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Abstract

We give a frequency-block enclosure for a constrained theta-weighted Weil operator. The retained paired-prime operator is translation invariant before the theta multipliers are applied. Splitting at that intermediate stage eliminates the cross-frequency blocks exactly. A finite-rank approximation of the remaining low-frequency map then has a complete operator error whose high-frequency term is proportional to the reciprocal digamma symbol, rather than its inverse square root. We retain the signed finite interaction matrix and enclose its resolvent Gram by full residuals. We prove matching leading asymptotics for both the raw resolvent tail and the positive spectral edge of the actual signed high-frequency block. The retained arithmetic coefficient is therefore asymptotically exact at every fixed prime cutoff and positive resolvent parameter. A compact-torus recurrence argument supplies the arithmetic peaks; an abstract conditional extension covers valid null-space deflations. An elementary theta estimate makes the multiplier constant self-contained.