Constructing RCA--WSIG Unified Metric via de Branges Phase Embedding:\\ Trajectory--Phase Metric, Geometric Reversibility, and Finite-Order NPE Error Closure

We propose a method to embed discrete spacetime trajectories of reversible cellular automata (RCA) into de Branges--Kreĭn phase geometry, defining a ``trajectory--phase metric'' d_{\rm TP}. This metric uses stable window families to window local trajectory segments as Hilbert observation vectors, obtains phase function \varphi(E) via Hermite--Biehler/de Branges embedding, and measures trajectory distance by weighted integral of phase difference on energy axis. Metric scale aligned by mother scale identity \varphi'(E)/\pi=\rho_{\rm rel}(E)=(2\pi)^{-1}tr\mathsf Q(E), where \mathsf Q(E)=-i S(E)^\dagger S'(E) is Wigner--Smith group delay matrix, \rho_{\rm rel} spectral shift density/relative state density. We prove that under tight/near-tight frame and bandwidth regularity conditions for window families, d_{\rm TP} is true metric; if RCA evolution conjugate to unitary scattering family satisfying geometric reversibility, then evolution isometric under d_{\rm TP}. Establish finite-order error closure of Nyquist--Poisson--Euler--Maclaurin (NPE) three-fold decomposition, upper bound controlled by aliasing term, Euler--Maclaurin boundary layer, and high-frequency tail term, yielding non-asymptotic consistency bound for discrete--continuous readouts. In multi-channel case, metric lifts to trace-type integration of eigenphase differences and group delay eigenvalues, remaining stable under Toeplitz/Berezin compression and Carleson/Landau conditions. Provide tight bound expressions and computable toy system examples in bandlimited/weak dispersion regimes.