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

Windowed Energy as Measure Theory (WEM: Windowed Energy as Measure) Auric (S-series / EBOC) Version 1.4 November 24, 2025 Abstract Establish self-consistent framework characterizing energy via first moment of windowed relative spectral density. Core scale chain holds almost everywhere on absolutely continuous spectrum: φ′(E) π=ρrel(E) = 1 2πtr Q(E), where φ(E) = 1 2Arg det S(E), Q(E) = −i S(E)†∂ES(E) is Wigner–Smith group delay matrix, ρrel relative spectral density. Define energy functional by weighting ρrel with window w: E[w] = ZR E w(E)ρrel(E)dE. This paper gives: covariant invariance and channel additivity under energy reparametrization– window pushforward; log det characterization based on Birman–Kre˘ın trace–phase formula and det2/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. 1 Notation & Axioms / Conventions Card-1 (Scale Identity Formula): Holds a.e. on absolutely continuous spectrum φ′(E) π=ρrel(E) = 1 2πtr Q(E),Q(E) = −i S†(E)∂ES(E). Single/multi-channel cases consistent with original “lifetime matrix” definition, computation and experimental pathways established in electromagnetic, acoustic and other systems. Card-2 (Finite-Order EM–NPE Discipline): All discrete approximations adopt only finite-order Euler–Maclaurin expansion; error decomposes as “alias + Bernoulli correction + tail”, constants depending only on endpoint derivatives and finite-order smoothness. Scattering–Spectral Shift Convention: In trace class scattering framework det S(E) = exp−2πi ξ(E),(log det S)′(E) = itr Q(E), 1 thus ρrel(E) = −ξ′(E); under Hilbert–Schmidt relative perturbation replace with Koplienko spectral shift ηand det2. Window and Windowed Measure: Window w∈L1(R)∩C1,w≥0, Rw= 1, R|E|w(E)dE < ∞; windowed relative spectral measure dµw(E) = w(E)ρrel(E)dE. 2 Framework and Basic Objects Set separable Hilbert space (H,⟨·,·⟩), self-adjoint operator pair (H0, H) wave operators exist and complete; on absolutely continuous spectrum exists differentiable scattering matrix E7→ S(E)∈U(N(E)). Define Q(E) = −i S†(E)∂ES(E), ρrel(E) = 1 2πtr Q(E), with E[w] = ZR E w(E)ρrel(E)dE as windowed spectral definition of “energy”. Single-channel S(E) = e2iδ(E)gives tr Q(E) = 2δ′(E), compatible with Friedel-type relation with state density difference (graph networks have local state corrections). 3 Axioms and Basic Properties Axiom 3.1 (Observability).E[w]depends only on windowed relative spectral measure dµw. Axiom 3.2 (Reparametrization Covariance).Set ϕ:R→Rstrictly monotone and C1. Define windowed relative spectral measure dµw(E) := w(E)ρrel(E)dE, and its pushforward dµϕ w:= ϕ∗dµw. Then covariant equivalence formula E(ϕ) S[w] := ZR ϕ(E)dµw(E) = ZR E dµϕ w(E). Axiom 3.3 (Channel Additivity).S=S1⊕S2⇒ρrel =ρrel,1+ρrel,2⇒ ES[w] = ES1[w] + ES2[w]. Axiom 3.4 (Regularized Extension).If S−I∈S2, maintain structure and characterization of E[w]using Koplienko spectral shift function ηand det2. Axiom 3.5 (Vacuum Truth).S≡I⇒ρrel ≡0⇒ E[w]=0. 4log det/det2Characterization and Relative Trace Theorem 4.1 (Trace Class Case).If S−I∈S1, then E[w] = 1 2πi ZR E w(E) (log det S)′(E)dE =−ZR E w(E)ξ′(E)dE. 2 Proof. By (log det S)′= tr(S−1S′) = itr Qand Card-1 directly derive; det S= exp(−2πi ξ) yields spectral shift version. Theorem 4.2 (Hilbert–Schmidt Case, Safe Statement).Set S(E)−I∈S2. Then exists Koplienko spectral shift function η∈L1 loc(R), such that for any f∈C2 c(R)have trf(H)−f(H0)−f′(H0)(H−H0)=ZR f′′(E)η(E)dE. In this framework, energy functional still defined as E[w] = RRE w(E)ρrel(E)dE. If further satisfy additional regularity assumption (e.g., det2S(E)exists non-tangential boundary value and a.e. differentiable), can define Ξ2(E) := 1 2πi ∂Elog det 2S(E), obtaining expression structurally consistent with trace class case E[w] = ZR E w(E) Ξ2(E)dE . 5 Variational Structure and Scale Window Family Under constraint Rw= 1, Gateaux derivative DE[w]·δw =ZR E ρrel(E)δw(E)dE, stationary points satisfy E ρrel(E) = λon support of w. Scale window family wλ(E) = λ−1wE λ, directional derivative d dλE[wλ]λ=1 =−ZR E ρrel(E)w(E) + E ∂Ew(E)dE . 6 Finite-Order Euler–Maclaurin (EM) Non-Asymptotic Error Closure For uniform grid En=E0+n∆ discrete approximation b E= R X n=−R Enw(En)ρrel(En) ∆, let f(E) = E w(E)ρrel(E)∈Cp, have E=b E − ∆ 2f(a) + f(b)−B2 2! ∆2f′(b)−f′(a)−B4 4! ∆4f(3)(b)−f(3)(a)− · · · Thus, without endpoint correction error leading term O(∆); when f(a) = f(b)=0 (window vanishes at endpoints) or using trapezoidal/midpoint symmetric rules, main error improves to O(∆2). Above decomposition still denoted 3 ∆NPE = ∆alias + ∆Bernoulli + ∆tail, embodying principle “smoother window better error”, giving computable bounds of endpointdominated terms. 7 Main Theorems (Selection) Theorem 7.1 (Reparametrization Covariance Consistency).For any strictly monotone C1 ϕhave E(ϕ) S[w] = ZR ϕ(E)dµw(E) = ZR E d(ϕ∗dµw)(E). Proof. Pushforward measure definition gives Rg(E)d(ϕ∗µ)(E) = Rg(ϕ(E)) dµ(E). Taking g(E) = Eimmediately yields conclusion. Theorem 7.2 (Finite-Order EM Stable Bounds–Unified Statement).Set f(E) = E w(E)ρrel(E)∈ Cp([a, b]), uniform grid En=E0+n∆covering effective support, discrete approximation b E= R X n=−R Enw(En)ρrel(En) ∆. Then exist constants C1, C2k(depending on endpoint derivatives up to order 2k−1), such that |E − b E| ≤ ∆ 2|f(a)|+|f(b)|+ ⌊p/2⌋ X k=1 C2k∆2k. Further, if satisfy any condition: (i) f(a) = f(b) = 0 or (ii) adopt trapezoidal/midpoint symmetric rules, then leading O(∆) vanishes and main order improves to O(∆2). 8 Discussion and Outlook This work establishes: 1. Windowed energy functional via first moment of relative spectral density 2. Covariant invariance under reparametrization and window pushforward 3. log det/det2characterizations in trace class and Hilbert–Schmidt cases 4. Non-asymptotic EM error closure with explicit bounds 5. EBOC embedding as observer-independent integrated invariant 6. RCA embedding via Koopman spectral correspondence Key formulas:  Energy functional: E[w] = RE w(E)ρrel(E)dE  Scale identity: φ′/π =ρrel = (2π)−1tr Q 4  EM error: |E − b E| ≤ O(∆) or O(∆2) depending on conditions Future directions:  Extension to dissipative and dispersive systems  Statistical theory for chaotic scattering  Numerical implementation and benchmarking  Applications to quantum graphs and photonic systems 5