T-012: Goebel's $n = 5$ packing is second-order rigid at fixed side

V3 C3 rigidity confirmed

2026-08-30 established · Levy · $n = 5$

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).

Significance, composition and next rung

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.
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.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The case

Case record

2.707

234

2.2363.236

\sqrt{n}

\sqrt{n} + 1

Proven

$s (5) =$ $2.707107$

optimal
exact
rigid

Citation record n-005

lowerGöbel 1979, Math. Centre Tracts 106

upperGöbel 1979, Squares in Squares

The case record

LowerUpper

Verified

2 + \frac{1}{2} \sqrt{2}

2 + \frac{1}{2} \sqrt{2}

Reported2.70710678…

2 + \frac{1}{2} \sqrt{2}

Gap0 solved: the verified bounds meet

Results on the case

4 results in the register on $n = 5$ , 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V0 C1 lower bound incomplete on this case, superseded
Nagamochi · Nagamochi 2005 · source · register
2026-08-30 established T-012 this result
Goebel's $n = 5$ packing is second-order rigid at fixed side
V3 C3 rigidity confirmed
Levy · register
2026-09-03 established T-014
Goebel's $n = 5$ optimum is rigid at fixed side: its pose is an isolated feasible point
V3 C3 rigidity confirmed
Levy · 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

T-012: Goebel's n=5n=5 packing is second-order rigid at fixed side

The case

Results on the case

Links

T-012: Goebel's $n = 5$ packing is second-order rigid at fixed side