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Unified Framework of Windowed Scattering, Causal Manifolds, and Filter Chains: Axiomatic Theory of Group Delay, Redshift, and Light Speed with Non-Asymptotic Error Closure

Haobo Ma, Wenlin Zhang

Published Nov 10, 2025• 0 citations
Publisher

Abstract

Within the unified context of ``universe = Generalized Light Structure (GLS),'' ``observer = filter chain (windowed compression $\to$ CP channel $\to$ POVM $\to$ threshold counting),'' and ``causality = lightlike cone preorder,'' we establish an axiomatic theory with the master scale $$ \boxed{\ \frac{\varphi'(E)}{\pi}=\rho_{\rm rel}(E)=\frac{1}{2\pi}\operatorname{tr}\mathsf{Q}(E),\qquad \mathsf{Q}(E)=-i\,S(E)^\dagger \tfrac{dS}{dE}(E)\ } $$ (phase derivative---relative density of states---Wigner--Smith group delay trace). \textbf{Causal red line:} Windowed group delay readout $T_\gamma[w_R,h]$ serves solely as operational time scale (can be negative in narrowband/resonance, having no universal size comparison with wavefront time $t_*$); causality determined exclusively by $t_*$ and lightlike cone partial order. Core results: (i) windowed group delay readout provides operational time scale, proven series-parallel additive and gauge covariant/relatively invariant (invariant when $U,V$ energy-independent or more generally $\partial_E(\arg\det U+\arg\det V)=0$); in vacuum links satisfying Nyquist discipline, high-frequency/alias-free limit aligns with wavefront velocity calibrating $c$; (ii) redshift characterized by spectral scaling with reciprocal time scaling law; (iii) light speed $c$ normed by vacuum wavefront, yielding wavefront lower bound and no-supercone propagation for arbitrary physical channels; (iv) ``GLS $\leftrightarrow$ causal manifold'' mutual construction outline with skeleton and consistency conditions---rigorous proof and naturality verification in separate work; (v) wave-particle duality and double-slit windowed complementarity $D^2+V^2\le 1$ unified in same ledger; (vi) spectral resolution enhancement (bandwidth narrowing) and redshift amplification are dual; conversely, temporal/image resolution enhancement ($\sigma_T$ decrease) and blueshift (bandwidth increase) are dual, consistent with $\sigma_T\gtrsim C_{w,h}/B$ in \S4.4, yielding Nyquist--Poisson--Euler--Maclaurin (NPE) finite-order error closure and engineering prescription; (vii) high-frequency platform bridge: under \S4 premises, blueshift limit of any fixed-shape normalized window family---single channel $\lim_{E_0\to\infty}T_\gamma=t_*$, multichannel $\lim_{E_0\to\infty}T_{\mathcal{C}}=N_{\rm eff}t_*$ and per-mode average $\lim_{E_0\to\infty}\overline{T}_{\mathcal{C}}=t_*$, treating causal wavefront $t_*$ as operational limit scale of $T_\gamma$ and $\overline{T}_{\mathcal{C}}$. Theory employs operator-measure-function language throughout (Toeplitz/Berezin compression $K_{w,h}$, readout = linear functional on spectral measure), unifying all time/density readouts under global unitarity postulate by $\mathsf{Q}=-iS^\dagger S'$; experimental non-unitarity traced back to master scale via unitary dilation $\widehat{S}$.

DOI
10.5281/zenodo.17574251

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We construct a unified framework of Windowed Scattering--Information Geometry (WSIG) centered on the ``operator--measure--function'' triple: observation is operationalized through Toeplitz/Berezin compression Kw,h=Πw Mh Πw\mathsf{K}_{w,h}=\Pi_w\,\mathsf{M}_h\,\Pi_wKw,h​=Πw​Mh​Πw​, reducing all readouts to linear functionals on spectral measures. The trinity scale identity φ′(E)/π=ρrel(E)=(2π)−1tr Q(E)=−ξ′(E)\varphi'(E)/\pi=\rho_{\mathrm{rel}}(E)=(2\pi)^{-1}\mathrm{tr}\,\mathsf{Q}(E)=-\xi'(E)φ′(E)/π=ρrel​(E)=(2π)−1trQ(E)=−ξ′(E) unifies phase, relative density of states, and Wigner--Smith group delay as master scales. For observer filter chains {Kθ}θ∈Θ\{\mathsf{K}_\theta\}_{\theta\in\Theta}{Kθ​}θ∈Θ​, we define an information action S\mathcal{S}S and induce an information metric via the Hessian gμν=∂μ∂νSg_{\mu\nu}=\partial_\mu\partial_\nu\mathcal{S}gμν​=∂μ​∂ν​S, interpreting connection and curvature as second-order non-commutative responses of readouts to observation parameters. Under the ``double-time separation'' postulate---where causal partial order is uniquely defined by frontier time t∗t_*t∗​ while windowed group delay Tγ[w,h]T_\gamma[w,h]Tγ​[w,h] serves only as operational scale---variation of the action A=(2κ)−1 ⁣∫(R−2Λ)∣g∣ dnθ+∫(Linfo+λμν[gμν−∂μ∂νSloc])∣g∣ dnθ\mathcal{A}=(2\kappa)^{-1}\!\int(\mathcal{R}-2\Lambda)\sqrt{|g|}\,d^n\theta+\int(\mathcal{L}_{\mathrm{info}}+\lambda^{\mu\nu}[g_{\mu\nu}-\partial_\mu\partial_\nu\mathcal{S}_{\mathrm{loc}}])\sqrt{|g|}\,d^n\thetaA=(2κ)−1∫(R−2Λ)∣g∣​dnθ+∫(Linfo​+λμν[gμν​−∂μ​∂ν​Sloc​])∣g∣​dnθ yields Einstein-type equations Gμν+Λgμν=κ Tμν\mathcal{G}_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa\,\mathcal{T}_{\mu\nu}Gμν​+Λgμν​=κTμν​. The right-hand stress-energy tensor Tμν\mathcal{T}_{\mu\nu}Tμν​ arises from spectral responses of Kθ\mathsf{K}_\thetaKθ​. Under band-limited and Nyquist--Poisson--finite-order Euler--Maclaurin (NPE) discipline, we achieve error closure with ``non-increasing singularity: poles = master scales''. Within the same geometric framework, we unify quantum measurement: superposition = convex superposition of compatible components in static blocks; collapse = anchor switching and π\piπ-semantic collapse; entanglement = inseparable information correlations in causal networks. Continuous monitoring satisfies GKSL dynamics and Spohn entropy production monotonicity, with Belavkin filtering providing conditional state updates. Core criteria and required external theorems are provided, along with one-dimensional potential scattering examples and complete proof chains.
We establish a rigorous theory unifying ``finite throughput--queueing delay--apparent dilation (redshift)'' within windowed scattering and information geometry (WSIG) scale system. In pure theoretical language of operator--measure--function, using ``trinity'' scale identity \varphi'(E)/\pi=\rho_{\rm rel}(E)=-1{2\pi}tr\mathsf Q(E) as mother scale, define observer window's Toeplitz/Berezin compression K_{w,h} and its readout functional, characterize ``finite throughput'' abstract form via passivity and causality-preserving scattering axioms. Prove three main theorems: (T1) Throughput--Delay Dilation Theorem: Under passive load monotonicity, any windowed group delay readout monotonically non-decreasing with load; when baseline readout \Phi_w[\lambda_0]>0, induces Mellin dilation of energy and time scales scale-equivalent to redshift factor 1+z_w=\Delta_w\ge1; (T2) Group Delay--Bandwidth Product Upper Bound: Windowed energy spectrum total readout upper-bounded by product of ``group delay flux'' and ``effective bandwidth constant'', with non-asymptotic error bound from Nyquist--Poisson--Euler--Maclaurin (NPE) three-term closure; (T3) Scale Gauge Equivalence: Cosmological expansion and internal resolution enhancement scale-equivalent to gauge pairing a(t)=R(t)^{-1}, all readouts remaining invariant on same mother scale. Further provide ``front slope--group delay'' correspondence at reversible cellular automaton (RCA) dynamics layer, embedding queueing--throughput--delay discrete dynamics into EBOC static block encoding. Throughout use only finite-order Euler--Maclaurin and Poisson discipline, maintaining ``singularity non-increasing/poles = principal scales'' error control; any numerical and experimental interfacing relegated to appendices.
Within abstract framework of de Branges--Kreĭn 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 non-increasing/pole = dominant scale'' under ``finite-order'' Euler--Maclaurin and Poisson discipline. Thus, in EBOC static block universe replaces external elapsed time with intrinsic T_{inv}; obtains unified metric under RCA reversible computation's isomorphic renormalization. Scale identity of this system holds almost everywhere on absolutely continuous spectrum: $ \ \frac{\varphi'(E){\pi} = \rho_{rel}(E) = 1{2\pi}trQ(E)\ }, where S(E)\in U\big(N(E)\big) scattering matrix, Q(E):=-i\,S(E)^\dagger \partial_E S(E) Wigner--Smith delay matrix, \varphi(E):=1{2}\Arg\det S(E), \rho_{rel} relative state density relative to reference channel/free Hamiltonian. Windowed readout defined by Toeplitz/Berezin compression map C_w and its covariant symbol K_w(E): T_w(E):=1{2\pi}tr\big(K_w(E)Q(E)\big),\qquad T_{inv}(I):=\int_I T_w(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--Kreĭn spectral shift--determinant formula, providing unified coordinate from phase to density, from scattering to measure, and stable benchmark for variational/optimization.
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Under the unified perspective of ``discrete--static blocks (EBOC)'' and ``continuous--causal manifolds/gauge systems,'' we construct an axiomatic framework for quantum measurement: wave functions are characterized as complex-amplitude superpositions of compatible SBUs (Static Block Units) in static blocks; collapse is characterized as observer anchor switching and π\piπ-semantic collapse from operator algebras to classical record algebras; entanglement is characterized as conditional mutual information and quantum Markov property in causal networks. Mathematically, quantum instruments and POVMs serve as core objects, with Stinespring/Naimark/Kraus representations, Lüders update, relative entropy minimization principle, and Belavkin filtering closing the ``measurement--update--record'' chain; at the information-geometric level, strong subadditivity and conditional mutual information control causal propagation of entanglement and ``no-signaling.'' At the readout--geometry interface, employing windowed compression Kw,hK_{w,h}Kw,h​ and ``trinity master-scale'' φm′π=ρrel=12πtr Q\displaystyle\frac{\varphi_{\text{m}}'}{\pi}=\rho_{\text{rel}}=\frac{1}{2\pi}\mathrm{tr}\,\mathsf{Q}πφm′​​=ρrel​=2π1​trQ unified readout framework, with non-asymptotic error closure under Nyquist--Poisson--Euler--Maclaurin (finite-order) discipline; this identity holds almost everywhere on absolutely continuous spectrum, thereby rigorously unifying ``phase derivative--relative spectral density--Wigner--Smith group delay.'' Davies--Lewis instruments/POVMs, postulated Lüders rule, Stinespring--Kraus--Naimark representations, and Belavkin continuous measurement stochastic filtering provide testable anchor points; Birman--Kreĭn identity, Kreĭn--Friedel--Lloyd bridge, and de Branges kernel diagonal formula provide spectral/functional-analytic basis for unified readout of ``windowed--spectral--scattering.''
Using the de Branges--Kreĭn (DBK) canonical system and weighted Mellin space as carriers, we explicitly incorporate the finite bandwidth/time window of real instruments into spectral measures, forming a windowed readout framework; we characterize ``commit (collapse/commit)'' through KL/Bregman information geometry; use scattering phase--spectral density--Wigner--Smith delay as energy scale; close non-asymptotic errors via Nyquist--Poisson--Euler--Maclaurin (three-term decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is $ \,\varphi'(E)=-\pi\,\rho_{\mathrm{rel}(E)=1{2}trQ(E)\,}\quad(a.e.) $ unifying (single/multi-channel) scattering phase derivative, relative local density of states (LDOS) and Wigner--Smith delay; under information geometry we obtain Born probability = minimal-KL projection (I-projection), and ``pointer basis'' is spectral minimum of windowed readout operator. Above criteria consistent with Herglotz--Weyl, Birman--Kreĭn, Wigner--Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable.