n = 6 provedO=
Proven
- optimal
- exact
Citation record n-006
lowerKearney & Shiu 2002, Electron. J. Combin. 9, #R14
Bounds
- Found by
- Michael Kearney, Peter Shiu 2001
- Construction
- hand
- Source
- [Kingbird]
- Evidence
E-kingbird-upper-register
3
- Proved by
- Michael Kearney, Peter Shiu 2002
- Kind
- unavoidable points
- Note
- Kearney and Shiu use two seven-point unavoidable lattices followed by geometric cases. The broad point-method classification does not mean a pure five-dot proof; Memo I already gives a geometric-helper proof in 1984.
- Source
- [Kearney–Shiu 2002]
- Evidence
E-n006-kearney-shiu-proof
0
Solved: the verified bounds meet.
Results in the register
T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
for
T-058 V3 C3 wand125 after Tokoharu, Daniel · 2026-09-29 · 100 cases
Rectangle-certificate ceiling
α·UB(n)proved for ..100;B·UB(n)on 64 grid rows
upper: replayed here; lower: external proof (not read here)
—
not rigid, numerically checked, numerical multiprecision
Evidence: E-translation-escape-not-rigid
Scope
Square 3 of the retained witness (witness id 4) translates 1 along (0, 1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 3 of its 6 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.
- Priority: s(6) = 3 (Walter Stromquist (1984, unpublished), Trevor Green (2000, unpublished))
4 evidence entries
E-kingbird-upper-register, E-basic-grid-upper, E-nagamochi-lower, E-n006-kearney-shiu-proof
- [Kingbird] record catalogue
- [Kearney–Shiu 2002] lower bound proof
- [Stromquist Memo I] lower bound proof
- [Friedman DS7] survey
— solved
. Kearney and Shiu (2002) gave the first located published proof. Stromquist’s archived Memo I contains an earlier proof, dated September 11, 1984.
The packing
Found by Michael Kearney and Peter Shiu in 2001, via a hand construction.
The lower bound
Kearney and Shiu’s §3 uses two seven-point unavoidable sets, related by a quarter turn and sharing the center. They separate whether the center is covered, then use geometric compatibility and segment-intersection bounds to exclude both cases.
Stromquist’s Memo I also needs a geometric helper: it excludes adjacent isolated dots, then forces one square to consume four of eight final dots. The memo review retains an independently checked finite allocation control and a precise obstruction to using only five unweighted dots.
Verification Code
The programs behind this case’s verified bounds, by their evidence.
The code column says how the code that ran stands to the code its producer used.
VERIFIERS.md says what each program is and whose it is.
| bound | evidence | run | code | programs |
|---|---|---|---|---|
| verified lower | E-n006-kearney-shiu-proof |
a published proof | no code | no verification code |
| verified upper | E-basic-grid-upper |
replayed here | independent | V-check-basic-bounds (first-party) |