Finite Throughput--Group Delay Dilation Principle:\\ Unifying Bandwidth, Time Delay, and Redshift via WSIG--EBOC--RCA's ``Window--Scale--Gauge''

Finite Throughput–Group Delay Dilation Principle: Unifying Bandwidth, Time Delay, and Redshift via WSIG–EBOC–RCA’s “Window–Scale–Gauge” Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.5 Abstract 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 φ′(E)/π =ρrel(E) = −1 2πtr Q(E) as mother scale, define observer window’s Toeplitz/Berezin compression Kw,h and its readout functional, characterize “finite throughput” abstract form via passivity and causalitypreserving 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 Φw[λ0]> 0, induces Mellin dilation of energy and time scales scale-equivalent to redshift factor 1 + zw= ∆w≥1; (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. Keywords: Finite throughput; Group delay dilation; WSIG; EBOC; RCA; NPE error closure; Redshift equivalence; Scale gauge 1 Notation & Axioms / Conventions 1. Scale Identity (Trinity): For absolutely continuous spectrum a.e. φ′(E) π=ρrel(E) = −1 2πtr Q(E),Q(E) = −i S(E)†S′(E), S(E)∈U(N), (1) where φis total scattering phase, ρrel relative state density (Birman–Kre˘ın perspective), QWigner–Smith group delay matrix. By Birman–Kre˘ın formula det S(E) = exp(−2πi ξ(E)) and −i ∂Elog det S(E) = tr Q(E) obtain tr Q(E) = −2π ξ′(E), thus above holds (ξspectral shift function). 2. Observer Window and Compression: Take non-negative weight window w∈ L1(R)∩L∞and smoother h(approximating identity, Carleson regular). Define Toeplitz/Berezin compression Kw,h acting on energy spectrum symbol F(E) windowed readout ⟨F⟩w:= ZR w(E) tr F(E)dE, ⟨F⟩w,h := tr(Kw,hF),(2) differing under NPE discipline only by finite-order Poisson+Euler–Maclaurin error. Characterization of Toeplitz/Berezin and Carleson embedding see recent developments in weighted Bergman/Hardy systems. 3. Passivity and Monotonicity: Load represented by unitarity-preserving scattering family Sλ(E)∈U(N) (λ∈[0,1]). Passivity axiom written as ∂λQλ(E)⪰0 a.e. (3) Throughout derivations use only this Loewner monotonicity, no longer claim sign equivalence with ∂λρrel. Supportable by monotonicity of spectral shift function ξ(E;λ) for monotonically coupled self-adjoint spectral pairs (Hλ=H0+λV, V ≥0) with λ. 4. NPE Three-Term Closure (Finite-Order): Discrete–continuous error decomposition Etotal =Ealias +EBL +Etail,(4) where Ealias from Poisson spectral aliasing, EBL finite-order Euler–Maclaurin boundary layer term, Etail controlled by window decay and target function step. Standard references for Poisson and Euler–Maclaurin (including Bernoulli constants) see cited sources. 5. Scale Gauge: Dual gauge of external scale factor a(t) and internal resolution constant R(t) a(t) = R(t)−1, κ(t) := ˙a/a =−˙ R/R, (5) implements Mellin dilation on energy mother scale, redshift satisfies 1+z=a(t0)/a(te) = R(te)/R(t0). 2 1 Windowed Readout, Group Delay Flux, and Effective Bandwidth Define windowed group delay flux Φw:= 1 2πZR w(E) tr Q(E)dE =−⟨ρrel⟩w.(6) Define window’s effective bandwidth constant Bw:= |w|L1· | bw|pack,(7) where bwcorresponding Fourier–Mellin transform, | · |pack controlled by Nyquist/Landau density (for bandlimited or tight frame window families can be given by upper/lower bound constants). If 0 ⪯F(E)⪯α(E)1, then ⟨F⟩w≤N|α|L∞(supp w)· |w|L1.(8) General case Qnot necessarily positive, thus cannot take F= (2π)−1Q; can use D1 2πQEw≤N 2πQL∞(supp w)|w|L1(9) as crude upper bound. Necessary conditions for sampling/frame density and bandwidth packing see Landau and subsequent generalizations. 2 Throughput Abstraction: Operator–Measure–Function Paradigm Let observation triple (H, w, S). Write scale measure dµQ:= (2π)−1tr Q(E)dE. Define energy readout Rw:= Zw(E)ρrel(E)dE =−1 2πZw(E) tr Q(E)dE =−Zw(E)dµQ(E) = −Φw. (10) “Finite throughput” abstracted as Carleson-type constraint on µQ: exists constant Cth and energy domain partition {Ωk}such that |µQ|(Ωk)≤Cth Λ(Ωk),(11) where Λ reference measure induced by window family (given by Nyquist density or frame density). Corresponding Toeplitz/Berezin type embedding in weighted function spaces characterizable by Carleson condition. 3 Main Theorem I: Throughput–Delay Dilation Theorem Theorem 3.1 (Load Monotonicity and Dilation Factor).Let Sλ(E)satisfy passivity axiom ∂λQλ(E)⪰0a.e., and window w≥0compactly supported or satisfying frame regularity and NPE conditions. Then for any 0≤λ0< λ1≤1, Φw[λ1]−Φw[λ0] = 1 2πZw(E) tr Qλ1(E)−Qλ0(E)dE ≥0.(12) 3 Define scale’s “delay dilation factor” (when Φw[λ0]>0) ∆w(λ1, λ0) := Φw[λ1] Φw[λ0]≥1.(13) General case always-valid statement is Φw[λ1]−Φw[λ0]≥0,(14) equivalent to Rw’s non-increasing (because Rw=−Φw). By Trinity, ∆wsimultaneously characterizes phase density derivative and relative state density elongation, inducing energy scale–time scale Mellin dilation, equivalent to redshift factor 1+zw= ∆w(λ1, λ0)≥1. Root lies in spectral shift function ξ(E;λ)monotonic with non-negative coupling, and tr Q(E) = −2π ∂Eξ(E). Proof. Loewner order monotonicity at each Eand w≥0 integration immediately yields non-decreasing. By Birman–Kre˘ın and Trinity linear relations, monotonicity equivalently transmitted between φ′and tr Q; on ρrel differs only by fixed negative sign (ρrel =−1 2πtr Q), thus above non-decreasing unified expressed as Φw. Corollary 3.2 (Window–Band Domain Locality).If wsupported on bandlimited region Ω, then zwis local redshift for that domain; for mutually non-overlapping {Ωj}with decomposition windows {wj}forming tight frame, 1 + zglobal =X j ωj(1 + zwj), ωj=Φwj[λ0] PkΦwk[λ0],(15) giving global dilation as weighted combination of local dilations; if baseline state µQ[λ0] is non-negative measure (equivalent to Φwj[λ0]≥0for all j), this combination is convex combination. 4 Main Theorem II: Group Delay–Bandwidth Product Upper Bound Theorem 4.1 (Finite Window Energy Spectrum Conservation Upper Bound).Under § 2’s Carleson-type throughput constraint, window family tight frame, and NPE validity, for any S(E)and window w Φw≤Cframe Cth Bw1 + εNPE,(16) where Cframe given by window family upper/lower bound constants, Bweffective bandwidth constant, εNPE finite-order error term satisfying |εNPE| ≤ C1Ealias +C2E(m) BL +C3E(β) tail.(17) Proof Sketch. Along Berezin compression commutator inequalities and Carleson embedding control µQ, window family frame upper/lower bounds connect local energy spectrum with global readout, finally use finite-order Poisson dealiasing and Euler–Maclaurin boundary layer absorb discrete–continuous difference; singularity non-increasing ensures principal scale poles not amplified. 4 Corollary 4.2 (Optimal Window’s Variational Tendency).Under fixed Cth and resource constraints |w|L1,|bw|pack,w⋆maximizing Φwtends toward tight frame and near-minimum uncertainty window; if bandlimited domain invariant, w⋆nearly constant amplitude within domain, minimum boundary layer cost at boundaries. Its frame/density conditions compatible with Wexler–Raz biorthogonality and Balian–Low obstructions. 5 Main Theorem III: Scale Gauge Equivalence and Redshift = Resolution Enhancement Theorem 5.1 (Gauge Equivalence).Let a(t), R(t)satisfy a(t) = R(t)−1and κ(t) = ˙a/a =−˙ R/R. On Trinity mother scale, for any window wand energy readout Rw(t) = Zw(E)ρrel(E;t)dE =−1 2πZw(E) tr Q(E;t)dE, (18) exists Mellin dilation invariance Rw(t0) = R(1+z)−1w◦D1+z(te),D1+z:E7→ E 1 + z,1 + z=a(t0) a(te)=R(te) R(t0).(19) Thus “expansion” and “resolution enhancement” merely gauge restatements under same mother scale, not changing any windowed readout’s ontological meaning. Corollary 5.2 (Redshift–Delay Pairing).If load λrelated to gauge asuch that ∂λQ⪰0 and dλ induces da/a =dλ ·η(λ)gain, then T1’s ∆wand T3’s 1 + zunify: 1 + zw= ∆w. 6 Discrete Dynamics Interface: RCA Front Slope and Group Delay Under EBOC static block encoding, let one-dimensional reversible cellular automaton A’s spacetime diagram as two-dimensional SFT. Let its front slope cRCA be average velocity of coherent defect propagation. WSIG scale embedding yields energy axis–lattice scale isomorphism, making cRCA =dx dt eff ∝1 Ntr Q−1,(20) i.e., group delay trace reciprocal determines RCA’s effective advancement rate; passive load increases tr Q, thus lowering cRCA, manifesting as discrete dynamics “deceleration” and continuous scale “redshift” isomorphism. Foundational properties of reversibility, information velocity, and light cone bounds see classical CA theory surveys. 7 Error Theory and “Singularity Non-Increasing/Poles = Principal Scales” Proposition 7.1 (Finite-Order NPE Closure).If window whas m-order smoothness and β-order decay, under Landau density condition, Rdisc ∗w−Rcont ∗w≤C(m, β) X ℓ=0 |bw(ℓfs)|+ m X j=1 |B2j| (2j)!|∂(2j−1)w|L1+Z|E|>Emax |w(E)ρrel(E)|dE!, (21) 5 where B2jBernoulli constants; if ρrel contains only finite poles and window operation does not excite new singularities, error terms do not elevate singularity, principal scale determined by poles. 8 Multi-Port Structure and Statistical Characterization Let {τj(E)}N j=1 be eigenvalues of Q(E) (proper delay times), then Φw=1 2π N X j=1 Zw(E)τj(E)dE, ∂λτj(E)≥0 a.e. (22) Statistical level distributions and moment generating functions in chaotic cavities and non-ideal coupling given by random matrix theory, supporting “total group delay = eigentime sum” decomposition and its extreme fluctuation laws. 9 Counterpoint with Communication Capacity Intuition (Pure Theoretical Restatement) Using µQ’s Carleson constraint to represent “finite throughput”, corresponding to finite total group delay per unit resource block. T1 gives “load ↑ ⇒ delay readout ↑”; T2 upperbounds total readout as “group delay flux ×effective bandwidth constant”. If using scale time constant Tw:= Φw/|w|L1as apparent period elongation, when Φw[λ0]>0 have Tw[λ1] Tw[λ0]= ∆w(λ1, λ0) = 1 + zw.(23) General case, only Tw[λ1]− Tw[λ0]≥0 (equivalent to Φwnon-decreasing) holds, giving scale equivalence “redshift = effective delay stretching”, no probabilistic assumptions or experimental vocabulary needed. 10 Gauge Group and Invariance Define Mellin–Heisenberg gauge group Gaction on window and readout g·w(E) = χ w(χE), g ·ρrel(E) = ρrel(E/χ),(24) then windowed readout under Trinity mother scale satisfies ⟨ρrel⟩g·w=⟨g·ρrel⟩w.(25) Scale gauge a=R−1is one-dimensional subgroup of G, equivalent to T3. 6 Appendix A: Theorem T2 Proof Outline Start with Berezin compression Kw,h and self-adjoint symbol F(E) = (2π)−1Q(E). Carleson embedding yields tr(Kw,hF)≤Cth |Kw,h|Car ≲Cth Cframe Bw.(26) where |Kw,h|Car controlled by window family frame density and bwpacking degree; then use finite-order Poisson dealiasing, Euler–Maclaurin absorb boundary layer and tail term, Φw=⟨F⟩w= tr(Kw,hF) + O(εNPE),Φw≤Cth Cframe Bw1 + εNPE.(27) When wbandlimited and window family nearly tight, |Kw,h|Car upper/lower bounds merge, bound can be tight. Appendix B: RCA’s Constructive Scale In one-dimensional reversible CA’s local rule and SFT encoding framework, introduce energy axis scale map Θ : Z2→R, making each characteristic direction’s discrete velocity satisfy with scale group delay cRCA(θ) = ∆x ∆t∝1 Ntr QΘ(θ)−1.(28) (If and only if choosing unified space/time unit gauge, proportionality constant can be normalized.) Passivity makes tr Qmonotonically increasing, thus cRCA monotonically decreasing; this decrease isomorphic with T1’s ∆w≥1, giving “non-acceleration–nonincrease” type reversible dynamics readout. Foundational theorems on reversibility, information light cone, and velocity upper bounds see cited sources. Appendix C: NPE Error Budget (Finite-Order Recipe) 1. Poisson Dealiasing: Choose sampling rate fssatisfying Landau density condition, control Ealias ≤X ℓ=0 bw(·+ℓfs).(29) 2. Euler–Maclaurin Boundary Layer (morder): Take msuch that ∂(2m)wand ∂(2m)ρrel integrable, error E(m) BL ≲ m X j=1 |B2j| (2j)! |∂(2j−1)w|L1|∂(2j−1)ρrel|L∞(supp w).(30) 3. Tail Term: Energy domain truncation Emax,E(β) tail ≤ |w·1|E|>Emax |L1|ρrel|L∞, controlled by w’s β-order decay to required precision. 7 Terminology Cards (Fixed)  Card A (Scale Identity): φ′(E) π=ρrel(E) = −1 2πtr Q(E); readouts uniformly executed on mother scale.  Card B (Finite-Order EM and “Poles = Principal Scales”): Use only finite-order Euler–Maclaurin and Poisson; error theory follows “singularity nonincreasing/poles = principal scales”. References [1] A. Pushnitski, arXiv:1006.0639v1 [math.SP], 2010. 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