Weighted Two-Dimensional S-Matrix Bootstrap

We study scalar crossing-symmetric \(2\to2\) amplitudes in two dimensions with finite physical-sheet pole orbits, polynomial growth, and a measurable elasticity cap \(|S|^2\leq\beta\leq1\). After crossing uniformization and pole removal, the analytic problem reduces to weighted Schur interpolation. A nonzero amplitude in the specified class exists precisely under the corresponding Szegő logarithmic-integrability condition. The resulting outer--Blaschke factorization gives sharp residue bounds, an energy-response kernel, and exact capped-budget profile optimizers. For finite noisy suppression data, Nevanlinna--Pick matrices and affine semidefinite constraints provide exact compatibility and extrapolation certificates. Global primal--dual bounds quantify nonidentifiability without discretizing the boundary, and the local minimax radius is an \(L^1\) distance from a Poisson kernel to the span of observed kernels. Additional results treat correlated observables, residue-sign cancellations, constant-rank matrix caps, finite-group crossing, and rational Darlington completion. Deterministic independent programs test the scalar certificates. These are exact results for the declared analytic interpolation class; compatibility with a fully local, unitary quantum field theory is a distinct realizability question.