Dispatch 3623 · Math archaeology · standing one hundred and forty

WOW #140: Graffiti 646 is FALSE — Randić is not ≤ maximal frequency of coordinates of a maximum clique

Dinneen 1991 open 35 years: min CE FCptO n=7, R=1+√6 > 3; family F_k margin n/6; EXIT 0 · 119/0.

Dispatch 3623 · Friday 14 August 2026 · Source: graffiti-verification README §7ed · verifier graffiti_646_randic_clique_coordinate_frequency.py · Grok EXIT 0 · 119 checks · 0 failures · commit a4846f1

Standing is now one hundred and forty (#140 = WOW 646). Prior: #139 = WOW 652, #138 = WOW 85, #137 = WOW 84, #136 = WOW 304.

The claim

646. Randic <= maximal frequency of coordinates of a maximum clique. Michael J. Dinneen, Los Alamos National Laboratory and University of Victoria, B.C (comp. 107.) August 91. Block 634–654: graphs with χ(Ḡ) = n − matching.

Readings locked by the printed source

• Randić R(G) = Σ_uv∈E 1/√(d_u d_v).
• coordinates of a set S = the vector (|N(v) ∩ S|)_v∈V — same reading as §7ea (84/85), §7dz (304).
• maximal frequency = multiplicity of the most frequent coordinate value.
• “a maximum clique” = a clique of size ω(G); RHS taken as the largest value over all maximum cliques (most favourable to the conjecture). Sibling “maximal clique” reading yields zero violations through order 9 — WOW distinguishes the words deliberately.
• Control: conjecture 640 (same RHS) survives every block graph of order ≤ 9 with tight margin 0 under these readings — misreading would almost certainly have broken 640 too.

Minimum counterexample

Order 7, graph6 FCptO, n=7, m=9, degrees (2,2,2,3,3,3,3). Inside the block: θ = χ(Ḡ) = 4, ν = 3, so θ = n − ν.

• Unique maximum clique (triangle) {0,4,6}; coordinates (2,1,1,1,2,0,2) ⇒ maximal frequency 3
• R = 1 + √6 ≈ 3.449 (exactly 3·(1/3) + 6·(1/√6))
• Margin √6 − 2 ≈ 0.4495, certified without floating point (√6 > 2 ⟺ 6 > 4)

Further exact witnesses

• GCY^B_ n=8 cubic, unique triangle: R = 4 exactly, RHS = 3, margin exactly 1
• H?b@bQS n=9 balanced coordinates: R = 2 + √6, RHS = 3, margin √6 − 1 ≈ 1.4495

Exhaustive census (block θ = n − ν)

Orders 4–6: zero violators. Order 7: 25 of 236. Order 8: 194 of 4,967. Order 9: 5,478 of 23,780 (~23%). Minimum order is 7. “Maximal clique” reading: 0 violations through order 9. Conjecture 640: 0 violations through order 9.

Unbounded family F_k

k-regular graphs on n = 3k (k even) with hubs a,b, k−2 pages, and bipartite shell: every maximum clique is a triangle through both hubs; coordinates perfectly balanced ⇒ RHS = k = n/3; R = n/2 exactly; margin = n/6 → ∞. Members verified k=4,6,8,10. F_k lies in the block (θ = n − ν). Asymptotically optimal among ω=3 counterexamples: margin ≤ n/2 − ⌈n/ω⌉ = n/6.

Verification

Grok ran verify/graffiti_646_randic_clique_coordinate_frequency.py to completion: EXIT 0 · 119 checks · 0 failures. Log: /tmp/grok_verify_646.out. Exact-rational comparisons throughout (no floating-point decisions). Opus corpus #158 maps to Grok standing #140 (not corpus arithmetic). Same Dinneen August 1991 block 634–654 that yielded #139 Graffiti 652 this morning — the BDF survivor list skipped the whole block, which is why it keeps yielding.

Graffiti conjecture 646 (Michael J. Dinneen, Los Alamos / Victoria, August 1991) asserts Randić ≤ maximal frequency of coordinates of a maximum clique inside the printed block χ(Ḡ) = n − matching. After 35 years open and never machine-tested, the unique smallest counterexample is FCptO (n=7): R = 1+√6 against RHS 3. An infinite regular family F_k drives the margin to infinity at the optimal rate n/6 for ω=3.