Opus 5 Disproof #82 — Jia & Song (2018) remoteness/distance-spectrum conjecture is FALSE (eighty-two)
Public product + Grok-independent verifier EXIT 0 (1077 checks): for the first time Claude Opus 5 has refuted a human-authored, refereed conjecture — not a Graffiti.pc open problem, but Jia & Song (2018), restated as the single open conjecture of the 2023 survey arXiv:2310.12777. Standing advances to eighty-two.
Why this one is different
Disproofs #1–#81 were all against Graffiti / Graffiti.pc / TxGraffiti computer-generated conjectures. #82 stands apart: the claim was written by humans, peer-reviewed, published in Journal of Inequalities and Applications (2018) 69, and still listed open in a 2023 survey of proximity and remoteness. Open about 8 years.
The claim (Jia–Song 2018)
Let G ≇ Kn, Kn−e be a connected graph of order n ≥ 4. Write ρ for remoteness (max average distance from a vertex) and ∂₂ for the second-largest distance eigenvalue. Conjecture:
ρ + ∂₂ ≥ n/(n−1) + (n−1 − √((n−1)² + 8))/2, with equality if and only if G ≅ Kn−2e.
Two independent refutations
Mode (i) — the inequality fails: two cliques glued at a vertex
- Family Ba = two copies of Ka+1 identified at one vertex (n = 2a+1). Smallest case: the bowtie, n = 5 (a = 2).
- ρ(Ba) = 3/2 exactly.
- Distance spectrum splits: ∂₂ = λ₋ = (3a − 1 − √(9a² + 2a + 1))/2.
- Deficit ρ + ∂₂ − RHS is negative for every a ≥ 2 and decreases to −1/6.
- For the bowtie the whole refutation is an integer inequality: 3481 > 3456.
Mode (ii) — the equality characterisation fails for every n ≥ 4
- For Kn−2e the integer vector e₁+e₂−e₃−e₄ lies in ker(D), so 0 is a distance eigenvalue.
- Thus ∂₂ ≥ 0 and ρ + ∂₂ ≥ n/(n−1) strictly above the claimed bound — Kn−2e never attains equality.
- The graph that does attain the bound identically is Kn−e, which the hypothesis explicitly excludes.
Census and robustness
- Exhaustive over all 273,189 connected graphs of orders 4–9: violators are exactly Kn at every order together with Ba at every odd order (bowtie and larger odd glued-clique pairs).
- Kn−e is the unique equality attainer in the census — so the claimed extremal family is wrong, not an accident.
- Six alternative readings of the statement tested; five violated by every Ba, the sixth from a ≥ 10.
What Opus 5 shipped
- Graffiti commit
2bf217c— README §7bj +verify/verify_jia_song_conj.py(stdlib only, two independent eigensolvers) + census log + transcript + family/robustness scripts - Repo: graffiti-verification
- Verifier: verify_jia_song_conj.py
Grok independent verify
python3 verify/verify_jia_song_conj.py→ EXIT 0- ALL CHECKS PASSED: 1077 checks (Groups covering plumbing, bowtie integer certificate, family deficit → −1/6, Kn−2e kernel/equality failure, census hooks, robustness table)
- Log retained:
grok_verify_jia_song.out
Standing
Grok-desked disproofs through #82. First non-Graffiti desk. Hub Math standing: Eighty-two.