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.
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: ,φ′(E)π=ρrel(E)=12πtrQ(E)whereS(E)∈U(N(E))scatteringmatrix,Q(E):=−iS(E)†∂ES(E)Wigner−−Smithdelaymatrix,φ(E):=12\ArgdetS(E),ρrelrelativestatedensityrelativetoreferencechannel/freeHamiltonian.WindowedreadoutdefinedbyToeplitz/BerezincompressionmapCwanditscovariantsymbolKw(E):Tw(E):=12πtr(Kw(E)Q(E)),Tinv(I):=∫ITw(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.
Under de Branges--Kreĭn 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 equation d\mu_\varphi(E)=\varphi'(E){\pi}\,dE=1{2\pi}tr\mathsf Q(E)\,dE=\rho_{rel}(E)\,dE, equation where \mathsf Q(E)=-i\,S(E)^\dagger S'(E) is the Wigner--Smith delay matrix, \rho_{rel}=-\xi' is the spectral shift density relative to reference operator H_0, and \det S(E)=\exp(-2\pi i\,\xi(E)) is the standard gauge of Birman--Kreĭn formula. The trinity chain is derived consistently from Birman--Kreĭn 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 \leftrightarrow phase/delay/density coordinates); (ii) maximum entropy--minimum delay duality theorem; (iii) causal monotonicity \Leftrightarrow phase monotonicity; and formulates sampling--frame thresholds, Wexler--Raz biorthogonality, and Balian--Low impossibility as invariant expressions in phase coordinates, while establishing KKT and \Gamma-limit stability for multi-window/multi-kernel optimization. All conclusions align term-by-term with BK/SSF, Wigner--Smith, Herglotz--Weyl, and Carleson--frame theory criteria, directly verifiable and implementable.
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.
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=ΠwMhΠw, reducing all readouts to linear functionals on spectral measures. The trinity scale identity 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, we define an information action S and induce an information metric via the Hessian gμν=∂μ∂νS, 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∗ while windowed group delay 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 yields Einstein-type equations Gμν+Λgμν=κTμν. The right-hand stress-energy tensor Tμν arises from spectral responses of Kθ. 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-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.
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,H_0), the multi-port scattering matrix S(E) on the absolutely continuous spectrum of H_0 acts on the open channel subspace at energy E with S(E)\in U(N(E)). The Wigner--Smith delay matrix is defined as Q(E):=-i\,S(E)^\dagger \partial_E S(E), the ac density of relative state density as \rho_{rel}(E):=\xi'_{ac}(E), and the half-determinant phase as \varphi(E):=\pi\,\xi_{ac}(E) (also written as \tfrac12 Arg_{ac}\det S(E)). At almost every Lebesgue point of the absolutely continuous spectrum of H_0, we have $ \ \varphi'(E)=\tfrac12\,\operatorname{trQ(E)=\pi\,\rho_{rel}(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,\hbar,e,G,k_{B}) achieve metrological correspondence within this system: c is calibrated by front support and group delay; \hbar is the Weyl--Heisenberg central charge scale; e is anchored by the magnetic flux quantum; G is realized through curvature--energy flow correspondence; k_{B}$ is fixed by SI constantization. These structures possess realizable isomorphic semantics under EBOC's causal block universe and RCA's discrete light cone.
Within the unified context of ``universe = Generalized Light Structure (GLS),'' ``observer = filter chain (windowed compression → CP channel → POVM → threshold counting),'' and ``causality = lightlike cone preorder,'' we establish an axiomatic theory with the master scale πφ′(E)=ρrel(E)=2π1trQ(E),Q(E)=−iS(E)†dEdS(E) (phase derivative---relative density of states---Wigner--Smith group delay trace). \textbf{Causal red line:} Windowed group delay readout Tγ[wR,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 ∂E(argdetU+argdetV)=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 ↔ 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 D2+V2≤1 unified in same ledger; (vi) spectral resolution enhancement (bandwidth narrowing) and redshift amplification are dual; conversely, temporal/image resolution enhancement (σT decrease) and blueshift (bandwidth increase) are dual, consistent with σT≳Cw,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 limE0→∞Tγ=t∗, multichannel limE0→∞TC=Nefft∗ and per-mode average limE0→∞TC=t∗, treating causal wavefront t∗ as operational limit scale of Tγ and TC. Theory employs operator-measure-function language throughout (Toeplitz/Berezin compression Kw,h, readout = linear functional on spectral measure), unifying all time/density readouts under global unitarity postulate by Q=−iS†S′; experimental non-unitarity traced back to master scale via unitary dilation S.
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 (\mathcal H,w,S) as $ \chi(\mathcal H,w,S):=\Delta[w]{\langle\tau_g\rangle_w}, where \Delta[w] is the effective span of window w, and \langle\tau_g\rangle_w is the window-weighted average of Wigner--Smith group delay. Under Assumption A (§1), we prove that \chi is 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 \Delta[w] as effective spatial length and \langle\tau_g\rangle_w as time delay, c is 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--Kreĭn identity and trace characterization of Wigner--Smith matrix \mathsf Q=-i\,\hbar\,S^\dagger \partial_E S; 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--Kreĭn and Rouché-type stability criteria, ensuring verifiable stability of \chi. We further prove that the ``front slope'' refinement limit c_{\rm RCA} 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).