WSIG--EBOC--RCA Unified Theory:\\ Universal Measure Coordinate and Gauge-Invariant\\ Time Scale via ``Window--Group Delay Scale''

Within abstract framework of de Branges--Kreĭn canonical systems and multi-channel scattering, this paper establishes unified system in pure theoretical language of ``operator--measure--function'', independent of experimental narrative: welding ``phase derivative--relative state density--Wigner--Smith group delay trace'' as universal measure coordinate of same parent scale, characterizing finite resources and observational choices via windowed readouts of Toeplitz/Berezin compression. On reversible observational transformation group generated by base automorphisms, phase gauge and reversible filtering, proves invariance of time scale within blocks constructed from windowed delay integral; gives non-asymptotic error closure and stable principle of ``singularity non-increasing/pole = dominant scale'' under ``finite-order'' Euler--Maclaurin and Poisson discipline. Thus, in EBOC static block universe replaces external elapsed time with intrinsic T_{inv}; obtains unified metric under RCA reversible computation's isomorphic renormalization. Scale identity of this system holds almost everywhere on absolutely continuous spectrum: $ \ \frac{\varphi'(E){\pi} = \rho_{rel}(E) = 1{2\pi}trQ(E)\ }, where S(E)\in U\big(N(E)\big) scattering matrix, Q(E):=-i\,S(E)^\dagger \partial_E S(E) Wigner--Smith delay matrix, \varphi(E):=1{2}\Arg\det S(E), \rho_{rel} relative state density relative to reference channel/free Hamiltonian. Windowed readout defined by Toeplitz/Berezin compression map C_w and its covariant symbol K_w(E): T_w(E):=1{2\pi}tr\big(K_w(E)Q(E)\big),\qquad T_{inv}(I):=\int_I T_w(E)\,dE, $ remaining invariant under reversible observational transformations, becoming universal time scale within EBOC blocks. Above trinity scale connected by Wigner--Smith delay matrix and Birman--Kreĭn spectral shift--determinant formula, providing unified coordinate from phase to density, from scattering to measure, and stable benchmark for variational/optimization.