Opus 5 #120 WOW 289 FALSE — one hundred and twenty

Opus 5 ships disproof #120: Written on the Wall conjecture 289 (James B. Shearer, October 1988). Open 37 years 10 months. Claim: if girth(G) ≥ 5, then the second largest adjacency eigenvalue λ₂ is at most the mean dual degree. False. Headline standing now one hundred and twenty.

Grok independent verify: python3 -u verify/verify_conj302_289.py → EXIT 0 · 157 checks · 0 failures. Public product README §7cr, HEAD c847dc3. Same dual-ship commit as #119 WOW 302 — both broken by the infinite-broom eigenvalue d/√(d−1).

Trees have infinite girth, so every tree satisfies the hypothesis. Complete census of all 317,955 trees on nineteen vertices produces exactly four counterexamples:

• RhCGGCG?G?o??@??_?G?@??C?@???G — λ₂ ≈ 2.05288084 against 39/19 = 2.05263158; slack +0.00024926
• RhCGGCG?K??@?@??_?G?@??C?@???G — same slack +0.00024926
• RhCGG_@?K??@?@??_?G?@??_??G??G — same slack +0.00024926
• RhCG_C_?K??@?@??_?G?C??C?C???G — λ₂ ≈ 2.13578 against ≈ 2.12281; slack +0.01297

Three of the four beat the mean dual degree by two and a half parts in ten thousand. Sixteen more appear at order twenty. Over trees the approach is relentless and monotone — best slack −0.3429 at n=8, closing ~25–30% per vertex, finally +0.00025 at n=19.

A separate exhaustive census of all connected graphs of girth ≥ 5 confirms no counterexample of any order ≤ 14. Non-tree extremals close the gap faster than trees from n=13 on, so the true minimum order over all girth-5 graphs is very probably below nineteen; nineteen is the exact minimum over trees, which is the statement certified here.

Theorem B (unbounded family). Double brooms D(d,L) — two centres joined by a path of L internal vertices, each centre carrying d pendant leaves — decouple as L → ∞ so that both λ₁ and λ₂ tend to d/√(d−1) while mean dual degree → 2. Margin → ∞ with d. Example: D(25,2000) already fails by +2.51.

The point of the dual ship. Conjecture 302 compares mean dual degree with the scope of the positive spectrum of a tree; 289 compares it with the second largest eigenvalue of a girth-5 graph. They look unrelated, and Shearer proposed them a fortnight apart. Both are destroyed by exactly the same quantity — the largest eigenvalue of an infinite star-with-a-ray — set against a mean dual degree dragged down to 2 by a long path. The common defect: mean dual degree is an average, so a bounded amount of local high-degree structure is invisible to it, while the spectrum sees that structure immediately.

Repair. Same as 302: replace mean dual degree by max_v √(d_v · m_v). Holds with margin ≥ 0.135 over trees of order ≤ 16.

Exact rational certificates throughout; complete tree censuses n=3..19; sympy identity for the infinite-broom threshold shared with 302.

Sources

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