WSIG--EBOC--RCA Unified Theory:\\ Trinity ``Universal Measure Coordinate'' Axiomatization, Change-of-Variables Consistency, and Error Theory

Under de Branges--Kreĭn canonical systems and multi-channel scattering theory, taking scattering phase derivative, relative state density, and Wigner--Smith group delay trace as three equivalent formulations of the same scale, we construct a ``universal measure coordinate'' and prove its change-of-variables consistency under windowed readouts and non-asymptotic error closure of Nyquist--Poisson--Euler--Maclaurin (NPE) three-fold decomposition. The core unification formula (almost everywhere on absolutely continuous spectrum) is equation d\mu_\varphi(E)=\varphi'(E){\pi}\,dE=1{2\pi}tr\mathsf Q(E)\,dE=\rho_{rel}(E)\,dE, equation where \mathsf Q(E)=-i\,S(E)^\dagger S'(E) is the Wigner--Smith delay matrix, \rho_{rel}=-\xi' is the spectral shift density relative to reference operator H_0, and \det S(E)=\exp(-2\pi i\,\xi(E)) is the standard gauge of Birman--Kreĭn formula. The trinity chain is derived consistently from Birman--Kreĭn formula and Wigner--Smith delay, applicable to multi-channel unitary scattering; sub-unitary systems admit compatible extension via generalized (complex) delay. The paper provides: (i) windowed change-of-variables consistency theorem (energy \leftrightarrow phase/delay/density coordinates); (ii) maximum entropy--minimum delay duality theorem; (iii) causal monotonicity \Leftrightarrow phase monotonicity; and formulates sampling--frame thresholds, Wexler--Raz biorthogonality, and Balian--Low impossibility as invariant expressions in phase coordinates, while establishing KKT and \Gamma-limit stability for multi-window/multi-kernel optimization. All conclusions align term-by-term with BK/SSF, Wigner--Smith, Herglotz--Weyl, and Carleson--frame theory criteria, directly verifiable and implementable.