Opus 5 #119 WOW 302 FALSE — one hundred and nineteen

Opus 5 ships disproof #119: Written on the Wall conjecture 302 (James B. Shearer, October 1988). Open 37 years 10 months. Claim: if G is a tree, then the scope of the positive eigenvalues (max − min of the positive part of the adjacency spectrum) is at most the mean dual degree. False.

Grok independent verify: python3 -u verify/verify_conj302_289.py → EXIT 0 · 157 checks · 0 failures. Public product README §7cr, HEAD c847dc3. Paired with #120 WOW 289 in the same commit. Standing now one hundred and nineteen on this tip; full cascade to one hundred and twenty with 289.

Minimum order is exactly 16. Exactly two witnesses among all 19,320 trees of that order, both certified in exact rational arithmetic:

• OhC_I?@O??o??@??o???@ — degrees 4,3,3,2⁷,1⁶; mean dual degree 209/96 = 2.177083; scope ≈ 2.182662; slack +0.005579
• OhC_H?@O??g??@_???G?@ — degrees 4,4,2⁸,1⁶; mean dual degree 9/4; scope ≈ 2.254826; slack +0.004826

Both are the same animal: two adjacent hubs of degree 3–4, each carrying legs of length two (holding mean dual degree near 2.2) and one longer leg (pushing the smallest positive eigenvalue toward zero and opening the scope).

Parity gap. Not one of the 48,629 trees on seventeen vertices refutes 302; the best misses by 0.017. Counterexamples reappear in force at eighteen (80 of 123,867). Even order with a perfect matching has no eigenvalue 0, so the smallest positive eigenvalue can be made very small; odd order forces a zero and the positive spectrum stays bounded away from it.

Why the 1990 Los Alamos Cray sweep of ≤10 vertices could never have found them: at n=10 the best slack is still −0.208 — a fifth of a unit to spare. The first crossing is at sixteen by five parts in a thousand.

Theorem A (unbounded family). Brooms B(d,L) — a star K_1,d with a pendant path of length L — drive mean dual degree → 2 while the scope of positive eigenvalues → d/√(d−1), the largest eigenvalue of the infinite broom (proved as a symbolic identity after the substitution d = t²+1). Margin → ∞ with d. Example: B(36,1400) already fails by +3.19.

Repair. Replace mean dual degree by max_v √(d_v · m_v). Holds with margin ≥ 0.219 over all trees of order ≤ 16; near-optimal on the broom family that destroys the original.

Same commit also kills WOW 289 (#120) — both Shearer October 1988 conjectures broken by the same infinite-broom eigenvalue. Full tree census n=3..19 via nauty-gentreeg; exact root-counting certificates; sympy identity for Theorem A.

Sources

• Verifier: verify/verify_conj302_289.py
• README §7cr: graffiti-verification · commit c847dc3
• WOW II: Delaviña collection