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

WSIG–EBOC–RCA Unified Theory: Trinity “Universal Measure Coordinate” Axiomatization, Change-of-Variables Consistency, and Error Theory Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.2 Abstract Under de Branges–Kre˘ın 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 dµφ(E) = φ′(E) πdE =1 2πtr Q(E)dE =ρrel(E)dE, (1) where Q(E) = −i S(E)†S′(E) is the Wigner–Smith delay matrix, ρrel =−ξ′ is the spectral shift density relative to reference operator H0, and det S(E) = exp(−2πi ξ(E)) is the standard gauge of Birman–Kre˘ın formula. The trinity chain is derived consistently from Birman–Kre˘ın 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 ↔phase/delay/density coordinates); (ii) maximum entropy–minimum delay duality theorem; (iii) causal monotonicity ⇔phase monotonicity; and formulates sampling–frame thresholds, Wexler– Raz biorthogonality, and Balian–Low impossibility as invariant expressions in phase coordinates, while establishing KKT and Γ-limit stability for multi-window/multikernel optimization. All conclusions align term-by-term with BK/SSF, Wigner– Smith, Herglotz–Weyl, and Carleson–frame theory criteria, directly verifiable and implementable. Keywords: Universal measure coordinate; WSIG; EBOC; RCA; Birman–Kre˘ın formula; Wigner–Smith delay; Sampling theory; Frame theory; Wexler–Raz; Balian–Low 1 1 Notation, Gauge, and Foundational Literature Scattering and Spectral Shift. For self-adjoint scattering pair (H, H0), on-shell multichannel scattering matrix S(E)∈U(N). The Birman–Kre˘ın formula (under trace-class assumptions) is det S(E) = exp−2πi ξ(E),(2) where ξis the spectral shift function; accordingly d dE Arg det S(E) = −2π ξ′(E). We stipulate relative state density ρrel(E) := −ξ′(E). Phase (Scattering Semi-Phase). For multi-channel scattering matrix S(E)∈ U(N), define φ(E) := 1 2Arg det S(E) (BK continuous branch),(3) then almost everywhere φ′(E) = 1 2tr Q(E) = π ρrel(E), where Q(E) = −i S(E)†S′(E), ρrel(E) = −ξ′(E). Wigner–Smith Delay. Define Q(E) = −i S(E)†dS dE (E). For unitary S, d dE log det S(E) = trS−1S′(E)= trS†S′(E),(4) thus d dE Arg det S(E) = tr Q(E) and 1 2πtr Q(E) = ρrel(E) (a.e.). For single-channel S= e2iφ, tr Q= 2φ′(E). For definition and properties of delay matrix, see surveys and computational literature. de Branges–Kre˘ın and Herglotz–Weyl. For canonical system and de Branges space H(E), its reproducing kernel diagonal satisfies K(x, x) = 1 πφ′(x)|E(x)|2,(5) where φis the phase function of Hermite–Biehler function E; simultaneously Weyl– Titchmarsh mis a Herglotz function, boundary imaginary part yields spectral density. Windowed Readout and NPE Three-Fold. Taking even bandlimited window wRand bandlimited kernel h, the numerical integration–summation reordering error of windowed readout RwR(E) (h∗ρ⋆)(E)dE can be composed of three parts from Poisson summation and finite-order Euler–Maclaurin (EM) formula correction plus out-of-window tail (alias + Bernoulli layer + tail), with aliasing term eliminated under Nyquist condition. 2 Axiomatization: Trinity Scale and Observable Operations Axiom 1 (Scale Unification).Almost everywhere at Lebesgue points of absolutely continuous spectrum, φ′(E) = 1 2tr Q(E) = π ρrel(E).(6) Derived from BK and delay matrix definition. Axiom 2 (Windowed Observability and NPE).For even bandlimited window wRand bandlimited kernel h, all summation–integration reordering controlled by finite-order EM and Poisson summation, error three-fold decomposition closes, alias vanishes under Nyquist. 2 Axiom 3 (Implementation–Semantics Covariance).There exist semantic embeddings of EBOC record geometry and RCA local reversible updates such that (φ, ρrel,1 Ntr Q) align respectively with “record page number–record density–readout cost” and “information increment–step delay”, maintaining consistency of causal cone and phase monotonicity. 3 Universal Measure Coordinate and Change-of-Variables Three Formulas Definition 3.1 (Universal Measure Coordinate).Take dµφ:= φ′(E) πdE, dτ := 1 Ntr Q(E)dE, dρ := ρrel(E)dE. (7) Proposition 3.2 (WSIG Change-of-Variables Consistency).Let F∈L1 loc, window wR∈ L1∩L∞. If on the window-dominated interval φ(or τ, ρ) is absolutely continuous and strictly monotone, then ZF(E)wR(E)dE =ZFE(φ)wRE(φ)dE dφ dφ, (8) with Jacobian dE dφ =1 2tr Q(E)−1 ,dE dτ =1 Ntr Q(E)−1 ,dE dρ =ρrel(E)−1.(9) Here E(φ) = φ−1(φ)exists and is measurable, change of variables guaranteed by absolute continuity and strict monotonicity. Here τ(E) := ZE E0 1 Ntr Q(s)ds,ρ(E) := ZE E0 ρrel(s)ds. Proof. Measurable change of variables and Lebesgue point property, branch guaranteed by BK continuous branch and windowed integrability. 4 Windowed Birman–Kre˘ın Identity and Error Closure Theorem 4.1 (Windowed BK Identity).Under resolvent difference trace-class, with h∈W1,1∩L∞(or C1 c), wR∈W1,1∩L1∩L∞(or C1 c),(10) thus (hwR)′=h′wR+h w′ R∈L1and (hwR)(±∞) = 0. Define Sspec(h;R) = Zh(E)ρrel(E)wR(E)dE, Sscat(h;R) = −1 2πi Zh′wR+h w′ Rlog det S dE, (11) then Sspec(h;R) = Sscat(h;R). Proof. Use identity ∂Elog det S(E) = tr S−1(E)S′(E),1 2πi ∂Elog det S=ρrel,(12) let f:= hwR, under f∈W1,1∩L1and f(±∞) = 0, integrate by parts for Rf ∂Elog det S, boundary terms vanish, conclusion follows. 3 Theorem 4.2 (NPE Three-Fold; Non-Asymptotic).Equispaced sampling approximation error for RFdecomposes as ER=Ealias +EEM +Etail,(13) where Ealias given by Poisson summation, Ealias = 0 under Nyquist condition; EEM is finite-order Euler–Maclaurin remainder (explicit Bernoulli sequence); Etail controlled by out-of-window exponential decay. 5 Information Geometry and “Maximum Entropy– Minimum Delay” Duality Definition 5.1 (Windowed Entropy).Denote output distribution for energy parameter as pE, windowed entropy HR:= ZH(pE)wR(E)dE. (14) Via change of variables in Proposition 2.2, relates to dE dφ = (1 2tr Q)−1in φ-coordinate. Theorem 5.2 (Maximum Entropy–Minimum Delay Duality).If E7→ pEsmooth, H strictly convex, and Rsufficiently large, then maximum point of HRand minimum point of windowed average delay R1 Ntr QwRdE align at same φ⋆(uniqueness modulo aliasing remainder). Proof Sketch. In φ-coordinate, extremal condition is ∂φ!H(pE(φ))dE dφ = 0; combining dE dφ with KL projection uniqueness and Ky–Fan minimal subspace consistency yields dual colocation. For Ky–Fan extremal properties and spectral subspace stability, see cited references. 6 Causality and “Phase Monotonicity” Equivalence Definition 6.1 (Loop Delay).For realizable energy loop γ, let ∆T(γ) := Hγtr Q(E)dE. Theorem 6.2 (Causal Monotonicity ⇔Phase Monotonicity).If for all realizable loops ∆T(γ)≥0, then Hγdφ =1 2∆T(γ)≥0; converse also holds. Both equivalent to nosignaling/realizability. Proof. Follows immediately from dφ =1 2tr QdE. 7 Sampling–Frame Thresholds, Carleson, and Wexler– Raz Theorem 7.1 (Phase Density and Landau Threshold).Under de Branges–Mellin unified framework, with dν0=φ′ πdx as geometric measure, necessary density threshold for sampling/interpolation given by Landau-type criteria; Paley–Wiener case reduces to classical Landau necessary density. For de Branges spaces, when φ′(x)dx is (locally) doubling measure, sampling–interpolation can be characterized by Beurling density. 4 Theorem 7.2 (Wexler–Raz Biorthogonality and Parseval Condition).Under Nyquist, multi-window {wα}generates Parseval tight frame if and only if frequency-domain energy balance identity holds; with aliasing, need to add periodic replication summation. Equivalent relation with dual window pointwise condition is Wexler–Raz identity. This condition is invariant in φ-coordinate. Theorem 7.3 (Balian–Low Obstruction; Critical Density).Single-window rectangular lattice at critical density cannot simultaneously have double-sided localization and generate Riesz basis/ONB; this obstruction remains invariant in phase coordinate. 8 Multi-Window/Multi-Kernel Optimization, KKT, and Γ-Limit Theorem 8.1 (Bandlimited Projection–KKT Equation and PSWF Structure).On even bandlimited subspace minimizing strongly convex functional of NPE upper bound, necessary optimality condition is kernel-type eigenequation after bandlimited projection, solution exhibits Prolate Spheroidal (Slepian–Landau–Pollak) structure, yielding optimal “principal scale–poles”. Theorem 8.2 (Multi-Objective Pareto Front and Stability).Minimal elements of multiwindow strongly convex multi-objective surrogate satisfy generalized Wexler–Raz and frame operator equations; have O(µ−1)Lipschitz stability under data/kernel perturbation, maintaining spectral invariance “poles = principal scales”. Frame operator diagonalization via Walnut representation can be used for stability estimates. Theorem 8.3 (Explicit Tight/Dual Construction for Non-Stationary Block Systems). In block non-stationary Weyl–Mellin systems, Walnut–Poisson diagonalization reduces frame operator to Calder´on sum multiplier, giving tight/dual frequency-domain closed form; no-aliasing condition 2Ωn≤∆τnequivalent to Nyquist. 9 Unification to EBOC and RCA Semantic Mapping Proposition 9.1 (EBOC Record Geometry Mapping).Interpreting φas “record page number”, ρrel as “record density”, τ=1 Ntr Qas “readout cost”, then “maximum entropy– minimum delay” duality corresponds to “shortest evidence path”, with reversible readwrite ordered by phase monotonicity. Proposition 9.2 (RCA Step Delay and Information Increment).Single-step update of reversible cellular automaton measured by ∆φin phase coordinate, discrete formulation of causal cone is phase monotonicity, system step delay addition equals P∆φ=1 2Rtr QdE. 10 Non-Unitary Extension and Semiclassical Change of Variables Proposition 10.1 (Complex Delay for Sub-Unitary Scattering).Under sub-unitary S, replacing tr Qwith trace/real part of complex delay preserves above change-of-variables and error theory structure; real parts of phase–density–delay trinity still give windowed observable. 5 Proposition 10.2 (Coordinatization of Egorov–Moyal).Using ∂φ= (1 2tr Q)−1∂Eto rewrite Egorov–Moyal series into phase coordinate, facilitates uniform estimates within Ehrenfest time on φ-uniform grid. 11 Main Theorems and Proof Summary Main Theorem A (Windowed Change-of-Variables Consistency and NPE Bound) Under Axioms A–B, windowed integrals in energy–trinity coordinates are equal, discretization/finitesum approximation error obeys NPE three-fold decomposition upper bound. Evidence chain. Measurable change of variables in Proposition 2.2 + windowed BK in Theorem 3.1 + NPE in Theorem 3.2. Main Theorem B (Maximum Entropy–Minimum Delay Duality) Under § 4 assumptions, maximum point of HRand minimum point of average delay align at same φ⋆(uniqueness modulo alias). Evidence chain. φ-coordinate extremal condition + KL/I-projection uniqueness + Ky–Fan minimal subspace; spectral subspace stability by Davis–Kahan. Main Theorem C (Causal Monotonicity–Phase Monotonicity Equivalence) See Theorem 5.2. Evidence chain. dφ =1 2tr QdE and delay additivity. 12 Mathematical and Implementation Checklist (Reproducible) 1. Scale Unification: Estimate dµφvia tr Qor Arg det S, perform Nyquist verification on φ-uniform sampling. 2. Pointer Verification: Spectral minimal subspace of window operator WR(Ky– Fan minimal sum) and stability under data perturbation controlled by Davis– Kahan. 3. Error Closure: Report (εalias, εEM, εtail), alias shuts off under bandlimited+Nyquist. 4. Sampling–Frame: Verify Landau necessary density, Wexler–Raz condition, and Balian–Low obstruction in φ-coordinate. 5. Multi-Window Optimization: Solve bandlimited projection–KKT equation, obtain PSWF-type solution; stability estimate and tight/dual construction under Walnut representation. 13 Conclusion With BK/SSF and Wigner–Smith as bridges, this paper unifies phase derivative, relative state density, and group delay trace as “universal measure coordinate”, providing changeof-variables consistency and non-asymptotic closure of NPE three-fold under windowed readouts. Phase coordinate offers coordinate-invariant formulation of sampling–frame thresholds, Wexler–Raz biorthogonality, and Balian–Low obstruction; on optimization side, bandlimited projection–KKT derives PSWF-type optimal windows, multi-window 6 frame stability and tight/dual construction characterized by Walnut–Poisson diagonalization. Non-unitary scattering and semiclassical Egorov–Moyal maintain isomorphic structure in this coordinate. EBOC and RCA semantic embeddings yield unified realizable interpretation of two dual pairs: “maximum entropy–minimum delay” and “causality– phase monotonicity”. References [1] Birman–Kre˘ın formula and spectral shift: B. Simon, Tosio Kato’s work on nonrelativistic QM (contains BK formula survey); D. R. Yafaev, On the spectral shift function; N. Athmouni et al. (BK derivation in discrete models). [2] Wigner–Smith delay: A. A. Patel, R. A. Michielssen (Q in electromagnetic scattering); G. Texier (time delay survey); ChaosBook (connection of tr Q with DoS). [3] de Branges kernel diagonal and phase derivative: J. P. Antezana, A. Marzo, J. Olsen, formula (K(x, x) = 1 πφ′(x)|E(x)|2). [4] Poisson and Euler–Maclaurin: NIST DLMF § 1.8(iv), § 2.10(i). [5] Landau necessary density and its extension to de Branges: H. J. Landau (1967); A. Marzo, S. Nitzan, J. Olsen (doubling phase case). [6] Wexler–Raz identity and density theorem survey: I. Daubechies et al.; C. E. Heil (history and density theorems). [7] Balian–Low theorem: Encyclopedia entry and subsequent quantitative improvements. [8] Davis–Kahan and Ky–Fan: B. Yu, T. Wang, R. J. Samworth (D-K variants); Ky Fan (1951). [9] Γ-convergence: G. Dal Maso; A. Braides. [10] Complex delay for non-unitary scattering: X. Chen et al. (complex time delay of sub-unitary scattering). [11] Egorov–Moyal and semiclassical: A. Bouzouina, D. Robert; G. Prouff (Egorov in Weyl–H¨ormander framework). [12] Walnut representation and non-stationary frames: C. E. Heil (textbook lecture notes); N. Holighaus et al. (Walnut-like representation for non-stationary Gabor). 7