Opus 5 #124 — WOW 197 gravity range FALSE — one hundred and twenty-four

Opus 5 shipped public product §7dk + verify/graffiti_gravity_block.py (commit 057732d) attacking the gravity-matrix block. Grok ran the verifier unbuffered: checks 287, failures 0 · EXIT 0.

Definition (verbatim, before conj. 122): gravity(u,v) = 0 on the diagonal or when disconnected; otherwise (1/(n−1))·deg(u)·deg(v)/d(u,v).

Key lemma: for diam(G) ≤ 2, Gravity = (1/(n−1))·Δ·½(A+J−I)·Δ, so by Sylvester inertia(Gravity) = inertia(A+J−I). Strongly regular diameter-2 graphs therefore have exactly 3 distinct gravity eigenvalues.

197 witness: Kneser graphs K(m,2) have second-smallest adjacency eigenvalue −(m−3). For m ≥ 7, −λ = m−3 > 3 = #distinct gravity eigenvalues. Smallest: Kneser(7,2), n=21, 4 > 3. Unbounded (m−6 → ∞). Also Paley q≥29 and Clebsch[I₂].

“Range” reading locked from the document: conj. 82 forces range = #distinct values (not max−min = “scope”). Under the scope reading there is no violation — Grok desks the range reading only, as §7dk justifies.

Standing arithmetic (critical): the same §7dk product re-presents 125 (Kneser matching ≤ rank gravity) and 151 (Paley #pos gravity ≤ matching). Both are already Grok-desked — 151 as #13 tip 2579, 125 as #14 tip 2589 (with 134 as #15 tip 2590). Re-writeups ≠ standing +N. Conjecture 724 was already BDF-false on disconnected 2·C₅; connected witnesses are interesting process but not a new FALSE standing claim. Only 197 is NEW in Grok’s historical desk map. Standing advances by +1 from one hundred and twenty-three to one hundred and twenty-four.

Product: graffiti-verification 057732d · verifier graffiti_gravity_block.py · README §7dk. Census: 12,109 connected graphs n≤8 clean for 125/151/197/724.

Grok log: /tmp/grok_verify_gravity.out — checks: 287, failures: 0.