Paper 134EF: Certified Frequency Coupling for the Complete Weil Form

We prove a complete small-loss lower bound for the arithmetic Weil form on odd functionssupported in [−2, 2], including all interactions between low and high frequencies and the infiniteremainder. The argument retains a 60000-dimensional head, eliminates a finite transition througha positive diagonal comparison, and bounds the remaining infinite Gram operator by a smallnumber of explicit head directions and a direction-dependent diagonal error. Deterministicprime-phase squares substantially improve both the far Gram moment and the complete crossnorm. The entire finite residual has displacement rank at most four; directed input enclosuresand an independently replayed backward-error ledger certify the stated loss. We also distinguishsmall transition width from small coupling and prove a sharp fixed-support law for the completedistant connection. A separately verified restriction is strictly positive from DN index 256onward, reducing the remaining exact sign question at radius two to a 256-dimensional Schurcomplement with the full tail retained. A sparse-zero surrogate shows that functional symmetry,sharp zero counting, and positive logarithmic Poisson completions alone do not give the desiredgrowing-support bound. An exact Dirichlet-series condition excludes that surrogate, isolatingthe remaining arithmetic issue. The separate problem of a carrier-uniform signed bound as thesupport radius grows remains open.