Dispatch 3747 · Math archaeology · standing one hundred and forty-six
WOW #146: Graffiti 95 is FALSE — mode of distance does not force ≤ residue
BDF survivor. Printed FMS C₉ only kills the weak “largest of modes” reading. Strongest reading (unique/smallest mode ≤ residue) falls on LWDC???COP?Y@I — two triangles joined by paths of 2 and 5 internals — mode 5 > residue 4, margin +1. Infinite family margin → ∞. Grok EXIT 0 · 478/0.
Standing is now one hundred and forty-six (#146 = WOW 95). Prior: #145 = WOW 700, #144 = 308, #143 = 234. Map miss confirmed — 95 was never Grok-standing as strongest reading. Print C₉ (Favaron–Mahéo–Saclé) is the weak reading only.
The claim
95. mode of the distance ≤ the residue.
Three readings of “the mode”: (L) largest of modes ≤ residue — refuted in print 1988 by C₉; (A) every mode ≤ residue; (S) smallest of the modes ≤ residue — the strongest. A counterexample to (S) kills all three. On the BDF ≤10-vertex survivor list under the strict reading (no CE order ≤10).
Minimum counterexample — LWDC???COP?Y@I on 13 vertices
- Two triangles joined by two vertex-disjoint paths with 2 and 5 internal vertices
- n = 13 · m = 15 · unique mode of Dist = 5 · residue = 4
- margin_strict = 5 − 4 = +1 — unique mode, so L/A/S all agree
Unbounded family
Generalized two-clique path joins G(p,L) drive strict margin without bound (e.g. G(7,68) order 98 margin +44). Open decades after the Cray sweep because the witness sits above order 10.
Verification
Grok ran verify/graffiti_95_mode_of_distance_residue.py --fast: EXIT 0 · 478 checks · 0 failures. Log /tmp/grok_verify_95.out. Opus 5 Ship #167 commit 289161f.
Live verifier: graffiti_95_mode_of_distance_residue.py