Windowed Energy as Measure Theory\\(WEM: Windowed Energy as Measure)

Establish self-consistent framework characterizing energy via first moment of windowed relative spectral density. Core scale chain holds almost everywhere on absolutely continuous spectrum: $ \ \frac{\varphi'(E){\pi}\;=\;\rho_{rel}(E)\;=\;1{2\pi}trQ(E)\ }, where \varphi(E)=1{2}\Arg\det S(E), Q(E)=-i\,S(E)^\dagger\partial_E S(E) is Wigner--Smith group delay matrix, \rho_{rel} relative spectral density. Define energy functional by weighting \rho_{rel} with window w: E[w]=\int_{R}E\,w(E)\,\rho_{rel}(E)\,dE. This paper gives: covariant invariance and channel additivity under energy reparametrization--window pushforward; \log\det characterization based on Birman--Kreĭn trace--phase formula and \det_2$/Koplienko regularization under Hilbert--Schmidt relative perturbation; non-asymptotic error closure under finite-order Euler--Maclaurin (EM) discipline; semantic embedding and Koopman spectral correspondence in EBOC (static block · observation--computation) and RCA (reversible cellular automata). Factual foundations include definition and multi-physics generalizations of group delay matrix, spectral shift function and relative trace, and EM error theory.