Null--Modular Double Cover and Overlapping Causal Diamond Chains: Total-Order Approximation Bridge for Quadratic Form Localization, Inclusion--Exclusion--Markov Splicing, and Parity Threshold for Distributional Scattering Calibration

We propose Null--Modular double cover carried by zero-measure boundaries of causal diamonds, decomposing modular Hamiltonians into local energy flux integrals on two null sheets in vacuum quadratic form sense. Through a total-order approximation bridge lemma, general diamonds are reduced to monotonic half-space family limits on the same zero-measure hyperplane, with quadratic form closedness and dominated convergence ensuring limit independence of approximation paths. We establish modular Hamilt