least covered mass in 38.7 s on the same box.

The margin was found and taken the following day: T-021 certifies 97/20 = 4.85 at $n = 20$ and $n = 21$ from a site set seeded with this certificate's own atoms, and supersedes this result at both sizes. What this result still holds alone is $n = 19$ , whose bound stays 24/5 because T-021's atoms are too heavy for it; the rung itself is retained at cases/n20_fractional_certificate/certificate-24-5.json and replays by name. On 2026-10-02 wand125's replayed rectangle certificate, $s (19) \geq 963 / 200$ (T-045), superseded it at $n = 19$ as well. This result stays true as stated at all three of its sizes.
Significance: Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value", and by parity with T-019, which has the same shape: one certificate moving three registered cases and displacing a published value.

This one displaces more. The movement is +0.21 at $n = 19$ against T-019's +0.0842, the largest single-case movement in the register; and at $n = 20$ and $n = 21$ it displaces the peer-reviewed 2005 closed form previously used in the verified register. The September 2026 source audit corrects the earlier claim that no size-specific bounds had been reported; it does not change these measured displacements of the verified fields.

Held below S5 for the reasons T-019 was: none of the three is a central open case, the weighted-resource lineage predates this project, the pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and no case changes character at the new value.

A calibration note for whoever scores the next one. This is the fourth result from one instrument in a day, and the S4 anchor is being carried here by the displacement rather than by the technique, which T-017 already banked. Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 19 n = 20 n = 21 register evidence 1 evidence 2

T-019

Claim: $s (17) \geq 459 / 100$ , and $s (18) \geq 459 / 100$ and $s (19) \geq 459 / 100$ , from this project's weighted fractional unavoidable-set certificate at container side 459/100 = 4.59.

This displaces Massaccesi's 22529/5000 = 4.5058 (T-015) and the two cases T-016 carried from it, all three of which had held that value until 2026-09-04. The movement past the published value is +0.0842 at each of the three sizes.

A stronger public candidate, anabologyco-maker's side 4.5705 (nine thousand one hundred forty-one two-thousandths) of 16 August 2026 (GitHub; not replayed here), was outside the record's search corpus when this was registered and was found on 2026-09-07; it lies 0.0195 below this result. Two rungs are retained below it, 451/100 and 229/50.

The September 2026 DS7 audit identifies a stronger reported n19 bound from Theorem 10 at k=4, approximately 4.6172815, with an unresolved theorem/table discrepancy and missing proof. T-019 does not improve that report; its public source displacement is supported at n17 and n18, while all three certified inequalities and their movements from the prior register remain valid.

The three sizes do not need a monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 423327/25000 certifies the side for every integer above it, which is 17 and upward; monotonicity would give the same thing and is not what is used.

It does not improve $n = 20$ , where Nagamochi's 1 + sqrt(13) = 4.6055... is larger -- the certificate does prove $s (20) \geq 459 / 100$ , but that is the weaker statement.
Composition: Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. Its mass 423327/25000 is below seventeen, and Condition 2 is the only condition that mentions n. The same certificate therefore applies directly at n17, n18 and n19; monotonicity would also prove the latter two inequalities but is unnecessary for this certificate.
Next rung: A second machine method decides it, the interval-certified decision, the first here that is not exact-algebraic. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

The bound itself: the retained certificate's total is 423327/25000 = 16.933080, leaving 0.066920 below seventeen, and that margin is what a further rung has to be found inside.

One figure from the ladder is worth carrying, because it contradicts the obvious reading. Margin is not monotone in the side. The 451/100 rung two below this one has total 16.593620 and margin 0.406380 -- an order of magnitude more headroom at a side 0.08 lower -- and the 229/50 rung between them has margin 0.034265, half of what 459/100 has at a higher side. So the covering value at a given side is set by the site set at least as much as by the side itself, and a better site set at a higher side can open the margin back up rather than close it. The $n = 12$ ladder shows the same shape in the same direction: margin 0.007175 at 197/50, then 0.029410 at the higher 79/20.

Massaccesi's own atoms are the reference point and not a limit: they give a covering value of 203/12 at 4.5058, tight by construction, which bounds nothing about what a different site set reaches at a larger side.

A further rung is therefore a search question, and the quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate.

One such question was asked and answered on 2026-09-04, at $n = 18$ and side 117/25 = 4.68, and it is recorded here as measurement rather than as a claim about tau*. Three site sets in succession -- 538, 578 and 618 orbits -- returned a restricted optimum of exactly 18.000000, the third of them after 157 row-generation rounds that grew the row set from 15888 to 27516 and cost 7056 s. Adding sites can only lower a restricted optimum, and across those three it did not move at all.

Two readings are open and the run did not separate them. Either the covering value at that side is at or above eighteen, in which case no certificate exists there; or the site sets are still short of it and the optimum is sitting on a degenerate vertex, which the collapse of pricing from 90 s to 1--3 s suggests -- a dual that sparse is what a degenerate basis looks like. Exactly round values are the known artefact signature in this pipeline, 18.0 having been one for $n = 17$ 's grid-31 optima, so the second reading cannot be dismissed.

What is not in doubt is the cost of settling it: two hours for the round that did not move. The run was stopped for that reason rather than for its answer, and the side is available to a later block with a cheaper decision path.

The method's own ceiling is not close here. A certificate for n cannot exist above ceil(sqrt(n)) * B, a wider container holding ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares whose masses Condition 5 forces past n; for $n = 17$ that is 5B = 4.9885, so 459/100 still has 0.3985 of runway. Unlike $n = 12$ , where the ceiling sits below the conjectured value and forecloses the case, nothing structural stops this one.

Where the next rung is worth spending on, though, is not $n = 17$ . Of the five conditions only Condition 2 mentions n, and the covering program behind the search does not contain n at all -- minimising total mass subject to every admissible B-square carrying mass at least 1 is a question about L, B and the net. So one atom set proves $s (n)$ >= L for every integer n above its mass, this certificate's 16.933080 reaching 17, 18 and 19 directly, and a larger n is strictly easier at the same side.

The headroom said the rest, when this was written. $n = 17$ had 0.0855 to its best known packing at 4.675530; $n = 18$ had 0.2329 to 4.822876 and $n = 19$ had 0.2956 to 4.885618, both still carrying this certificate's side in the verified register; and $n = 20$ had 0.3944 to the trivial 5, holding Nagamochi's closed-form 1 + sqrt(13) = 4.605551. A run at a side whose covering value lands between 17 and 18 would raise $n = 18$ and leave $n = 17$ where it is; one landing between 19 and 20 would displace a published closed form.

The resumed search ran the same day and is registered as T-020, at side 24/5 with retained certificate mass 18.922620, between eighteen and nineteen. It supersedes this result at $n = 19$ and moves $n = 20$ and $n = 21$ as well. What this result still holds alone is $n = 17$ and $n = 18$ : T-020's atoms are too heavy for either, which is the reach rule above read in the other direction. This result stays true as stated at all three of its sizes and keeps its rungs.

For $n = 17$ the reading is unchanged and the runway is 0.3985 to the ceiling and 0.0855 to the packing. For $n = 18$ , T-030 now certifies 4679/1000; T-029's 1871/400 remains as the previous rung. 117/25 remains the open probe this next_rung named.
Significance: Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value". One certificate moves three registered cases, and it replaces a published value rather than an inherited or closed-form one -- the only bound this project has displaced that someone else had put in print.

A stronger public candidate found afterwards, anabologyco-maker's 9141/2000 = 4.5705 of 16 August 2026 (GitHub; not replayed here), was outside the search corpus at scoring time, so the displacement at n17 and n18 is 0.0195 over that public report and 0.0842 over the one this record had adopted. At n19, the DS7 audit now retains a stronger source-reported bound than T-019; its 0.0842 movement refers to the prior register. The family-level score is unchanged.

Held below S5 because none of $n = 17$ , 18, 19 is a central open case, the movement is 0.0842 rather than a qualitative change, weighted resource counting predates this project, the recent pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and the case does not change character at the new value.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 17 n = 18 n = 19 register evidence 1 evidence 2

T-018

Claim: $s (11) \geq 381 / 100$ , by this project's weighted fractional unavoidable-set certificate at container side 381/100 = 3.81. This improves Stromquist's 2 + 4/sqrt(5) = 3.788854..., stated in 1984 (Memo III, p. 10) and published in 2003.

No intervening improvement was found by the recorded search, and $n = 11$ was then the smallest open case. The movement was +0.021146, and the case stayed open against Trump's 1979 packing at 3.877084: the interval narrowed from 0.088230 to 0.067084. Two rungs are retained below it -- 19/5, the value that first passed Stromquist, and the 189/50 calibration rung below him.
Composition: Primary: one certificate, one verifier, one accepted verdict. The bound is the certificate's container side directly, with no monotonicity or composition step. It raises no other case: every n > 11 already carries a larger verified bound.
Next rung: A second machine method decides it, the interval-certified decision, which is independent in method and, deciding coverage on the doubled net, never invokes the D4 reflection and so does not need Condition 1 at all. One adversarial AI review is retained, PR 78's, ported here as a dated record on 2026-09-05; V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. An outside mathematical review and an archival release would strengthen the priority record without changing this rung.

The bound itself: 3.81 converged with margin -- objective 10.8603 against eleven, rationalised to 434547/40000 = 10.863675 -- so the next rung is a search question rather than a limit of the method. Getting there took two attempts at the same side, and the difference between them is worth keeping. The first stopped with its row loop still finding violated placements (final least 0.9764) and was refused by the exact sweep at directions 55 to 63, least cell 199531/200000. The second ran the loop to convergence -- three row rounds, 25318 rows, final oracle least mass exactly 1 -- and was accepted at every direction. An objective below n proves nothing on its own; the loop's stopping condition is the thing to read.

Where it stops: 3.82 was then attacked from both sides and neither route closes. The covering LP was run on two independent site sets, and the two stop at exactly eleven for different reasons, which is worth separating because only one of them converged.

The grid-built set, 6637 sites and 14820 rows at the end, ran its row loop to convergence and finished at an objective of exactly 11.000000, descending to it from 11.6 over twelve rounds and never crossing below.

The set seeded from this certificate's own 1121 atoms, 24069 sites, has not converged and does not need to. Its objective has stood at 11.000000 through twenty-four row rounds and 30240 rows while the loop's least covered mass climbed from 0.8490 towards 1, reaching 0.9997 at the round the run was stopped -- so violated placements remained and the loop had not exhausted. Adding rows can only raise a restricted optimum, so the optimum for that site set is at least eleven whatever the loop does next, and no certificate exists on it. Reading the objective alone would have understated this: the loop's stopping state is what makes the conclusion independent of finishing.

A certificate needs mass strictly below eleven, so neither site set yields one. The rejection route does not close either, and it is much further from closing than the search's own progress figures suggest. H-061's object was built and decided exactly: the converged dual is 76 squares, 608 after the D4 images, with a raw total of exactly 11, and sqpack.fractional.ceiling checked all 1650944 vertices of their arrangement, 272244 of them in exact arithmetic. The maximum pointwise depth is 1925/1152 = 1.671007, so the feasible total -- the raw total scaled to make the family a packing -- is 1152/175 = 6.5829 against the eleven a ceiling needs.

The gap between that and the search's running estimate is the thing to carry forward. Column generation prices against a grid sample and reported a depth of 12/11 = 1.0909, which would have made the feasible total 121/12 = 10.08. The exact maximum is 53 per cent higher, because depth peaks at vertices of the arrangement that no grid samples. A ceiling must therefore be judged on the exact check and never on the sampled depth, which flatters it.

One thing 3.82 is not: the method's ceiling. A certificate for n cannot exist above ceil(sqrt(n)) * B, because a wider container holds ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares whose masses Condition 5 forces past n; for $n = 11$ that is 4B = 3.9908, leaving 0.1808 of runway above the retained 381/100. The wall at 3.82 is a covering wall, met far below where the method stops working.

If tau*(3.82) is exactly eleven then this is the one configuration where neither pre-registered route can close, since a certificate needs mass below n and a ceiling needs the scaled dual to reach n, and both fail by an infinitesimal at exactly n. That is a limit on the reach of the rejection route rather than a stalled search. It is recorded as measurement: two LP runs agreeing to six decimals is evidence about the restricted optima, not a proof that tau* equals eleven, and nothing here rules out a site set neither run found.
Significance: Scored against the rubric's own anchor for S5, "movement on a central open case". $s (11)$ is the smallest open case and a central one for this project; the recorded public search found no movement beyond Stromquist's value, stated in 1984 and published in 2003, and this displaces a bound from the refereed literature rather than an inherited or closed-form one.

What S5 does not claim: the weighted-resource lineage predates this project and the pure-atomic direction-net architecture follows Burns and Massaccesi, so what is new is an instance and the generator that found it; apparent novelty is not absolute priority; and the remaining gap to Trump's 1979 packing is 0.067, which this approach does not close. The calibration rung at 189/50, below Stromquist's bound, was run first by design and is retained.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1 evidence 2

T-017

Claim: $s (12) \geq 99 / 25$ , by this project's weighted fractional unavoidable-set certificate at container side 99/25 = 3.96.

This is the first lower bound specific to $n = 12$ in the retained corpus: the case previously held 2 + 4/sqrt(5) = 3.788854..., which is Stromquist's $s (11)$ bound inherited by monotonicity and says nothing about twelve squares in particular. The movement is +0.171146, and the case stays open against its conjectured optimum of 4, now by 0.04. Seven rungs are retained below it, 19/5 through 79/20.

It also separates the two cases for the first time. The retained side exceeds Trump's 1979 packing of eleven squares at 3.877084, so $s (12)$ > $s (11)$ strictly, by at least 0.082916. Monotonicity gives only $s (12)$ >= $s (11)$ ; the strict inequality did not follow from anything on record before, because $n = 12$ 's previous lower bound was $s (11)$ 's own 3.788854 and sat below Trump's packing.
Composition: Primary: one certificate, one verifier, one accepted verdict. The bound is the certificate's container side directly, with no monotonicity or composition step between the artifact and the claim.
Next rung: Two machine methods decide it. The independent re-derivation agrees but is exact-algebraic like the first; the second method is the interval-certified decision, which bounds coverage by branch and bound over boxes of centres with directed rounding: the two routes share the certificate and the closed-form conditions, and share no part of how Condition 5 is decided. The register shows the two methods beside the rung. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

The bound itself: the ladder 19/5, 77/20, 97/25, 39/10, 393/100, 197/50, 79/20, 99/25 was climbed once the separation oracle was corrected to score the cells the verifier decides (D-434), and every rung is retained.

The retained certificate's total is 149987/12500 = 11.998960, leaving 0.001040 below twelve -- the tightest margin of any rung in the record, and it survived on the sign of a rounding rather than by design. The next tightest, two rungs below at 197/50, is about seven times wider.

Rationalisation rounds each orbit up to a multiple of 1/scale, so its cost is at most atoms/scale; at 2097 atoms and scale 200,000 it took 0.005314, five times what remained. Raising the scale twentyfold would have cost about 0.00027 and would not have changed the atom count. That is a defaults question and it is BC-191's.

Margin does not shrink monotonically as the ladder climbs -- the 197/50 rung two below this one has 0.007175 and 79/20 has 0.029410, so a tight rung is evidence about that run and not about how much room is left.

This rung also cost the exhaustive tier, when it was retained: 2097 atoms is 1.8 times the next largest retained certificate, the 1184-atom $n = 17$ rung, and the exact sweep is quadratic in the atom count, so deciding it took 4866 s where the interval route took 110 s. The sweep was rewritten the same day to decide in integers on the weights' common scale, in parallel over directions, and returns the same verdict in well under a minute; BC-195 owns whether the tier can still afford what it holds.

What is left is bounded, though, and by the method rather than by the search. A certificate for n cannot exist above ceil(sqrt(n)) * B, since a container wider than that holds ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares, each carrying mass at least 1 by Condition 5, which forces the total past n and breaks Condition 2. Here that ceiling is 4B = 3.9908 on this certificate's own B, and it is what binds: 4/(1 + D) = 3.990816 is the supremum over every shrink the 181-direction net admits, and the grid packing's 4 sits above both. So the retained 99/25 has 0.0308 of runway left.

The consequence for this case is structural and worth stating exactly: the grid packing gives $s (12) \leq 4$ , the ceiling sits strictly below 4, and 4 is the conjectured value -- so no single certificate of this shape certifies $s (12) \geq 4$ , however fine the net or the site set. What the ceiling does not exclude is a proved family of certificates with sides tending to 4 and a limit argument on top of it; whether such a family exists is a question about the covering value, and an earlier version of this sentence overstated the ceiling into a method-wide impossibility (PR 78's adversarial review, F14). What remains is how close to 4 a certificate can get.
Significance: Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value". What is retained is not one bound but an eight-rung ladder -- 19/5, 77/20, 97/25, 39/10, 393/100, 197/50, 79/20, 99/25 -- by a generator that applies at every n and that has since produced the $n = 11$ result as well, so this is a bound family rather than a case result.

It is also the first bound located in the retained corpus that is proved about $n = 12$ rather than inherited from $n = 11$ : the frontier case body had recorded that nothing specific to $n = 12$ had been proved there.

Held below S5 because $n = 12$ is not a central open case, weighted resource counting predates this project, the recent pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, the second exact check shares a method family with the first, and the gap to the conjectured 4 is 0.04, which this approach does not close.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 12 register evidence 1 evidence 2

T-014

Claim: For s = 2 + sqrt(2)/2 and Goebel's labeled pose P0 in C = (R^2 x S^1)^5, P0 is an isolated point of Feas(s) -- closed unit squares in [0, s]^2, pairwise disjoint interiors -- equivalently there is no nonconstant continuous feasible path from P0 and no sequence of distinct feasible poses converging to it; hence the $n = 5$ optimum is rigid at fixed side in the catalogue's sense.

Proved exactly over Q(sqrt 2) in one intrinsic half-angle chart, from a complete exact accounting of all 400 elementary inequalities, by semialgebraic curve selection on the punctured feasible set and an induction on a putative arc's Taylor coefficients through order 2m that T-012's non-negative self-stress contradicts.
Composition: Compound, and the minimum is set by the part no machine checks. The exact local system -- the injective chart, all 400 base margins, the 20 active rows, the 128 strict conditions that cut out the neighbourhood, T-012's first-order cone and its non-negative self-stress with w . q < 0 -- is machine-confirmed here with the producer's code, devtools.assess_n5_rigidity, and independently rebuilt by the reviewer (machine-checked fragments, replayed here).

The two steps that close the argument, the cited Nash curve-selection lemma and the order-2m coefficient induction, are an audited proof rather than a computation, and no instrument decides isolation, so the whole claim declares V3.

Confirmation is C3: every exact quantity the proof consumes carries a certificate and a passing replay in this repository, and no cited part rests on anything weaker. BC-153's independent review of the proof is retained and mapped; it is one adversarial AI review, by a reviewer whose model is unstated, and not a human oversight record.
Next rung: V4 needs the closing steps mechanised, rather than only the quantities they consume, and with C4 a second adversarial AI review by a distinct reviewer and a human oversight record. BC-153's review, installed as a mapped review document on 2026-09-03, is the one retained; the round record carries it as an amendment as well. Rung 5 needs a proof-assistant formalization reviewed by human experts. Reaching the printed BCR page, or citing the Basu-Pollack-Roy plus Puiseux derivation the review writes out, would close the one cited hypothesis this rests on.
Significance: The first exact proof of fixed-side local rigidity of Goebel's $n = 5$ optimum -- a property Kingbird asserts with no argument, Goebel does not state, and Friedman does not annotate -- at the smallest case where tilting beats the grid. A case result, S3 rather than S4: the closing principle is the classical second-order sufficient optimality condition and the induction has the shape of Connelly-Whiteley 1996 Theorem 4.3.1, neither claimed as new; what is new is the exact accounting that reduces the local feasible set to twenty inequalities on an explicit neighbourhood, which is what lets that principle reach corner-on-line and corner-on-wall contacts.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 5 register evidence 1 evidence 2

T-005

Claim: Bentz 2010, Lemma 10 is false as printed -- the middle replacement point (1, 1.74) is refuted by an exact escape certificate, and the published page image carries the same transposed text -- and true under the corrected reading (1.74, 1), with all three corrected replacement covers certified exactly.
Next rung: Communicate the erratum to the author or journal; nothing mechanical remains on our side.
Significance: An erratum-level finding about the published record: real and citable, settled to the journal's own page image, changing no theorem.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 13 register evidence 1 evidence 2 source

T-003

Claim: The sixteen-point set's unavoidability ceiling lies in [4426213/1000000, 4427/1000): certification at the left endpoint, an exact escaping pose at the right, with the top strips' a + 2b <= 2*sqrt(2) hypothesis identifying the closing mechanism at 753/250 + sqrt(2), inside the bracket.
Composition: Compound: the lower side of the bracket carries T-001's two methods and the refutation side rests on the interval audit alone. Both are machine-replayed here, so the bracket is C3; a second method on one side does not change the rung. The equality reading is deliberately not this claim.
Next rung: think-iye2: the shared certifier generalized to a Q(sqrt 2) scalar certifies at the ceiling exactly and an exact escape family above it turns the bracket into an equality at C3.
Significance: Sharpens what T-001's construction can and cannot give; the exact equality claim at 753/250 + sqrt(2) remains analysis until the Q(sqrt 2) certificate lands.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 17 n = 18 register evidence 1 evidence 2

T-002

Claim: $s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001 (a packing of 18 unit squares contains a packing of 17).
Composition: Derived: the minimum over T-001 and the monotonicity step, which is one recorded line (any 18-packing contains a 17-packing) carried in the evidence limitations and the n-018 case body; both inputs sit at V3/C3, so the derived claim keeps the rungs.
Next rung: Rises with T-001; a bespoke $n = 18$ set stronger than the inherited bound would be a new result, not a rung change.
Significance: The same movement carried to $n = 18$ , above Nagamochi's 4.316625; the derivation adds nothing beyond T-001 but the case's frontier field moved for the first time since 2005.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 18 register evidence 1 evidence 2

T-001

Claim: Sixteen points make [0, 4426213/1000000]^2 unavoidable for open squares of side above one, so $s (17) \geq 4426213 / 1000000$ = 4.426213.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; the packet to review is the statement, hypotheses, proof narrative, code map and replays. Rung 5 needs a proof-assistant formalization reviewed by human experts. The stronger side at the exact ceiling is T-003's business.
Significance: The first verified lower-bound movement in the register since Nagamochi 2005 at these n, and the strongest theorem this repository has proved itself to date; a two-case improvement in a specialist literature, 0.019 below Green's reported but sourceless value.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 17 register evidence 1 evidence 2

T-013

Claim: Goebel's $n = 40$ packing is infinitesimally flexible -- seven verified independent first-order flexes turn the sixteen-square tilted block -- and every retained flex is refused at second order by a verified non-negative self-stress, exactly over Q(sqrt 2), so no first-order argument can establish rigidity here.
Next rung: Characterise the admissible cone past dimension 45 (the corner-touching branch obstruction, D-391) or establish local rigidity by another route; the rigidity property stays undetermined until then.
Significance: The first separation of infinitesimal from finite rigidity in this literature's cases: the sources assert rigidity without the distinction, and this refines rather than contradicts them.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 40 register evidence 1

T-012

Claim: Goebel's $n = 5$ optimal packing is not infinitesimally rigid but is second-order rigid at fixed side: the cone of infinitesimal motions is exactly the middle square's rotation about its own centre, and that one direction is refused at second order by a verified non-negative self-stress, all exactly over Q(sqrt 2).
Next rung: Local rigidity is discharged, not open: X-007's curve-selection argument was written out in full as X-012, checked against a complete exact accounting of the 400 local inequalities, independently reviewed by BC-153 and registered as T-014, which is where the $n = 5$ frontier property now rests. What remains here is the review record: two adversarial AI reviews and a human oversight record for rung 4, and for rung 5 a proof-assistant formalization reviewed by human experts.
Significance: A structural theorem at the smallest nontrivial case, proved exactly where the catalogue asserts 'Rigid.' with no definition or argument and the corpus contains none of the machinery for deciding it.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 5 register evidence 1

T-009

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.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. An exact-algebraic confirmation of the same witness would be shown beside the rung as a second machine method; the standing reported-value promotion question is the owner's evidence-contract decision, not a rung.
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

T-049

Claim: $s (12) \geq 15680 / 3951$ = 3.9686155..., by Evan Daniel's weighted point certificate, published on 25 August 2026 and first seen here on 27 September. It raised the verified bound by 851/98775, about 0.0086, over T-017.

The certificate is 1,736 D4-invariant rational points of total weight 11.9738036 < 12 such that every closed unit square in [0, 15680/3951]^2, at every angle, captures weight at least one, decided over the rational angle net 2 arctan(k/6000) with a per-bin shrink of 1/(cos d + sin d).

It was replayed here on 27 September 2026 by the source's Rust arrangement-sweep verifier at $N = 6000$ , least captured weight 10000056/10^7 at bin 0 as the source records, and by its exact Python re-check on 50 bins. It was also decided completely by this repository's native parent-core interval branch and bound over all 2,486 rows, which shares nothing with the source's sweep and gives the strict $s (12) > 15680 / 3951$ .

Evan Daniel, evand/square-packing (formerly square-packing-12), building on Sam Burns's and Gustavo Massaccesi's weighted exact-rational covering method. The source's CREDITS.md says the work was produced with an AI agent under human direction.
Composition: Primary at $n = 12$ : one certificate, no monotonicity. Two entries whose methods differ decide it, the source's exact arrangement sweep (exact-algebraic, replayed here) and this repository's directed-rounding interval coverage of every row (interval-certified); that is C3, with the two methods shown beside the rung. Both rest on the source's certificate and the same counting theorem, and the thin reader that maps the source's integer file onto parent-core rows has had no review reading of its own.
Next rung: A review reading of the native reader (devtools.verify_evand_angle_net_native) would close the one unread link in the second method's route. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; whether a mapped same-project review of another author's result counts toward them is the owner's decision. Rung 5 needs a proof-assistant formalization reviewed by human experts. The exact value remains open; the conjecture is $s (12) = 4$ .
Significance: Raised the verified lower bound at $n = 12$ by 0.0086 to within 0.0314 of the grid's 4, on two methods. The method is Burns's and Massaccesi's, as this repository's own $n = 12$ ladder is; the instance is new. A substantive case result, S3. It entered its source ten days before T-017 was scored, which bears on T-017's "first lower bound specific to $n = 12$ " (the plan's open question 3) but not on this score.
Novelty: previously-published Present in an identified source
Records: n = 12 register evidence 1 evidence 2 evidence 3 source 1 source 2 review 1 review 2

T-010

Claim: $s (11) \geq 2 + 4 / \sqrt{5}$ , by a source-distinct repair of Stromquist 2003's Figure 14 point set: the replacement G' = (79/100, 37/20) restores the complete Figure 13 localization, A-triple forcing, repaired unavoidability, and 3+9 capacity chain, certified exactly. The printed argument does not close (exp-016) and is not relied on.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second independent mechanism, the pose-space interval audit built for T-001 run at side 2 + 4/sqrt(5) over the repaired twelve-point set, would be shown beside the rung. Rung 5 needs a proof-assistant formalization reviewed by human experts.
Significance: Resolves the proof status of the subject's flagship case: the $n = 11$ lower bound was stated in 1979 and cited as proved since, its printed argument was falsified here, and this restores it with a source-distinct repair.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps
Records: n = 11 register evidence 1

T-016

Claim: $s (18) \geq 22529 / 5000$ and $s (19) \geq 22529 / 5000$ , by monotonicity from T-015 (a packing of n >= 17 unit squares contains a packing of 17). At $n = 19$ this replaces Nagamochi's 1 + sqrt(12) = 4.4641... since (22529/5000 - 1)^2 = 307265841/25000000 > 12; at $n = 20$ the certificate does not improve 1 + sqrt(13) = 4.6055... since that square is below 13, and nothing changes.
Composition: Derived: the minimum over T-015 and the monotonicity step, which is one recorded line carried in the evidence limitations and the n-018 and n-019 case bodies; T-015 sits at V3/C3, so the derived claim keeps the rungs.
Next rung: Rises with T-015; a bespoke $n = 18$ or $n = 19$ certificate stronger than the inherited bound would be a new result, not a rung change. Superseded as the verified lower bound on 2026-09-04 by T-019, which carries 459/100 to $n = 18$ and $n = 19$ by the same monotonicity step this result uses, from a larger base. This result stays true as stated and keeps its rungs.
Significance: The same movement carried to $n = 18$ (+0.079587 over T-002) and $n = 19$ (+0.0416984 over Nagamochi's 1 + sqrt(12), the first movement of that case's verified lower bound since 2005); the derivation adds one line beyond T-015 and inherits its source and credit.
Novelty: previously-published Present in an identified source
Records: n = 18 n = 19 register evidence 1 evidence 2

T-015

Claim: $s (17) \geq 22529 / 5000$ = 4.5058, by Massaccesi's 168-atom fractional unavoidable-set certificate on Burns's architecture: total mass 203/12 < 17 and mass at least 1 in every closed unit square of [0, 22529/5000]^2, reduced exactly to 181 rational directions and finitely many event cells.

It was replayed here by the source verifier and by an accumulation-independent repository instrument.

Gustavo Massaccesi, building on Sam Burns, in a blog post of 21 August 2026. Massaccesi marks the value "(?)".
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.

A method-distinct certificate would be shown beside the rung as a second machine method: a pose-space interval branch-and-bound over (theta, x, y) with the true unit square and weighted atoms, on the pattern of cases/green17/interval_audit.py (note that this certificate is tight, with every row minimum exactly 1 on open cells, so a naive box-splitting audit will not terminate at event lines and needs the semicontinuity argument built in); or, short of that, retain a from-scratch whole-check evaluator (the BC-149, BC-150 or BC-151 scratch evaluators) as a repository case with a replay so the reduction is no longer single-sourced on the record. Rung 5 needs a proof-assistant formalization of the eleven-lemma argument in the BC-150 packet, reviewed by human experts.

Superseded as the verified lower bound on 2026-09-04 by T-019, which certifies 459/100 at $n = 17$ with this project's denser certificate. This result stays true as stated and keeps its rungs; what changed is that the register no longer reads 22529/5000 as the strongest available at this size.
Significance: Raises the verified lower bound at $n = 17$ by 79587/1000000 = 0.079587 over this project's sixteen-point certificate T-001 and closes the gap to the reported record to 0.1697; Massaccesi's result, which this repository replayed, checked by an independent accumulation, and audited lemma by lemma in two reviews.
Novelty: previously-published Present in an identified source
Records: n = 17 register evidence 1 evidence 2

T-087

Claim: Bašić and Slivková 2018, Theorem 7 with Proposition 8: no more than B(x) unit squares fit in a square of side x, where B(x) counts the points of an equilateral-lattice piercing set, floor(x)(m + 2) plus floor((m + 2)/2) when frac(x) >= 1/2, with m = floor((2/sqrt 3)(x + 1 - 2 sqrt 2)). Just below 7 sqrt(3)/2 + 2 sqrt(2) - 1 it is 60, so $s (61) \geq 7$ sqrt(3)/2 + 2 sqrt(2) - 1, about 7.8906 (their Theorem 10); just below 5 sqrt(3)/2 + 2 sqrt(2) - 1 it is 36, so $s (37) \geq 5$ sqrt(3)/2 + 2 sqrt(2) - 1, about 6.1586, which the paper does not state.

Both are above Karakuş's general floor, T-083; at every other case the theorem is weaker than the verified floor already held. The proof uses nothing of Nagamochi 2005. It was read and re-derived here and is not machine-checked; its arithmetic is replayed exactly for every case by devtools/check_piercing_lower_bounds.py. Bašić and Slivková, Discrete Applied Mathematics 247 (2018).
Next rung: C2 and above need a replay of the geometry, not only the arithmetic: a machine check that the lattice of Theorem 7 pierces every unit square in the square of side x, or a formalization of the three-case reduction in Theorem 3's proof.
Significance: Registered as the verified lower bound at $n = 37$ and $n = 61$ , the only two cases where the piercing bound beat the floor its line of the register held; the merge of 3 October 2026 brought stronger replayed bounds to both, so it holds neither now. S3 by the anchor "a substantive case result or machine audit"; the score is the theorem's, not ours.
Novelty: previously-published Present in an identified source
Records: n = 37 n = 61 register evidence 1 source

T-008

Claim: $s (46) = 7$ : the lower half by T-004's audited unavoidable set, the upper half by the exact $7 \times 7$ grid packing of 46 squares.
Composition: Compound: lower = T-004 (exact-algebraic audit), upper = the grid bound replayed exactly by the basic-bounds checker. Under epistemics.md's rule for an equality whose upper half is an exactly replayed packing, the lower half sets the rung: it is machine-replayed here, so the equality is C3.
Next rung: Rises with T-004: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A method-independent second confirmation of the lower half would be shown beside the rung.
Significance: The first equality from the literature whose both halves are machine-confirmed here end-to-end.
Novelty: previously-published Present in an identified source
Records: n = 46 register evidence 1 evidence 2 evidence 3 source

T-006

Claim: $s (13) = 4$ (Bentz 2010, Theorem 9). Proved again, without cases, by Evan Daniel's weighted closed cover of [0,4]^2.

The second proof was kernel-checked in Lean 4 here on 30 September 2026: SquarePacking.s13_eq_4 : minSide 13 = 4, depending only on propext, Classical.choice and Quot.sound, built from the retained source with Mathlib's official cache. The cover's zero-margin property was also certified here by the source's zmx2 over every root of its D4 region.

The theorem is Bentz's. The case-free proof is Evan Daniel's, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as his CREDITS.md says.
Composition: Two independent proofs of one value. Bentz's is compound, with Sections 3.1-3.2's case analysis read rather than machine-checked. Evan Daniel's case-free proof is complete as it stands: the Lean kernel checked s13_eq_4 from the definitions, including every one of the 209 cover chunks (E-n013-evand-casefree-cover-lean-kernel). A kernel check is machine evidence at V3: V5 also needs a human expert's review of the formalization, and the retained review of s13_eq_4 was written by an AI lane. The confirmation rung comes from the machine-method replay E-n013-evand-casefree-cover-zmx2-replay (C3); whether a kernel check itself counts toward C is the owner's open question.
Next rung: One named human expert's review of the formalization restores V5; C5 needs two, with the open-review pointer, beside the rebuild already done here. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method on the same cover, for example a complete zeromargin.py sweep (1 h 42 min on 8 processes at the source) beside the zmx2 replay, would be shown beside the rung. Bentz's own route would reach C3 by machine-checking Sections 3.1-3.2's staged sets (typed on think-1o1f), which would also make it the second fully audited theorem of the paper.
Significance: A published exact value in the m^2 - 3 family; the score is the theorem's, not ours.
Novelty: previously-published Present in an identified source
Records: n = 13 register evidence 1 evidence 2 evidence 3 evidence 4 source 1 source 2 review

T-004

Claim: Bentz 2010, Theorem 8: the printed 45-point unavoidable-set argument for $s (46) \geq 7$ is correct as printed, machine-audited in full.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second independent mechanism, such as the pose-space interval audit generalized to Q(sqrt 2, sqrt 3) sides, would be shown beside the rung as a second machine method. Rung 5 needs a proof-assistant formalization reviewed by human experts.
Significance: As far as the archived corpus shows, the first machine verification of a published unavoidable-set proof in this literature; the theorem is Bentz's, the audit is the contribution.
Novelty: previously-published Present in an identified source
Records: n = 46 register evidence 1 evidence 2 source

T-007

Claim: For every integer $4 \leq N \leq 324$ , Nagamochi 2005, Theorem 2 gives $s (N)$ >= min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1).
Next rung: A checkable proof of Theorem 1, or of Theorem 2 by another route, restores a rung: for the m^2 - 2 family a measure on [0,m]^2 of total at most m^2 - 2 that gives every square of side in (1, 1.01] more than one, or a replay of chelokot's Lean compensation proof with its axiom receipt; for the general floor, a proof at full strength. The read that found the defect is what stands, and the transcribed values are still re-derived in the gate.
Significance: The workhorse lower bound until 2026-10-02, and no less important for being unproved: 95 of the hundred case records cite this result, and since that date it supplies the operative verified lower bound in 0 of them. It supplied 88 until 2026-08-31, when this project's certificates began displacing it; replayed external certificates displaced 18 more case values on 2026-09-22, one more, $n = 32$ , on 2026-09-27, and another, $n = 45$ , on 2026-09-29, leaving 63 at n <= 100 and 287 in all when its proof was found incomplete. Replayed certificates recorded in parallel and merged on 2026-10-03 have since raised 51 of those floors, 38 of them at n <= 100, above the ones that replaced it. Its values remain recorded as reported bounds.
Novelty: previously-published Present in an identified source
Records: n = 4–324 register evidence 1 source

T-011

Claim: Trump's 1979 packing is exactly valid: 11 unit squares in a square of side the published degree-8 algebraic number 3.877083590022814..., with 14 of 55 pairs in exact zero-separation contact and 20 corner coordinates exactly on the boundary, so $s (11)$ <= that side.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second independent exact verification of the same degree-8 witness would be shown beside the rung (the retained rational control proves a slightly weaker side and does not confirm this claim). Rung 5 needs a proof-assistant formalization reviewed by human experts.
Significance: Confirms the record witness exactly, deciding the 14 zero-gap contacts no finite numerical evaluation can: real and citable, changing no theorem.
Novelty: previously-published Present in an identified source
Records: n = 11 register evidence 1

Every Result

Verification Ladders