A Single Completion-Locked Generating Functional with Dual Canonical Projections

We construct an explicit structural bridge between arithmetic–spectral rigidity diagnostics and responsetype invariants by defining a single completion-locked spectral generating functional Z(Δ_𝐺; Lock) associated to a unique symmetry-native self-adjoint generator Δ_𝐺 ≥ 0 and an admissible holonomy/twist deformation family. Under standard spectral regularity hypotheses (trace-class heat kernel, meromorphic continuation of the spectral zeta function, and a well-posed zeta-regularized determinant), we prove that Z admits two canonical projections with shared normalization: a heat/Mellin/trace projection (arithmetic–spectral) and a holonomy/response projection (stiffness). Once the completion lock is fixed upstream, quantities extracted from these projections cannot be normalized independently within the admissible class: agreement or rigidity constraints in one projection constrain admissible normalization and parameterization in the other, and any mismatch is a diagnostic of hypothesis failure rather than a post hoc scheme choice. License note: Distributed under CC BY-NC-ND 4.0.