Every Result

Verification Ladders

Each result has an identifier, a claim, and three ratings, defined in epistemics.md: S, its significance; V, the strongest verification its original evidence supports; and C, what this repository has confirmed independently. For results by others, this repository assigns V and C under its review policy. Each ladder links to its rubric; each rung has a tooltip with its full meaning.

Level

Significance How significant is the result?

Verification How was it originally verified?

Confirmation How has it been confirmed?

Level 5

Moves a central open case, or broad adoption

V5Formal, expert-reviewed

C5Formal, replayed here, open, two experts

Level 4

Reusable technique, bound family, or settled value

V4Mechanized, AI-reviewed, human-overseen

C4Mechanized confirmation, AI-reviewed, overseen

Level 3

A substantive case result or machine audit

V3Checkable; review record pending

C3Machine-replayed; review record pending

Level 2

A citable detail that changes no theorem

V2Proof asserted, not recoverable

C2Replayed, no machine certificate

Level 1

Bookkeeping or a routine consequence

V1Numerically checked

C1Read

Level 0

V0Claimed or recorded only

C0Recorded

Evidence and Exactness

The ladders grade a result. Its evidence carries an assurance label: reported, a named source’s claim not checked here; numerically checked, a finite-precision calculation with its precision, rounding and tolerance recorded; or verified, an exact check, rigorous interval certificate or complete proof covering the claim and its preconditions. A verified packing proves an upper bound; optimality also needs a matching verified lower bound.

Finite precision is not enough where squares touch exactly. A tolerance that accepts a true zero-gap contact also accepts a small overlap, so a contact-heavy packing requires exact algebraic signs or outward-rounded intervals (why). Schadt’s $n = 29$ packing passes its 300-digit numerical check, while the verified interval witness proves a slightly weaker side. Trump’s $n = 11$ packing is verified exactly over a degree-eight number field, including fourteen zero-gap contacts.

These checks also audit published work: T-004 and T-008 check Bentz’s 2010 Theorem 8, including both halves of $s (46) = 7$ , and T-011 checks Trump’s 1979 packing for eleven squares. The theorem remains the source’s; this repository adds an exact machine check.

Results and Credits

A result’s kind is lower bound, upper bound, optimality (settling an exact value), or a kind that bounds no case, such as rigidity, case exclusion or simplification (a simpler proof of a value another result establishes).

Credits name the source’s authors; this project’s results are credited to Joshua Levy as Levy. X after Y means X’s result rests directly on Y’s proof, method or tool.

Status and Supersession

A result’s status is recorded (registered from its source without review or replay here), reviewed (its argument read here), confirmed (its certificate replayed successfully) or incomplete (a known defect remains open). Of the 146 results, 132 are confirmed; the rest are 11 recorded, 2 reviewed and 1 incomplete. Status follows the confirmation rung and is never set by hand; the status policy defines it. In analysis marks a review or replay under way here; waiting on marks a request with the source or another party.

A bound is superseded when no current case bound rests on it and its cases match or beat it. A better bound not yet adopted by its case records remains unmarked. For other kinds, superseded means the register records a later result implying the whole claim; superseded in part means it implies only part, so the original remains current. The table names the most recent replacement, with “and others” when needed; the full record lists every replacement.

The overview highlights recent major results.

All Results

Significance Verification Confirmation Kind Status Hide supersededSource Case n Max age days146 results

Date	Result	n	Credit	Rungs	Status	ID
2026-10-10 published	Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists	70, 102, 103, 123, 129, 146, 153, 236, 258, 263, 269, 292, 295, 302, 303	Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy	V3 C3upper bound	confirmed	T-146
2026-10-10 published	$s (40) > 1340000 / 199529 = 6.7158157 \dots$ , by a continuous-pose interval kernel	40	Guzhou0806 after wand125, Tokoharu, Levy, Stromquist, Burns, Massaccesi	V3 C3lower bound	confirmed	T-145
2026-10-10 published	Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported	122–126	wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V0 C0lower bound	recorded	T-144
2026-10-10 published	Mixed rectangle-measure lower bound $s (30) \geq 2357 / 400$ reported, on a 2,073-direction declared net	30	wand125 after Tokoharu and Levy	V0 C0lower bound	recorded	T-143
2026-10-10 published	Mixed rectangle-measure lower bound $s (28) \geq 2297 / 400$ reported, on a 2,073-direction declared net	28	wand125 after Tokoharu and Levy	V0 C0lower bound	recorded	T-142
2026-10-10 published	Exact rational certificates at 132 and 308 from Evan Daniel's third record hunt	132, 308	Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy	V3 C3upper bound	confirmed	T-141
2026-10-10 published	Six more exact rational certificates from Francisco Couzo	132, 175, 209, 237, 270, 305	Couzo after Chaoweeraprasit, Mishapolk, Xu, Daniel, Ellsworth, Levy	V3 C3upper bound	confirmed	T-140
2026-10-10 published	An exact rational refinement of Ryan Xu's packing of 70 squares	70	Deleeuw after Xu, Levy	V3 C3upper bound	confirmed	T-137
2026-10-10 published	A packing of 308 squares below the grid, from Kevin Fang	308	Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead	V3 C3upper bound	confirmed	T-135
2026-10-10 published	$s (40) > 335427 / 50000 = 6.70854$ , by a clipped-corner transfer on a 401-direction net	40	Guzhou0806 after wand125, Tokoharu, Levy, Stromquist, Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-145	T-133
2026-10-10 published	Mixed rectangle-measure lower bound $s (29) \geq 291 / 50$ reported, on a 2,073-direction declared net	29	wand125 after Tokoharu and Levy	V0 C0lower bound	recorded	T-132
2026-10-09 published	A more precise certificate of Ryan Xu's 126-square arrangement, from Evan Daniel	126	Daniel after Xu, Chaoweeraprasit	V3 C3upper bound	confirmed	T-139
2026-10-09 published	Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered	84, 86, 103, 105, 108, 127, 131, 132, 175, 180, 258, 267, 270, 302, 303, 306	Mishapolk after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy	V3 C3upper bound	confirmed	T-138
2026-10-09 published	Seventeen exact rational packings from SQUISH's third request	131, 153, 154, 207, 209, 232, 236, 237, 259, 263, 269, 270, 292, 302, 303, 305, 307	Chaoweeraprasit after Couzo, Xu, Goebel, Schadt, Ellsworth, Hajba, Levy	V3 C3upper bound	confirmed	T-136
2026-10-09 published	Six exact rational certificates from Francisco Couzo	132, 237, 263, 267, 270, 303	Couzo after Mishapolk, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy	V3 C3upper bound	confirmed	T-134
2026-10-09 published	$s (132) \leq 11.987099332245063 \dots$ from Evan Daniel's record hunt	132	Daniel after Couzo, Chaoweeraprasit, Levy	V3 C3upper bound	confirmed	T-131
2026-10-08 published	Five follow-up rational refinements from Francisco Couzo	84, 86, 105, 175, 270	Couzo after Xu, Daniel, Ellsworth, Levy	V3 C3upper bound	confirmed	T-130
2026-10-08 published	Eight complete rational refinements from Francisco Couzo	105, 108, 127, 131, 155, 180, 228, 306	Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy	V3 C3upper bound	confirmed	T-128
2026-10-08 published	Fourteen rational refinements, seventeen complete source cases	88, 108, 123, 129, 130, 153, 154, 179, 180, 199, 207–209, 236–239	Gupta after Chaoweeraprasit, Daniel	V3 C3upper bound	confirmed	T-127
2026-10-08 published	Undilated n51 construction over Q(sqrt2)	51	ry-xu	V3 C3upper bound	confirmed	T-126
2026-10-08 published	Complete rational construction reports at 25 counts	51, 70, 84, 86, 88, 102, 103, 105, 108, 123, 126, 127, 129–131, 146, 153, 175, 179, 236, 258, 261, 263, 267, 295	ry-xu	V3 C3upper bound	confirmed	T-125
2026-10-07 published	Three dated Daniel certificate reports with historical source custody	105, 130, 292	Daniel after Couzo, Levy	V0 C0upper bound	recorded superseded by T-127 and others	T-129
2026-10-07 published	Reported non-strict local minima for 178 source configurations	1–4, 6–9, 11–16, 20–25, 28, 30–36, 42–49, 56–64, 72–81, 90–100, 111–121, 133–144, 157–169, 183–196, 212–225, 242–256, 274–289, 308–324	Daniel after Couzo	V0 C0restricted optimality	recorded	T-124
2026-10-07 published	Reported `s(177) ≤ alpha_177` with a degree-32 exact side form	177	Daniel after Couzo	V0 C0upper bound	recorded	T-123
2026-10-07 published	Reported `s(152) ≤ alpha_152` with a degree-40 exact side form	152	Daniel after Couzo	V0 C0upper bound	recorded	T-122
2026-10-07 published	Reported `s(106) ≤ alpha_106` with a degree-32 exact side form	106	Daniel after Couzo	V0 C0upper bound	recorded	T-121
2026-10-07 published	Reported `s(102) ≤ alpha_102` with a degree-8 exact side form	102	Daniel after Couzo	V0 C0upper bound	recorded superseded by T-125	T-120
2026-10-07 published	Three exact feasible upper bounds at n=266,270,272 from new source arrangements	266, 270, 272	Daniel after Levy, Ellsworth, Couzo, Stead	V3 C3upper bound	confirmed	T-119
2026-10-07 published	Exact rational ceiling refinements at $n = 68$	68	Rehwaldt after Couzo and earlier contributors	V3 C3upper bound	confirmed	T-118
2026-10-07 published	Exact rational ceiling refinements at $n = 105$ , 292	105, 292	Rehwaldt after Couzo and earlier contributors	V3 C3upper bound	confirmed	T-117
2026-10-07 published	$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings	88, 108, 179, 180, 199, 207, 236, 263, 302	Chaoweeraprasit after Ellsworth, Couzo and SQUISH	V3 C3upper bound	confirmed	T-116
2026-10-07 published	$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings	123, 126, 129, 154, 155, 179, 208, 237–239, 258, 263	Chaoweeraprasit after Couzo and Ellsworth	V3 C3upper bound	confirmed	T-115
2026-10-07 published	$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing	153	Chaoweeraprasit after Ellsworth	V3 C3upper bound	confirmed superseded by T-127	T-114
2026-10-07 published	$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings	108, 126, 129, 130, 154, 155, 180, 209, 238, 303	Chaoweeraprasit after Ellsworth	V3 C3upper bound	confirmed	T-113
2026-10-06 established	Trump's packing is the only optimal packing of eleven squares, up to symmetry	11	Levy after Ahmed	V3 C2uniqueness	confirmed	T-112
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	T-111
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	T-110
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	T-109
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	T-108
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	T-107
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	T-106
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	T-105
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	T-104
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	T-103
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	T-102
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	T-100
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	T-099
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	T-101
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	T-098
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	T-097
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	T-096
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	T-095
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	T-094
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	T-091
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	T-090
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-127 and others	T-092
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	T-082
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	T-081
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	T-080
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	T-079
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	T-078
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-104 and others	T-077
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	T-076
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	T-075
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	T-073
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	T-072
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-094 and others	T-071
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	T-074
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	T-069
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	T-068
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	T-067
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	T-066
2026-09-30 published	$s (17) > 18641771 / 4000000 = 4.66044275$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed	T-093
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	T-064
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	T-084
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	T-083
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	T-061
2026-09-29 published	Trump's eleven-square packing is globally optimal	11	Ahmed after Levy, Kleddamag	V3 C3optimality	confirmed	T-060
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	T-059
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	T-058
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-111 and others	T-070
2026-09-28 published	$s (61) = 8$ , by monotonicity from T-062	61	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed	T-063
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	T-062
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	T-055
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	T-054
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	T-048
2026-09-28 published	$s (17) > 116511 / 25000 = 4.66044$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093	T-043
2026-09-28 published	$s (17) > 233009 / 50000 = 4.66018$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093	T-042
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 superseded by T-127 and others	T-056
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	T-053
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	T-052
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-145 and others	T-046
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-145 and others	T-045
2026-09-27 published	$s (17) > 466001 / 100000 = 4.66001$	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3lower bound	confirmed superseded by T-093	T-041
2026-09-26 published	$s (32) = 6$	32	Daniel after Burns, Massaccesi	V3 C3optimality	confirmed	T-051
2026-09-26 published	$s (17) > 232001 / 50000 = 4.64002$	17	Kleddamag after Levy, Mira, Guzhou0806	V3 C3lower bound	confirmed superseded by T-093	T-040
2026-09-25 published	$s (17) > 231001 / 50000 = 4.62002$	17	Guzhou0806 after Kleddamag, Mira, Levy	V3 C3lower bound	confirmed superseded by T-093	T-039
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	T-089
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	T-088
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-112 and others	T-036
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	T-035
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	T-050
2026-09-23 established	$s (21) \geq 122 / 25 = 4.88$	21	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-052	T-034
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-109 and others	T-047
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-145 and others	T-044
2026-09-22 published	$s (11) > 31 / 8 = 3.875$	11	Kleddamag after Levy, Guzhou0806, Mira	V3 C3lower bound	confirmed superseded by T-060	T-037
2026-09-22 established	$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$	11	Levy	V3 C3lower bound	confirmed superseded by T-060	T-033
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	T-065
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	T-038
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	T-032
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	T-031
2026-09-19 established	$s (18) \geq 4679 / 1000 = 4.679$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102	T-030
2026-09-19 established	$s (18) \geq 1871 / 400 = 4.6775$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102	T-029
2026-09-19 established	$s (18) \geq 187 / 40 = 4.675$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102	T-028
2026-09-18 established	$s (18) \geq 467 / 100 = 4.67$	18	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-102	T-027
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	T-057
2026-09-09 established	$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$	11	Levy	V3 C3lower bound	confirmed superseded by T-060	T-026
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	T-025
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	T-024
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	T-023
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	T-022
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-104 and others	T-021
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	T-086
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	T-085
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-104 and others	T-020
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-103 and others	T-019
2026-09-04 established	$s (11) \geq 381 / 100 = 3.81$	11	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-060	T-018
2026-09-04 established	$s (12) \geq 99 / 25 = 3.96$	12	Levy after Burns, Massaccesi	V3 C3lower bound	confirmed superseded by T-095	T-017
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	T-014
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	T-005
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	T-003
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	T-002
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	T-001
2026-08-30 established	Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order	40	Levy	V3 C3rigidity	confirmed	T-013
2026-08-30 established	Goebel's $n = 5$ packing is second-order rigid at fixed side	5	Levy	V3 C3rigidity	confirmed	T-012
2026-08-29 established	$s (29) \leq 5.933833 \dots$ , by a Krawczyk interval certificate	29	Levy	V3 C3upper bound	confirmed	T-009
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	T-049
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	T-010
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-103 and others	T-016
2026-08-21 published	$s (17) \geq 22529 / 5000 = 4.5058$	17	Massaccesi after Burns	V3 C3lower bound	confirmed superseded by T-093	T-015
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-069 and others	T-087
2010-09-13 published	$s (46) = 7$	46	Bentz	V3 C3optimality	confirmed	T-008
2010-09-13 published	$s (13) = 4$	13	Bentz; Daniel after Burns, Massaccesi	V3 C3optimality	confirmed	T-006
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	T-004
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	T-007
1979 published	Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$	11	Trump	V3 C3upper bound	confirmed	T-011

Significance

Verification V0 V1 V2 V3 V4 V5 Confirmation C0 C1 C2 C3 C4 C5

What each rung means