RH Toolkit 1: Deferral Operators on ℓ² and Their Continuous Realization

This paper develops a functional-analytic framework for kernel-defined linear operators on discrete Hilbert sequence spaces, called deferral operators. Under Hilbert-Schmidt admissibility conditions, these operators are bounded and compact, and under kernel symmetry they are self-adjoint with the standard compact spectral decomposition. The paper constructs continuous L2L^2L2 realizations through orthonormal indexed families, preserving compactness, self-adjointness, and nonzero spectral data by unitary equivalence. It also records structural invariances including Fourier diagonalization, gauge reduction via cycle flux, support-graph decomposition, scale-inversion covariance, normalization, and stability under composition. The appendices provide finite reservoir models, localized theta-structured kernels, positivity-preserving semigroup/resolvent benchmarks, comparison families, projected near-zero reduction, and a conditional theta-convex residual-absorption interface for later work.