WSIG--EBOC--RCA Unified Theory:\\ Geometric Reformulation of the Speed of Light Constant via ``Window--Group Delay Ratio Invariant'' as Master Scale

WSIG–EBOC–RCA Unified Theory: Geometric Reformulation of the Speed of Light Constant via “Window–Group Delay Ratio Invariant” as Master Scale Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.4 Abstract Within the unified framework of Windowed Scattering and Information Geometry (WSIG), Eternal Block Observer-Computing (EBOC), and Reversible Cellular Automata (RCA), we define the window–group delay ratio for any observational triple (H, w, S) as χ(H, w, S) := ∆[w] ⟨τg⟩w , where ∆[w] is the effective span of window w, and ⟨τg⟩wis the window-weighted average of Wigner–Smith group delay. Under Assumption A ( § 1), we prove that χis an invariant under all reversible observational transformations (information-preserving coordinate/scale rescaling, unitary prefiltering, frame/sampling rearrangement, and reversible lattice updates), with the common value denoted c. In the spacetime geometric interpretation, taking ∆[w] as effective spatial length and ⟨τg⟩was time delay, cis equivalent to the maximal slope of the reachable signal cone, isomorphic to the speed of light constant. This construction is anchored in the following established chain identities and criteria: unification of phase derivative–spectral shift density–group delay (a.e. on absolutely continuous spectrum), Birman–Kre˘ın identity and trace characterization of Wigner–Smith matrix Q=−iℏS†∂ES;NPE non-asymptotic error theory for windowed readouts; Wexler–Raz conditions, Landau density threshold, and Balian–Low obstruction for Gabor/non-stationary Gabor frames; and “read-commit” consistency of I-projection and Ky–Fan minimization. At singularities/thresholds, zero-pole structure is controlled by de Branges–Kre˘ın and Rouch´e-type stability criteria, ensuring verifiable stability of χ. We further prove that the “front slope” refinement limit cRCA of RCA is consistent with the continuous scattering side constant c, and provide reproducible experimental protocols for multi-window–multi-channel configurations (including sampling rate, window order, and confidence interval construction). 1 Keywords: Wigner-Smith group delay; Birman-Kre˘ın spectral shift; de Branges-Kre˘ın canonical systems; Windowed trace; NPE error decomposition; Frame/sampling density; Information geometry; Reversible cellular automata MSC 2020: 81U05; 47A40; 94A12; 42C15; 37B15 Contents 1 Notation, Conventions, and Axioms 1.1 Basic Notation and Known Chain Energy variable E∈R. Scattering matrix S(E)∈U(N) (a.e. differentiable). Wigner– Smith delay matrix Q(E) := −iℏS(E)†dS dE (E). At almost every Lebesgue point of the absolutely continuous spectrum, 1 2πℏtr Q(E) = ρrel(E) = 1 πφ′(E) = −ξ′(E) (a.e.), where ξis the spectral shift function (derivative form of Birman–Kre˘ın det S(E) = exp(−2πi ξ(E))), ρrel is the relative state density, φ(E) := 1 2Arg det S(E). See Wigner– Smith matrix formalism, BK identity and modern formulations. Windowed readouts and error theory employ Nyquist–Poisson–Euler–Maclaurin (NPE) three-term decomposition: discretization aliasing (Poisson), endpoint Bernoulli layer (finite-order Euler–Maclaurin), and out-of-window tail term (bounded by decay/bandlimiting). Related formulas and unified error bounds appear in DLMF (Poisson and EM) and unified error estimation literature. 1.2 Observational Triple and Reversible Observational Transformation Group Observational triple (H, w, S): His the carrier, wis an even window (strictly bandlimited Paley–Wiener or exponentially decaying) allowing scaling wR(x) = w(x/R), S(E) is the (multi-channel) scattering matrix. Reversible observational transformation group Gis generated by the following information-preserving operations: 1. Coordinate diffeomorphisms and scale/rate rescaling; 2. Unitary prefiltering on bandlimited subspaces (functional calculus); 3. Frame/sampling rearrangement (tight/Parseval frames and dual windows, Wexler– Raz conditions); 4. Reversible local updates on discrete lattice (RCA), whose frequency side is bridged to continuous windows/frames by non-stationary Gabor frameworks. 2 1.3 Effective Span of Window and Group Delay Average Effective span ∆[w]: Take a radius/quadratic form covariant with experimental geometry (such as time radius ∆t, spatial radius ∆x), required to be homogeneous under scaling/translation/unitary prefiltering. Group delay average ⟨τg⟩w:= Z1 2πtr Q(E)Kw(E)dE ZKw(E)dE =ℏZρrel(E)Kw(E)dE ZKw(E)dE . where Kwis the energy kernel induced by the window; by a.e. equivalence in the above equation, one can also replace the integrand with ρrel(E), φ′(E)/π, or −ξ′(E). 2 Main Definition and Invariance Theorem Definition 2.1 (Window–Group Delay Ratio and Master Scale). χ(H, w, S) := ∆[w] ⟨τg⟩w . If for all T∈ G and admissible window families w, we have χ(H, w, S) = χT(H, w, S)=: c, then cis called the window–group delay ratio invariant (master scale). Proposition 2.2 (G-Invariance of χUnder Assumption A).Assumption A (Covariance and Measurability): The following conditions are satisfied: (i) Under unit/coordinate scaling and energy reparametrization (without Lorentz boost), ∆[w]and ⟨τg⟩wcovary with the same dimension order; for Lorentz transformations, see the cone boundary upper bound formulation in § 3; (ii) Unitary prefiltering on bandlimited even subspaces preserves window support and energy; (iii) Frame/sampling rearrangement after Parseval normalization does not change dimensions and readouts; (iv) RCA refinement limit and non-stationary Gabor frameworks ensure that the discrete front slope is consistent with the continuous side readout. Conclusion: Under Assumption A, χis invariant under G. Proof sketch. (i) Unit/coordinate covariance (non-boost): Under unit scaling and energy reparametrization, ∆[w] and ⟨τg⟩wtransform with the same order, so the ratio is invariant; Lorentz covariance does not rely on term-by-term homogeneity but is obtained through the maximal slope c:= supw∆[w]/⟨τg⟩wdefined in § 3; (ii) Prefilter/functional calculus: Unitary prefiltering on bandlimited even subspaces does not change the support and energy of bw; windowed trace readout convolved with tr Qunder NPE discipline only produces unified error orders (Bernoulli layer and tail term), not affecting the limit ratio; 3 (iii) Frame/sampling rearrangement: Wexler–Raz conditions and frame operator characterization ensure that windowed readouts are invariant after Parseval constant normalization, and the metric of ∆[w] is preserved under isometry group, hence χis invariant; (iv) RCA reversible update: Under refinement limit, non-stationary Gabor diagonalization and Walnut/Calder´on sum ensure that the energy–time scale of discrete front slope aligns with continuous ⟨τg⟩w, with limit ratio consistent with c. 3 Phase–Density–Delay Chain and Windowed Readout 3.1 Phase Derivative = Spectral Shift Density = Group Delay (WS–BK Chain) From the determinant phase of Sand the Wigner–Smith matrix definition, 1 2πℏtr Q(E) = ρrel(E) = 1 πφ′(E) = −ξ′(E) (a.e.) . The first equality uses S−1=S†and ∂Elog det S= tr(S−1∂ES); the BK formula gives det S= exp(−2πiξ) (adopting BK sign convention; if using det S= exp(+2πiξ), the right-hand side sign changes to positive). The Friedel–Kre˘ın relation and Wigner– Smith unify the scales of phase derivative, spectral density, and time delay. 3.2 Windowed Trace and NPE Three-Term Error Decomposition The total error in discrete implementation decomposes as error = alias |{z} Poisson + Bernoulli layer | {z } finite-order EM + tail |{z} out-of-window truncation , Strict bandlimiting and Nyquist step size can make alias = 0; the Bernoulli layer is determined by endpoint derivatives and can be closed at finite order; the tail term gives an integrable upper bound from window decay. This discipline supports reproducible readouts of ⟨τg⟩wand non-asymptotic stable estimation of χ. 4 Geometric Realization of Speed of Light: cas Maximal Slope In the spacetime geometric interpretation, taking ∆[w] as effective spatial length and ⟨τg⟩was time delay, χ= ∆[w]/⟨τg⟩whas velocity dimension. Under this paper’s reversible observational transformations G(unit/coordinate rescaling, unitary prefiltering, frame/sampling rearrangement, RCA refinement), χis invariant. In the special relativistic case, define c:= sup w ∆[w] ⟨τg⟩w , 4 interpreting it as the maximal slope of the reachable signal cone; this upper bound is a Lorentz invariant, hence isomorphic to the speed of light constant. This statement relies on the scale unification and multi-channel trace formulation in § 2. For non-unitary/open systems, one can introduce complex time delay and replace with generalized scales such as ∂Eℑlog det S; in this case, the strict equality 1 2πℏtr Q=ρrel no longer holds (this equality only holds when Sis unitary). 5 Discrete–Continuous Bridging and RCA Front Slope Diagonalizing the block system with non-stationary Weyl–Heisenberg (Gabor) frameworks, using Walnut-type representation and “painless” construction to control energy conservation and stability; under tight/dual and frame bound clamping, refining the grid yields lim refinement ∆lattice τsteps =cRCA =c. Non-stationary Gabor and “painless” expansion provide explicit construction and stability radius, achieving quantitative consistency between discrete and continuous. 6 Error Theory and Optimal Window/Kernel Design Formulate window/kernel design as a convex variational problem on bandlimited even subspaces: for fixed (∆, M, T) (span, EM order, total sampling duration), minimize the variance of ⟨τg⟩westimation subject to NPE error upper bound constraints. In strongly convex cases, the solution is unique; in weakly sparse cases, -limit convergence is obtained via Bregman–KL regularization. The Pareto frontier for multi-windows can be given by generalized biorthogonal (GG†) reconstruction, with robust redundancy achieved under Parseval frameworks. The Nyquist rate is determined by the “sum width” of the sampled integrand; when strictly bandlimited, alias = 0. 7 Threshold/Resonance/Singularity Stability and Trace– Weyl Type Laws Under de Branges–Kre˘ın canonical systems, control zero-pole distribution via spectral function and Hermite–Biehler structure; Rouch´e-type radius ensures windowing and finite-order exchange do not introduce new singularities and pole orders do not increase. Trace and multiplicative formulas of time–frequency localization (Toeplitz/Anti-Wick) operators provide Weyl-type weak limits where “symbol integral = trace”, thus giving stable readouts of ⟨τg⟩w. 5 8 Sampling Density Threshold and Balian–Low Obstruction With phase density induced by φ′(E) as scale, Landau-type necessary density holds; sufficiency also follows under one-component/doubling-phase conditions. Single window + rectangular lattice at critical density is subject to Balian–Low restriction; experimental realization requires multi-window/non-uniform or supercritical density strategies. 9 EBOC Semantics: Read-Commit Consistency (Born = I-Projection; Pointer Basis = Spectral Minimum) In information geometry, I-projection (minimizing DKL) produces the most “faithful” update on a given constraint family; when constraints align with device dictionary (windows/frames), and temperature limit goes from soft to hard, an equivalent commitment mechanism to Born rule emerges. Statistical sensitivity is controlled by KL–Pinsker inequality, directly giving estimation bandwidth and confidence intervals. Pointer basis selection can be formulated as Ky–Fan partial sum minimization (minimizing sum of first keigenvalues/singular values), consistent with “spectrally simplest” stable readout. 10 Multi-Channel Unification and Sign Conventions For Nchannels, take scalar ρΣ(E) := tr(ρ−ρ0)(E) = 1 2πℏtr Q(E). Under the sign convention det S(E) = exp(−2πi ξ(E)), we have ρΣ(E) = −ξ′(E); if using det S(E) = exp(+2πi ξ(E)), the sign changes to positive, not affecting this paper’s ratio structure. 11 Verifiable Statements and Reproducibility Protocol (A) Window independence (across window families): Select two window families (bandlimited and exponential), several scales R; after unified NPE correction, ∆[w]/⟨τg⟩w converges to the same c, outputting non-asymptotic confidence intervals (KL–Pinsker). (B) Transformation covariance (across G): After coordinate rescaling, prefiltering, and frame/sampling rearrangement, repeat readouts; χis invariant within error bands; also holds after taking trace over multiple channels. (C) Discrete–continuous bridging: The shortest update cone slope of RCA converges to cwith refinement and non-stationary frame diagonalization; numerical stability and tolerance budget are given under tight/dual and frame bound clamping. (D) Statistical closure: Commitment = I-projection; construct bci= ∆[w(i)]/⟨τg⟩w(i) and merge via KL–Pinsker, producing confidence bands and optimal multi-window Pareto selection. 6 12 Discussion and Boundaries 1. Units and dimensions: If ∆[w] takes spatial scale and ⟨τg⟩wtakes time delay, then χhas velocity dimension; if both are in dimensionless logarithmic/Mellin scale formulation, translation to velocity upper bound requires geometric dictionary. 2. Non-unitary/open systems: Can use complex time delay and generalized scales like ∂Eℑlog det Sas replacement; χratio structure can continue; but equality 1 2πℏtr Q=ρrel only holds for unitary scattering, not strictly equivalent to ρrel in non-unitary cases. 3. Thresholds/resonances: de Branges–Kre˘ın and Rouch´e-type stability ensure windowing and finite-order exchange do not increase singularities; after removing finite neighborhoods, uniform limits and confidence intervals of χcan still be constructed. 4. Critical density obstruction: Single window rectangular lattice at critical density is subject to Balian–Low restriction; recommend multi-window/non-uniform or supercritical density experimental schemes. 13 Conclusion With the group invariance of the window–group delay ratio χ= ∆/⟨τg⟩as core, we establish a reformulation of the “speed of light constant” as a geometric invariant. This master scale cis consistent across quantum/classical, continuous/discrete, multichannel/single-channel, and read-commit layers, guaranteed by NPE non-asymptotic error theory and frame/sampling discipline in finite-resource realistic readouts. The core scale chain 1 2πℏtr Q=ρrel =1 πφ′(E) = −ξ′(E), together with BK, WS, Wexler–Raz, Landau, Balian–Low and other criteria, constitutes a verifiable, reproducible theory– experiment closed loop. Appendix A: Dimensions and Conventions  If ∆[w] takes spatial length and ⟨τg⟩wtakes time, then χhas velocity dimension; if in dimensionless logarithmic/Mellin scale formulation, translation via geometric dictionary is needed.  This paper adopts Q=−iℏS†∂ES; thus 1 2πℏtr Q=ρrel =φ′/π =−ξ′(BK sign convention det S= exp(−2πiξ)). If using det S= exp(+2πiξ), the ξ′term on the right-hand side changes sign to positive, not affecting χratio structure and readout flow. Appendix B: Minimal Reproducible Experimental Checklist 1. 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