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

Abstract Within abstract framework of de Branges–Kre˘ın 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 nonincreasing/pole = dominant scale” under “finite-order” Euler–Maclaurin and Poisson discipline. Thus, in EBOC static block universe replaces external elapsed time with intrinsic Tinv; obtains unified metric under RCA reversible computation’s isomorphic renormalization. Scale identity of this system holds almost everywhere on absolutely continuous spectrum: φ′(E) π=ρrel(E) = 1 2πtr Q(E), where S(E)∈UN(E)scattering matrix, Q(E) := −i S(E)†∂ES(E) Wigner–Smith delay matrix, φ(E) := 1 2Arg det S(E), ρrel relative state density relative to reference channel/free Hamiltonian. Windowed readout defined by Toeplitz/Berezin compression map Cwand its covariant symbol Kw(E): Tw(E) := 1 2πtr Kw(E)Q(E), Tinv(I) := ZI Tw(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–Kre˘ın spectral shift–determinant formula, providing unified coordinate from phase to density, from scattering to measure, and stable benchmark for variational/optimization. 1 Notation & Axioms / Conventions 1. Observational triple (H, w, S): HHilbert space; S(E)∈UN(E)scattering matrix at energy scale E(a.e. on absolutely continuous spectrum); wwindow, inducing analysis–synthesis map Πwand Toeplitz/Berezin compression map Cw[X] := ΠwXΠ† w; covariant symbol Kw(E) := Πw(E)†Πw(E)≥0, determining readout functional. Window families take bandlimited or exponential decay classes, satisfying regularity in reproducing-kernel context. 2. Window family normalization (Parseval/tight frame, component-wise): In direct integral decomposition of absolutely continuous spectrum and channel fibers, choose window family such that within each threshold regular component Jhave tr Kw(E)≡ NJ(a.e. E∈J). This normalization compatible with reproducing-kernel regularity, ensuring windowed readout phase gauge terms only produce endpoint constants. 3. Threshold set and regular domains: Denote threshold set T:= {E:N(E+)= N(E−)}. For all integrals in Theorem 4.2, default I∩ T =∅, or equivalently first subdivide Ialong Tthen componentwise integrate and aggregate. 1 4. Phase branch and differentiability: Fix continuous branch of Arg det S(E) on each threshold regular component J, thus φ′(E) exists a.e. on J; across thresholds and discrete spectrum treat in distributional sense (Levinson-type transitions). 5. Scale identity card: Axiomatize φ′(E)/π =ρrel(E) = (2π)−1tr Q(E),Q:= −i S†∂ES. Equivalence between phase derivative and tr Qfrom Wigner–Smith definition and determinant differential identity; equivalence with ρrel given by Birman–Kre˘ın formula and spectral shift function differential connection. 6. Finite-order EM+Poisson card: For sum–integral transformation and energy discretization, uniformly use finite-order Euler–Maclaurin and Poisson summation for non-asymptotic error closure; explicit bound constants depend on finite norms of window and symbol; singularity non-increasing and “pole = dominant scale”. 7. Language and objects: Window/readout uniformly treated as “operator–measure– linear functional” objects; avoid experimental procedure narrative. Toeplitz/Berezin compression and Berezin transform used to map function symbols in energy–phase analytic platform (such as de Branges space, Paley–Wiener/Mellin models) to operators. 8. Notation: det! denotes regularized (Fredholm) determinant; tr trace; Pac absolutely continuous spectral projection; “a.e.” all refer to almost everywhere on absolutely continuous spectrum. 2 Scattering Phase, Group Delay and Spectral Shift: Trinity Coordinate Let Hand reference H0self-adjoint, satisfying usual traceable perturbation conditions making S(E) exist and unitary. Define Q(E) := −i S(E)†∂ES(E) and φ(E) := 1 2Arg det S(E). Wigner–Smith gives Hermiticity of Qand its relation with energy derivative of S; trace satisfies tr Q(E) = ∂EArg det S(E)=2φ′(E). On other hand, Birman–Kre˘ın formula det S(E) = exp(−2πi ξ(E)) connects scattering determinant with spectral shift function ξ, thus ξ′(E) = −1 2πtr Q(E) = −φ′(E)/π. Taking ρrel(E) := −ξ′(E) yields scale identity. Corollary 2.1. On absolutely continuous spectrum a.e., measures induced by three objects satisfy dµφ=dµρ=dµQ, and dµQ(E) = (2π)−1tr Q(E) dE. This provides parent scale for subsequent windowed readout and transformation consistency. 3 Windowed Readout and Toeplitz/Berezin Compression Take reproducing-kernel space H(such as de Branges, Paley–Wiener or Mellin–Hardy) as energy–phase analytic platform. Window winduces analysis–synthesis map Πw. Define compression map Cw[X] := ΠwXΠ† w, and its covariant symbol Kw(E) := Πw(E)†Πw(E)≥0. 2 Definition 3.1 (Channel Fiber Compression).Under direct integral decomposition of absolutely continuous spectrum Hac ≃R⊕CN(E)dE, analysis map Πw(E) : CN(E)→CN(E)gives covariant symbol Kw(E) := Πw(E)†Πw(E)∈CN(E)×N(E), Kw(E)≥0. Thus windowed density and readout Tw(E) := (2π)−1tr Kw(E)Q(E),Tinv(I) := RITw(E) dE, under Parseval normalization, for each threshold regular component J, satisfy tr Kw(E)≡NJ (a.e. E∈J). For energy-local matrix symbol A(E), define windowed trace ⟨A⟩w:= Rtr Kw(E)A(E)dE. Windowed density of group delay defined as Tw(E) := (2π)−1tr Kw(E)Q(E), thus Tinv(I) := RITw(E) dE. Toeplitz/Berezin system ensures positivity and regular limits of Kw, and consistency on symbol algebra. 4 Reversible Observational Equivalence and Gauge Invariants Definition 4.1 (Reversible Observational Transformation).Reversible observational transformations generated by: (i) Automorphism Uof H(fixing energy scale); (ii) Phase gauge:S7→ eiθ(E)S, where θ∈W1,1(I)∩C0(I) and for each component interval endpoint θ(Ej,±) = 0; (iii) Reversible window renormalization (energy-independent): On each threshold regular component Jtake fixed channel basis U∈U(NJ). Window renormalization w7→ ewinduces Πew= ΠwU†, Kew(E) = UKw(E)U†. Theorem 4.2 (Gauge Invariance of Windowed Delay–Normalized Version).Under conditions of normalization and Definition, for any threshold regular finite union interval I=FJ j=1[Ej,−, Ej,+]⊂R(i.e., I∩ T =∅), quantity Tinv(I) := ZI 1 2πtr Kw(E)Q(E)dE invariant under reversible observational transformations (automorphism, phase gauge, reversible window renormalization). Proof. Use trace and similarity invariance get tr(UKwU†·UQU†) = tr(KwQ). Phase gauge contributes term θ′(E) tr Kw(E); by component-wise normalization tr Kw(E)≡Nj(E∈ [Ej,−, Ej,+]) and endpoint condition θ(Ej,±) = 0, get RIθ′(E) tr Kw(E) dE=PjNj[θ]Ej,+ Ej,−= 0, gauge term vanishes, invariance holds. Corollary 4.3 (Universal Time Scale).Tinv independent of observational representation, constitutes intrinsic time scale in EBOC static blocks. 3 5 Universal Measure Coordinate and Transformation Consistency Proposition 5.1 (a.c. Three-Measure Consistency and Distributional Extension).Let S(E) satisfy usual traceable perturbation and limiting absorption conditions. Then on absolutely continuous spectrum a.e. have dµac φ=dµac ρ=dµac Q, where dµφ(E) = φ′(E) πdE,dµρ(E) = ρrel(E) dE,dµQ(E) = 1 2πtr Q(E) dE. If incorporating discrete spectrum/thresholds into full spectrum, three consistent in distributional sense: dµρcontains δ-masses at discrete spectrum, φexhibits phase jumps (Levinson-type), Qtakes boundary values. Therefore any windowed readout comparable and transformable under same coordinate, transformation error bounded by unified constants of Section 5. 6 Stable Error Theory: Finite-Order Euler–Maclaurin and Poisson Let wbelong to bandlimited class or exponential class, a(E) sufficiently smooth energy symbol, {En}energy partition (generated by window or spectral tube). For energy domain I=FJ j=1[Ej,−, Ej,+], exists m∈Nand constants Cm, C′ m(depending only on window family and finite-order derivative seminorms) such that X n a(En)−ZI a(E) dE− m X k=1 B2k (2k)! J X j=1 a(2k−1)(E)Ej,+ Ej,−≤CmRm(a, w), X k=0 ba(2πk)bw(2πk)≤C′ mPm(a, w). where B2kBernoulli numbers, Rm,Pmerror functionals composed of finite seminorms. For a= tr Q, φ′, ρrel apply same constant chain; obtain unified error budget on windowed readouts of three objects; and singularity non-increasing and “pole = dominant scale” hold. 7 EBOC Intrinsic Time and RCA Reversible Computation Isomorphic Renormalization Definition 7.1 (Intrinsic Time Scale).For energy domain Idefine Tinv(I) = ZI 1 2πtr Kw(E)Q(E)dE, as relational progression time scale in EBOC static blocks; invariant under reversible observational transformations. Theorem 7.2 (RCA Isomorphic Renormalizability).Embed step depth of reversible cellular automaton Uinto Tinv metric: if two observational triples (Hi, wi, Si)reversibly equivalent, then RCA “depth” corresponding to their boundary–channel coupling measured by same time scale. Proof depends on invariance of Theorem 4.2 and unified coordinate of scale identity, obtaining isomorphic renormalization under different bases/encodings. 4 8 Discussion and Outlook This work establishes: 1. Trinity scale unification φ′/π =ρrel = (2π)−1tr Qvia Wigner–Smith delay and Birman– Kre˘ın formula 2. Windowed readout framework via Toeplitz/Berezin compression with covariant symbol Kw(E) 3. Gauge invariance of intrinsic time scale Tinv under reversible observational transformations 4. Non-asymptotic error closure via finite-order Euler–Maclaurin and Poisson summation 5. Connection to EBOC static block universe and RCA reversible computation 6. Frame-theoretic foundations via Wexler–Raz, Balian–Low, Landau density 7. de Branges–Kre˘ın analytic platform and Herglotz–Nevanlinna structure Key formulas:  Scale identity: φ′/π =ρrel = (2π)−1tr Q  Windowed time: Tinv(I) = RI(2π)−1tr(KwQ) dE  Invariance: Tinv unchanged under (U, θ, w) transformations Future directions:  Numerical implementation and benchmarking  Extension to open quantum systems and non-Hermitian scattering  Connections to quantum information and complexity theory  Applications to quantum gravity and emergent spacetime 5