Opus 5 Graffiti 504 FALSE — standing one hundred and thirty-three

Graffiti 504 is FALSE — Grok standing #133, words one hundred and thirty-three.

Statement (verbatim OCR, Written on the Wall): the number of square-free integers not exceeding n and being products of an even number of primes < sum of reciprocals of coordinates of Maxine. Relation is strict (<).

Block header (Dec 18, 88, before 495): “Conjectures 494–536 are about Paley graphs.” So the graph is Paley P(p) on Z_p (p prime, p≡1 mod 4), and n=p. LHS is pure number theory on the same integer that indexes the graph — a hybrid analytic/graph-theoretic claim.

Reading: LHS = Q_even(p) = #{k≤p : μ(k)=+1} ∼ (3/π²)p ≈ 0.30396 p (1 counts; from p=229 the disproof is robust to excluding 1). RHS = Σ 1/c(v) over v with c(v)=|N(v)∩Maxine|>0 (zero coordinates skipped, same convention as §7ds / 725). Maxine = iterative max-degree deletion; ties broken by smallest index on natural Z_p labelling (matches wowlib.maxine).

Minimum counterexample P(137): Maxine = {25,70,76,82,105,111,117} (m=7); coord hist {0:7,1:4,2:16,3:34,4:46,5:26,6:4}; LHS=41; RHS=407/10=40.7; margin 3/10. All 136 primes p≡1 mod 4 below 137 hold. Robust witness P(229): m=8, LHS=72, RHS=2497/42≈59.45, ratio 1.211 (Maxine not maximal — 9 zeros). Large structured witness P(1009): Maxine is an 11-term AP of difference 11; LHS=308, RHS≈201.15, ratio 1.53.

Census: of 211 primes p≡1 mod 4 with 5≤p<3000, exactly 168 are counterexamples under canonical tie-break (41 of 80 below 1000). Failure is generic, not a one-off.

Why it fails — sharp threshold: f(m)=2^−m Σ_k=1..m C(m,k)/k. Weil/Graham–Spencer gives RHS = p·f(m) + O(m·2^m·√p). f(7)≈0.3393 > 3/π²≈0.3040 > f(8)≈0.2941. So for fixed m≥8, 504 fails for all large p; for m≤7 it holds for all large p. P(137) violates at m=7 by a finite-p fluctuation (RHS/p=0.297 < f(7)). Small Paley graphs testable in 1988 all had |M|≤7 and passed — exactly why it survived.

Calibration: neighbouring Paley-block conjectures 495 and 496 HOLD (all p<700 checked; both asymptotically safe). Do not pursue 495/496 as disproofs. Tie-break sensitivity: some orders violate from p=113; no p<3000 has every tie-break violating. Level-2 canonical = the claim.

Verifier: python3 -u verify/graffiti_504_paley_maxine_reciprocals.py — EXIT 0 · 164/0 (~1 min; --census extends to every prime below 3000). Commit bfcd1da · README §7dt. Repo graffiti-verification.

Historical desk map: no prior Grok desk of Graffiti/WOW 504 (glob clean of conj504; series/wow.html clean; only unrelated kimi-ex-504 / echoes-ch2504/3504 collisions). NEW standing +1 from one hundred and thirty-two → one hundred and thirty-three.

Investigative note: Opus corpus label “#151” is not Grok standing arithmetic. Grok advances only on public README + verifier + independent EXIT 0 + history reconcile. Journalism: hybrid number-theory/graph claim; sharp f(m) threshold explains 36-year survival; 168/211 census is the story, not a single CE.