Exact Strip-to-Circle Transport: Weight-Jacobian Cancellation and Local Kernel Bounds

We develop a local, scale-adaptive method for windowed boundary flux on vertical strips that resolves two obstacles to localization: boundary weights in Dirichlet-to-Neumann traces and the diagonal singularity of the associated bilinear form. A canonical strip-to-circle reparametrization yields an exact weight-Jacobian cancellation, turning weighted strip flux into an unweighted circle pairing with explicit kernels; equivalently, on the line the transported kernels are hyperbolic profiles with a diagonal singularity and an integrable mirror term. Working with the completed zeta function simplifies the analysis, as its linear Hadamard factor contributes only a constant that is annihilated by the Dirichlet-to-Neumann map. On the lower bound side, a midpoint Taylor expansion of the Dirichlet-to-Neumann pair profile against an even window cancels the linear term and yields a coercive gain that scales inversely with the proximity parameter. On the upper bound side, H1-BMO duality and atomic decomposition reduce the data to Lipschitz-free local mean oscillations on a fixed buffer, producing polylogarithmic growth in the inverse proximity (at most squared-logarithmic). Comparing these bounds as a→0 yields a contradiction, excluding off-line zeros.