Compact‑Tail Reduction and the Remaining Finite Schur Obstruction for a Fixed‑Window Weil Form at Log 8

Abstract This manuscript records a functional‑analytic reduction step in a computer‑assisted spectral program for a fixed‑support Weil form at log 8. The scope is deliberately narrow: it separates purely mathematical claims from those dependent on archived theorem‑producing code, and it identifies the precise point where the current proof program remains incomplete.

In the parity‑adapted coefficient representation, the diagonal contribution simplifies to d(k)=ℜψ(1/4+ik/2)−log⁡π, after an exact cancellation between the analytic tail constant and a Lerch‑transcendent term. We prove strict monotonicity d′(k)>0 for k>0 and logarithmic divergence d(k)=log⁡(k/2π)+O(k−2). Consequently, on a sufficiently far coefficient tail, D−1/2 is compact. If the remaining interaction V=K+P is bounded, then T=D−1/2VD−1/2 is compact self‑adjoint. A certified tail inequality with D≥δI and ∣V∣≤M<δ excludes threshold‑one defects in the tail.

We also provide an explicit finite‑rank projection estimate, formulate the primitive constraint with the correct tail projector, and prove compact resolvent for the raw tail operator. Under bounded head‑tail coupling, a Schur inertia reduction transfers all remaining sign and kernel information to a finite block. At the endpoint L=log⁡8, the logical prime‑power ledger contains six directions {2,3,4,5,7,8}, but the compressed q=8 block vanishes identically, leaving five effective directions.

Finally, we derive a residual inverse‑energy identity showing that for a positive projected operator D and residual columns E, any approximate solve X yields E∗D−1E−(X∗E+E∗X−X∗DX)=R∗D−1R≥0. This reduces the unresolved endpoint problem to a finite directed Schur matrix once the coercivity bound is rebound to the same projected operator used in the endpoint identity. That rebind, and the resulting finite sign/kernel decision, remain open.

Keywords Weil explicit formula; compact‑tail reduction; Schur complement; validated numerics; interval arithmetic; residual Gram identity; Riemann Hypothesis

Mathematical Status Compact‑tail and finite‑Schur reduction theorem established. Threshold‑one positivity: incomplete. Global status: RH proof not obtained.