T-057: $s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record

V3 C3 upper bound confirmed superseded by T-098

2026-09-16 published · de Winter · $n = 211$

$s (211) \leq 14.99796070496771500150$ < 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.

Significance, composition and next rung

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.
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.
Novelty: previously-published Present in an identified source

The case

Case record

14.5414.998

141516

14.52615.526

\sqrt{n}

\sqrt{n} + 1

Proven

$14.54457$ $\leq s (211) \leq$ $14.997961$

numerical

Citation record n-211

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperde Winter & Daniel, GitHub (confirmed T-098)

Open

optimality
exact value

The case record

LowerUpper

Verified14.54457190…14.99796070…

Reported14.56465996…14.99796070…

Gap0.45338879…

Results on the case

5 results in the register on $n = 211$ , oldest first, each with what it established and how it stands now.

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 correction confirmed
Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-16 published T-057 this result
$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record
V3 C3 upper bound confirmed superseded by T-098
de Winter · de Winter n211 2026-09-16 · packet · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
V3 C3 upper bound confirmed
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

Links

On this site: Case record, $n = 211$ · Frontier row, $n = 211$ · T-057 in the results table

On GitHub, at main

Register: T-057 in results.yaml, line 4996
Evidence: E-n211-de-winter-report · E-n211-de-winter-exact-replay · E-n211-de-winter-interval-replay
Proofs and certificates: certificate n-211-rational.yaml.gz · certificate n-211.yaml
Sources: de Winter n211 2026-09-16 (its own site, retained copy)
Source packet: resources/web/de-winter-square-packing-211-2026-09-16/README.md
Artifacts: 14 artifacts and controls
resources/web/de-winter-square-packing-211-2026-09-16/README.md resources/web/de-winter-square-packing-211-2026-09-16/acquisition/sources.json resources/web/de-winter-square-packing-211-2026-09-16/receipts/certification.json resources/web/de-winter-square-packing-211-2026-09-16/receipts/interval-certification.json witnesses/de-winter-2026/n-211-rational.yaml.gz devtools/upper_bound_packets.py devtools/check_rational_witness_independent.py devtools/upper_bound_intervals.py docs/project/reviews/review-2026-09-29-issue-227-upper-bound-packings.md docs/project/reviews/review-2026-09-30-interval-route-t056-t057.md resources/web/franciscouzo-square-packing-2026-09-27/receipts/negative-controls.json resources/web/de-winter-square-packing-211-2026-09-16/receipts/interval-negative-controls.json tests/test_upper_bound_packets.py tests/test_upper_bound_intervals.py
Case file: frontier/n-211.md (verified lower, verified upper, reported lower, reported upper)

T-057: s(211)≤14.99796070496771500150<15s(211)≤14.99796070496771500150<15, the first packing of 211 squares below the grid on record

The case

Results on the case

Links

T-057: $s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record