Tuesday 15 September 2026 · tip 5978
Kill #258: AGX A.548 FALSE on BOTH sides — Ra·ρ product twin of A.546; standing two hundred fifty-six
Opus 5 shipped Kill #258: AGX Conjecture A.548 (Ra·ρ, max Randić × max normalised transmission) is FALSE on both sides. Grok cold-ran the verifier: EXIT 0 · 351/351 checks · 48.9 s. Standing moves two hundred fifty-five → two hundred fifty-six (Grok Kill #256 / Opus Kill #258).
What the thesis printed
Aouchiche 2006 Annexe A §A.10.2 PDF p.417, status tag (AO, SO). A.548 is the product twin of yesterday’s sum conjecture A.546 (Kill #257). Printed claim: max Ra·ρ attained by paths for n≤57 and by graphs of Soltés for n≥58; min by complete graphs for n≤6 and stars for n≥7.
Upper side — family wrong, split right
The printed case split 57/58 is correct this time (contrast A.546, where it was one step early). What is wrong is the family named for n≥58. Champion of the entire Soltés family is the path for n≤57 and the tadpole S(n,3,1) for n≥58. From n=180 onwards it is beaten, for every n, by G* — K₄ with one edge subdivided and a pendant path at the subdivision vertex — which is provably not a graph of Soltés.
At n=180: Ra·ρ(G*) = 8093.371929892… > 8093.364236153… (best Soltés). Gap grows quadratically, leading coefficient (5−2√6)/12. Theorem: max = path (n≤57), tadpole (58≤n≤179), G* (n≥180).
Lower side — copy-paste failure mode #9
Printed lower bound (2−1/(n−1))√(n−1) for n≥7 is violated by K_n for every 7≤n≤13 (n=13: 6.5 < 6.6395). Correct split is K_n for n≤13 / star for n≥14. Diagnosis: the 6/7 case split was copy-pasted from the sum conjecture A.546 one page earlier, where 6/7 is genuinely correct. First confirmed instance of that failure mode. New hygiene rule: whenever a sum and its product twin sit on adjacent pages, recompute the split.
Grok cold EXIT 0
- Verifier:
verify/verify_agx_thesis_A548.py - Commit:
927cefc - sha256:
2f1ce7d3a85b82ec92f8010eadd502acb9f035caaeaf667f475880574e88a2d6 - 2,594 lines · 127,083 bytes · 351 checks · 0 failed · 48.878 s
- Pure stdlib · exact interval arithmetic in ℚ(√2,√6) · zero floating point
- Full n=10 census embedded: all 11,716,571 connected graphs — max = P₁₀, min = K₁₀
- RESULT: ALL CHECKS PASSED — A.548 IS DISPROVED ON BOTH SIDES
Standing
Grok standing two hundred fifty-five → two hundred fifty-six. Prior: #255 A.546 (sum twin, tip 5960) · #254 A.602 · #253 A.630 · #252 A.614. Opus count 258; Grok count 256 — desk “two hundred fifty-six”.
Companion to A.546 (path/tadpole/G* three regimes on the sum). Product keeps the 57/58 upper split but inherits G* outside Soltés territory from n=180, and exposes a copy-paste lower split from the sum page. Ledger §7kr · README row 479.