Quantum--Classical Unification Theory Within the WSIG--EBOC--RCA Framework

Quantum–Classical Unification Theory Within the WSIG–EBOC–RCA Framework Haobo Ma1 Wenlin Zhang2 1Independent Researcher 2National University of Singapore (Windowed Scattering & Information Geometry · Eternal-Block Observer-Computing · Reversible Cellular Automata) November 19, 2025 Version: 1.15 (2025-11-02, Asia/Dubai) Abstract Using the trinity of phase–relative state density–group delay from Windowed Scattering–Information Geometry as the unique master scale for the energy axis, this paper establishes isomorphic semantics between static block geometry and reversible cellular automata, achieving quantum–classical unification. For a scattering pair (H, H0), the multi-port scattering matrix S(E) on the absolutely continuous spectrum of H0acts on the open channel subspace at energy Ewith S(E)∈ U(N(E)). The Wigner–Smith delay matrix is defined as Q(E) := −i S(E)†∂ES(E), the ac density of relative state density as ρrel(E) := ξ′ ac(E), and the half-determinant phase as φ(E) := π ξac(E) (also written as 1 2Argac det S(E)). At almost every Lebesgue point of the absolutely continuous spectrum of H0, we have φ′(E) = 1 2tr Q(E) = π ρrel(E). The semiclassical limit is closed to classical Hamiltonian flow, Poisson brackets, and Liouville dynamics through Egorov’s theorem, Moyal deformation, and Wigner measure propagation. Readouts employ the Nyquist–Poisson–Euler–Maclaurin (NPE) error ledger to provide non-asymptotic upper bounds, with a variational optimality framework for single/multi-window–multi-kernel configurations (details in § 6). The constants (c, ℏ, e, G, kB) achieve metrological correspondence within this system: c is calibrated by front support and group delay; ℏis the Weyl–Heisenberg central charge scale; eis anchored by the magnetic flux quantum; Gis realized through curvature–energy flow correspondence; kBis fixed by SI constantization. These structures possess realizable isomorphic semantics under EBOC’s causal block universe and RCA’s discrete light cone. Keywords: Quantum-classical unification; WSIG; EBOC; RCA; Wigner-Smith delay; Semiclassical limit; Egorov theorem; Birman-Kre˘ın formula MSC 2020: 81Q20; 81S30; 47A40; 37N20; 35Q40 Contents 1 1 Axioms and Objects A1 (Causal–Front): The world is a causal block (M, g). The null cone ds2= 0 determines front velocity c. In linear time-invariant (LTI) systems, if the impulse response h(t) is supported on t≥0 and h∈L1(R) (or more generally is a tempered distribution with frequency response H(ω) in the Hardy class), then H(ω) is analytic in the upper half-plane and satisfies Kramers–Kronig; conversely, if His bounded analytic in the upper half-plane with appropriate growth/decay, the corresponding his causal. For time-varying systems, causality is stated via support of the retarded Green’s function. A2 (Master Scale–WSIG): Define Q(E) := −i S(E)†∂ES(E), τWS(E) := ℏtr Q(E). Further define the per-port average group delay for multi-port cases as τ(E) := ℏ N(E)tr Q(E), N(E) := number of open channels (at that E). Unless otherwise specified, trace and average are taken over the current open channel subspace; far from channel thresholds, N(E) is constant. Let the scattering pair (H, H0) satisfy definable wave operators, with multi-port scattering matrix S(E) on the absolutely continuous spectrum of H0; for each energy E, S(E) acts on the open channel subspace with S(E)∈U(N(E)). BK Condition (ensuring applicability of spectral shift and determinant phase): Assume (H, H0) forms a trace-class perturbation pair in the Birman–Kre˘ın sense, e.g., (H−i)−1−(H0−i)−1∈S1, then the Kre˘ın spectral shift function ξexists, and at almost every Lebesgue point E∈ σac(H0), det S(E) = exp2πi ξ(E),−i ∂Elog det S(E) = 2π ξ′ ac(E). Denote the ac density of relative state density as ρrel(E) := ξ′ ac(E). To avoid multivaluedness ambiguity in Arg, define φ(E) := π ξac(E)also written as 1 2Argac det S(E). Then at almost every Lebesgue point of the absolutely continuous spectrum of H0, tr Q(E) = 2π ρrel(E), φ′(E) = 1 2tr Q(E) = π ρrel(E). Regularity and domain: The following equations involving ∂ES(E) and Q(E) are understood on E∈σac(H0) where S(E) is locally absolutely continuous (or has weak derivative) with respect to E. This paper defaults to working energy windows far from channel thresholds and branch points; when discussing across thresholds, replace with traces/derivatives on the N(E)-dependent open channel subspace. A3 (Bridge Constants): ℏis the central parameter of Weyl–Heisenberg; cis realized via front support and group delay metrological correspondence; eis anchored by the magnetic flux quantum; Gis realized through curvature–energy flow correspondence; kB is fixed by SI constantization. 2 A4 (Readout–Error): Any readout is a “window Ö kernel” weighted average, subject to NPE three-term error closure: aliasing/Poisson, finite-order Euler–Maclaurin remainder, bandwidth tail term. A5 (Realizability–RCA, Reversibility Criterion): The influence domain of a radius-rreversible cellular automaton (RCA) is a discrete light cone. The necessary and sufficient criterion for reversibility is: the global map is bijective and its inverse is also a cellular automaton. 2 Trinity Master Scale and Main Theorem 2.1 Definition and Notation Let the scattering pair (H, H0) give S(E) on the absolutely continuous spectrum of H0 (acting on the open channel subspace at E), with S(E)∈U(N(E)). Take Q(E) := −i S†∂ES, τWS(E) := ℏtr Q(E), ρrel(E) := ξ′ ac(E), det S(E) = exp2πi ξ(E),−i ∂Elog det S(E) = 2π ξ′ ac(E) (a.e. on σac(H0), under BK condition). 2.2 Main Theorem (Trinity) Theorem 2.1 (Trinity Identity). φ′(E) = 1 2tr Q(E) = π ρrel(E) holds at almost every Lebesgue point of the absolutely continuous spectrum of H0. Proof. From det S(E) = exp2πi ξ(E)(BK condition), at almost every Lebesgue point E∈σac(H0), −i ∂Elog det S(E) = 2π ξ′ ac(E). Unitarity gives ∂Elog det S(E) = trS†∂ES. Thus on E∈σac(H0), tr Q(E) = −itrS†∂ES= 2π ξ′ ac(E)=2π ρrel(E), and from the BK chain, φ′(E) = π ξ′ ac(E) = 1 2tr Q(E) at almost every Lebesgue point of the absolutely continuous spectrum of H0. Single-channel verification: If S(E) = e2iδ(E), then tr Q(E) = 2 δ′(E), ρrel(E) = δ′(E) π, φ′(E) = δ′(E), consistent with the Friedel relation. 3 3 Semiclassical Bridge: Egorov–Moyal–Wigner Measure (OpℏNotation) Egorov (leading order): For Weyl quantization H= Opℏ(p) and classical Hamiltonian flow Φt, U† tOpℏ(a)Ut= Opℏ a◦Φt+O(ℏ), which can extend to Ehrenfest time under appropriate regularity (with log(1/ℏ) corrections in chaotic cases). Moyal deformation: Weyl calculus maps i ℏ[ˆ A, ˆ B] to the Moyal bracket, and {A, B}M={A, B}+O(ℏ2). Wigner measure propagation: The Wigner measure of a sequence of states propagates along the classical flow, converging to Liouville/Vlasov-type transport equations. 4 EBOC: Front Support, KK Causality, and Metrological Closure Front support of retarded Green’s function: For the three-dimensional wave equation, the retarded Green’s function is Gret(t, r) = δt− |r|/c 4π|r|, with support exactly on the front t=|r|/c. When § 1.A1 conditions are satisfied (h∈L1or more generally tempered distribution with frequency response Hin Hardy class with appropriate growth/decay), strict causality is mutually equivalent to upper half-plane analyticity and Kramers–Kronig dispersion relations; if these conditions fail, only directional implications hold or additional regularization is needed. Decomposition and validity conditions for metrological closure: Suppose external links are uniform, and the free propagation phase for each open channel n= 1, . . . , N(E) can be written as diagonal factor Dk(E) := diageikn(E)L, kn(E) = E ℏvp,n(E). Taking decomposition S(E) = Dk(E)U(E), from dkn dE =1 ℏvg,n(E)we obtain tr Q(E) = L ℏ N(E) X n=1 1 vg,n(E)−itrU†∂EU. If all channels satisfy vg,n(E)≈vg(E) in the given energy window (or −itr(U†∂EU) has been eliminated with reference link), then τ(E) = ℏ N(E)tr Q(E)≈L vg(E)in vacuum, vg(E) = c. If external links exhibit dispersion or reflection, or the scattering region contains localized states decoupled from ports, separate the continuous part and correct the above decomposition. 4 5 RCA: Reversibility Criterion, Discrete Light Cone, and Floquet Spectrum On a finite alphabet and Zdshift, a continuous global map commuting with shift is a cellular automaton; it is a reversible cellular automaton if and only if the global map is bijective and its inverse is also a cellular automaton. For radius-r, lattice spacing a, time step ∆t, the influence domain at tsteps is ±rt, with discrete “speed of light” cdisc = ra/∆t; the continuum limit aligns with the EBOC front. The quasienergy spectrum of one-step evolution under periodic driving is given by Floquet–Sambe formalism, with energy scale E=ℏωcompatible with group delay readouts. 6 NPE Error Ledger and Window/Kernel Optimization Poisson–Nyquist (aliasing term): When band-limiting satisfies the Nyquist–Shannon condition, the aliasing term vanishes; for general windows, the aliasing term can be quantitatively bounded via Poisson summation. Euler–Maclaurin (unified remainder bound): If f∈C2m∩W2m,1(R) with f(k)(±∞) = 0 (0 ≤k≤2m−1), the even-order truncation remainder satisfies R2m≤2ζ(2m) (2π)2mZRf(2m)(x)dx, m ∈N. If decay/integrability fails, truncate with window function and incorporate boundary terms into R2m, modifying this bound accordingly. NPE readout notation and total error: Denote convolution [κ⋆f](E) := RRκ(E− E′)f(E′)dE′. For any window–kernel pair (wR, κ), the readout is written XW=1 2πZR wR(E) [κ⋆f](E)dE, with composite error estimated as Err =O(ℏ) + O(εalias) + O(εEM) + O(εtail), where εalias = 0 (when band-limited), εEM is bounded by the above, and εtail is controlled by out-of-band mass and |tr Q|∞. Multi-window–multi-kernel optimization: Under Parseval tight frames or Gabor frameworks, fit tr Qwith target kernel κand minimize functional penalizing Err; feasibility and stability are guaranteed by biorthogonality relations and density theorems. 7 Typical Models and Unified Inferences Free propagation link (unified formulation, including multimode): If external links are uniform and −itrU†∂EU= 0 (or this term has been canceled via reference link), then tr Q(E) = L ℏ N(E) X n=1 1 vg,n(E), τWS(E) = L N(E) X n=1 1 vg,n(E), τ(E) = 1 N(E) N(E) X n=1 L vg,n(E). 5 If all channels satisfy vg,n(E)≈vg(E), this reduces to τWS(E)≈N(E)L vg(E)and τ(E)≈ L vg(E). If these conditions fail, retain the correction term −itrU†∂EU. Discrete light cone of RCA: For radius-r, lattice spacing a, time step ∆t, the RCA influence domain at tsteps is ±rt, with discrete “speed of light” cdisc =ra/∆t. In the continuum limit, it locally matches the group velocity of classical dispersion only in a linearized energy/wavenumber neighborhood E0: cdisc E≈E0 −−−−−−−−−→ continuum limit vg(E0), with vacuum linear dispersion giving the special case cdisc →c. Potential scattering DOS–phase–delay: If S(E) = diag e2iδj(E), then ρrel(E) = 1 πX j δ′ j(E) = 1 2πtr Q(E), with delay peaks characterizing resonance lifetimes. Quantum–classical dynamics reduction: d dt⟨ˆ A⟩=i ℏ⟨[ˆ H, ˆ A]⟩=⟨{H, A}⟩ +O(ℏ2), with Wigner measure providing macroscopic transport limit. 8 Falsifiable Exits and Interfaces Given multi-port S(E) or port data, the trinity chain provides consistent predictions for φ′(E), ρrel(E), tr Q(E); any systematic deviation indicates window–kernel or model assumption mismatch. When § 7 conditions (−itr(U†∂EU) = 0 or baseline canceled) are satisfied, multi-window regression τ(E)→L/vg(E) rate is controlled by § 6 bounds, experimentally verifiable. RCA prototype cdisc-calibration in continuum limit and linearized energy neighborhood E0locally matches vg(E0); quasienergy spectrum readouts should be consistent with continuous links in that neighborhood. 9 Appendix: Technical Lemmas and Proof Outlines 9.1 BK–Kre˘ın–WS Chain Let the scattering pair (H, H0) give S(E) on the absolutely continuous spectrum of H0 (with S(E)∈U(N(E))). Then det S(E) = exp2πi ξ(E)⇒ −i ∂Elog det S(E)=2π ξ′ ac(E) (a.e. on σac(H0)), ∂Elog det S(E) = trS†∂ES,Q(E) = −i S†∂ES, ρrel(E) = ξ′ ac(E), φ(E) := π ξac(E)=1 2Argac det S(E), φ′(E) = 1 2tr Q(E) = π ρrel(E) (a.e. on σac(H0)). 6 9.2 Egorov–NPE Composite Error If ft=f◦Φt+O(ℏ), then for XW=1 2πZR wR(E) [κ⋆ft](E)dE we have Err =O(ℏ) + O(εalias) + O(εEM) + O(εtail), where εEM is given by the Euler–Maclaurin bound in § 6. 10 Conclusion This paper establishes a quantum–classical unification framework across WSIG–EBOC– RCA: the trinity of phase–relative state density–group delay as the unique master scale for the energy axis; Egorov–Moyal–Wigner measure propagation implementing semiclassical reduction; NPE error ledger providing non-asymptotic verifiable bounds; front support and metrological closure calibrating bridge constants (c, ℏ, e, G, kB). 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