Modular Reconstruction of the Proof Chain: The Z / L / P Modules

Breaks the proof of Claude's paper into three independently trackable modules. Module Z (Zero Side): restricts the Weil Hermitian form to a finite-dimensional test family, then uses the functional equation to symmetrically decompose it into P⪰0 plus a hyperbolic block Q controlled by off-line pairs; adding the two gives the compression G̃=P+Q. Module L (Linear Algebra): Claude's Lemma 3.2 uses the von Neumann trace inequality to give a rank lower-bound inequality in terms of rank P≤r and n₊(Q)≤b, which takes a concrete form when c=2. Module P (Prime Side): uses the explicit formula to turn the traces of the compression into prime-power/archimedean integrals, unconditionally giving tr G̃~N and tr G̃²~(1/λ+λ/3)N, with λ≤1 as the key structural boundary. Together, the three modules give the baseline flat window value of 2/3 at λ=1; after optimising the window density in §7.1 (the Euler extremal problem yields v*_λ(s)=cos(√2λs)), this reaches c₁*=0.753296… at λ=1, corresponding to 67.25%. It also notes that the existing unconditional main proof really only uses low-order moments — if fourth moments were known, under the conditional HL*(4,λ) hypothesis this could reach 13/18≈72.22%, showing that P_70 has a very concrete conditional-moment route. The page closes with 5 Proof Obligations (PO-01 through PO-05) that each require independent re-verification, including the 0.68185 constant (PO-05), whose complete derivation has still not been located in the main text.