T-015: $s (17) \geq 22529 / 5000 = 4.5058$

V3 C3 lower bound confirmed superseded by T-093

2026-08-21 published · Massaccesi after Burns · $n = 17$

$s (17) \geq 22529 / 5000$ = 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 "(?)".

Significance, composition and next rung

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.
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Superseded as the verified lower bound on 2026-09-04 by T-019, which certifies 459/100 at $n = 17$ with this project's denser certificate.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.
Novelty: previously-published Present in an identified source

The case

Case record

4.664.676

456

4.1235.123

\sqrt{n}

\sqrt{n} + 1

Proven

$4.66044$ $\leq s (17) \leq$ $4.675531$

exact

Citation record n-017

lowerGuzhou0806 after Kleddamag et al. 2026, GitHub (confirmed T-093)

upperBidwell 1998, Squares in Squares (confirmed T-065)

Open

optimality

The case record

LowerUpper

Verified

\frac{18641771}{4000000}

4.67553009…

Reported

\frac{18641771}{4000000}

4.67553009…

Gap0.01508734…

Results on the case

17 results in the register on $n = 17$ , 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 on this case, superseded by T-093
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-015 this result
$s (17) \geq 22529 / 5000 = 4.5058$
V3 C3 lower bound confirmed superseded by T-093
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-31 established T-001
$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set
V3 C3 lower bound confirmed superseded by T-093
Levy after Bentz · register
2026-08-31 established T-003
The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$
V3 C3 method limit confirmed
Levy after Bentz · register
2026-09-04 established T-019
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
V3 C3 lower bound confirmed superseded by T-093
Levy after Burns, Massaccesi · 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
2026-09-20 published T-032
$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , and beneath it Mira's $s (17) \geq 4613 / 1000$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806, Mira after Levy, Burns, Massaccesi · n17 weighted certificates 2026-09-20 · packet · register
2026-09-21 published T-038
$s (17) > 461300 / 99853 = 4.6197910929 \dots$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-21 published T-065
$s (17) \leq 4.6755300936045509 \dots$ , Bidwell's packing certified exactly
V3 C3 upper bound confirmed
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-25 published T-039
$s (17) > 231001 / 50000 = 4.62002$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R052 · packet · source · review · register
2026-09-26 published T-040
$s (17) > 232001 / 50000 = 4.64002$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 4.640020 · packet · source · review · register
2026-09-27 published T-041
$s (17) > 466001 / 100000 = 4.66001$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register
2026-09-28 published T-042
$s (17) > 233009 / 50000 = 4.66018$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R067 · packet · source · review · register
2026-09-28 published T-043
$s (17) > 116511 / 25000 = 4.66044$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R068 · packet · source · review · register
2026-09-29 published T-058
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
V3 C3 method limit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · 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 on this case, superseded by T-093
Karakuş · Karakuş 2026 · source · register
2026-09-30 published T-093
$s (17) > 18641771 / 4000000 = 4.66044275$
V3 C3 lower bound confirmed
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R071 · packet · source · review · register

Links

On this site: Case record, $n = 17$ · Frontier row, $n = 17$ · T-015 in the results table

On GitHub, at main

Register: T-015 in results.yaml, line 763
Evidence: E-n017-massaccesi-source-replay · E-n017-massaccesi-h052-agreement
Proofs and certificates: certificate massaccesi-verify-n17-lower-bound-4_5058.py · certificate exp-059-h-052-n17-fresh-successor-completion.json
Sources: Burns–Massaccesi n17 (its own site, retained copy)
Source packet: resources/web/n17-lower-bounds-2026/README.md
Artifacts: 10 artifacts and controls
resources/web/n17-lower-bounds-2026/massaccesi-verify-n17-lower-bound-4_5058.py resources/web/n17-lower-bounds-2026/massaccesi-lower-bound-4_5058.html resources/web/n17-lower-bounds-2026/burns-n17-lower-bound-4.4811.md cases/n17_weighted_certificate/independent.py cases/n17_weighted_certificate/source_faithful.py cases/n17_weighted_certificate_successor/run.py campaign/series/series-000-smoke-and-calibration/results/exp-059-h-052-n17-fresh-successor-completion.json campaign/hypotheses/H-052-n17-independent-certificate-agreement.md tests/test_n17_weighted_certificate.py tests/test_n17_weighted_certificate_successor.py
Case file: frontier/n-017.md (verified lower, verified upper, reported lower, reported upper)

T-015: s(17)≥22529/5000=4.5058s(17)≥22529/5000=4.5058

The case

Results on the case

Links

T-015: $s (17) \geq 22529 / 5000 = 4.5058$