Q Quantum Theory II-B: Electrodynamics - Technical Companion

This paper develops the mathematical analysis of finite-response electrodynamics, from a common whole-Q response datum through conserved-history geometry and gauge-complete Wilson descendants to branch-resolved fixed-order Abelian ultraviolet convergence and a one-loop Pauli calculation. The operator/Fock, TOPT/Minkowski, optical and infrared reconstruction is separated into Paper II-C, while experimental inference is separated into Paper II-D. The analysis comprises the ultraviolet estimates and the complete one-loop Pauli calculation. First, the parent tangent metric and current conservation define the physical cycle projector; the exact branch evaluates one non-Gaussian response on the complete conserved-cycle norm, while the generic branch uses a same-datum cylindrical lift and a nonpropagating auxiliary determinant response. Second, the one-loop Pauli sector is reduced to the complete fourteen-term $111+21_L+21_R+3$ family (three ordinary vertices, left/right mixed two-plus-one contacts, and the direct three-photon contact) with parent-derived Wilson phases and term-resolved histories. Third, the response-number sums are removed analytically by coherent-semigroup kernels, after which the Hamiltonian-matched transverse plus constrained-Coulomb loop integral is subjected to independent hard-tail, infrared, spectral, path, radial and angular convergence conditions. The unresolved global limit returns the Schwinger coefficient and the finite-response production grid quoted in Paper II-A.