67.25% and 68.185%: The Scope of Two Ceilings

Why the two must be kept separate: you cannot argue that because 67.25%<68.185%, “optimising Claude's window a bit further would reach 68.185%.” This page precisely dissects the scope of each — 67.25% is the extremal value of the §7.1 window-optimisation/two-trace subframework (block structure + two traces + primes up to T), for which CCLM17 Corollary 14 guarantees that the Montgomery–Taylor kernel is extremal within this subframework. 68.185% is the upper bound of the broader bandwidth-one, configuration-by-configuration certificate class given by Remark 1.1 — using only mean density, pair-correlation data with Fourier support (-1,1), and multiplicity integrality, required to hold for every configuration — but the main paper's text never expands it into a directly recomputable closed-form formula the way it does for c₁*, so it is flagged OPEN-RECONSTRUCTION-01: the research task is not to doubt this number but to find the extremal law's complete definition, variables, constraints, optimizer, and a path to numerical reproduction. §7.5 also gives a finite-dimensional dimension cap (rank P≤d≈λN), and notes that a certificate relying only on the first/second trace is not even positive when λ≤1/2, while unconditional higher moments are in turn constrained by the Rudnick–Sarnak range kλ<2 — showing that “just computing more moments” is not free even in the unconditional bandwidth-one regime. Conclusion: P_70 already lies outside 70%>68.185%, so if Remark 1.1's certificate-class assumptions apply in full, P_70 must violate at least one of these constraints — more support, more correlation information, or a richer, non-configuration-wise global structure.