EBOC--WSIG Unified Axiomatization:\\ Windowed Group Delay Definition of Speed of Light and Fusion of ``Spacetime/Time/Space'' Axioms

EBOC–WSIG Unified Axiomatization: Windowed Group Delay Definition of Speed of Light and Fusion of “Spacetime/Time/Space” Axioms Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 3.17 Abstract Under the sole premises of causal front (speed of light constant c) and spacetime geometry (M, g), we establish an acyclic system from geometry–causality to measurability–calibration: taking the trinity scale of scattering phase derivative φ′(E), relative state density ρrel(E), and Wigner–Smith group delay trace 1 2tr Q(E) as the unique master scale of the energy axis; implementing non-asymptotic measurement via windowed readouts with Nyquist–Poisson–Euler–Maclaurin (NPE) three-term error closure; defining the four bridge constants ℏ, e, kB, G within the system; accordingly packaging derived quantities and providing triple equivalence of force (worldline non-geodesicity = momentum flux divergence = minimal coupling to curvature/connection). The speed of light constant is not redefined; the following provides windowed group delay readout as metrological equivalence to the constant cin axiom A1, forming fourfold alignment with the trinity scale, causal front and information light cone, and metrological realization. This system guarantees existence, uniqueness, and window/kernel independence, with directed acyclic graph (DAG) of units–calibration eliminating circular dependencies. Keywords: Windowed group delay; Wigner-Smith delay matrix; Trinity scale; NPE error ledger; Four bridge constants; Reversible cellular automata; Metrological equivalence; Directed acyclic graph MSC 2020: 81U05; 47A40; 94A12; 83C05; 37B15 Contents 1 Notation and Implementation Discipline Terminology and abbreviations:WSIG: Refers to “operational protocol and metrological correspondence based on windowed phase/group delay readout with NPE error ledger”. EBOC: Refers to geometric–dynamical framework of “static blocks + leaf decomposition + suspension flow” (realized in text as leaf decomposition of 2D subshift 1 X⊂ AZ2with induced metric h). RCA: Reversible cellular automaton, referring to discrete reversible evolution system Y⊂ BZwith bijective sliding block code F:Y→Y. Notation distinction: Qalways denotes Wigner–Smith matrix −iS†∂ES(dimension E−1); Qdenotes “charge” in minimal coupling; these two have different meanings and dimensions and must not be confused. Phase and group delay: Let S(E)∈U(N), Φ(E) := Arg det S(E), φ(E) := 1 2Φ(E). Define Q(E) := −i S(E)†dS dE , Φ′(E) = tr Q(E), φ′(E) = 1 2tr Q(E). Phase branch and derivative convention: Φ(E) = Arg det S(E) takes continuous phase branch; Φ′(E) is understood as distributional derivative, equal to tr Q(E) almost everywhere on absolutely continuous spectrum. Trinity formula (almost everywhere on absolutely continuous spectrum): φ′(E) = π ρrel(E) = 1 2tr Q(E). Group delay dimension (channel-averaged/single-channel): Channel-averaged group delay ¯τWS(E) := ℏ1 Ntr Q(E) (time); single-channel (eigenphase θα)τg,α(E) := ℏ∂Eθα(E). Relation to semi-determinant phase φ(E) := 1 2Arg det S(E): ¯τWS(E) = 2ℏ Nφ′(E). Frequency–energy map and group refractive index: Let E=ℏω, in isotropic, static, local linear media take dispersion relation k(ω) = n(ω)ω/c. Define group velocity and group refractive index vg(ω) := ∂ωk(ω)−1=dω dk , ng(ω) := c vg(ω)=n(ω) + ω ∂ωn(ω). In general, ng(ℓ, E) denotes local group refractive index along path γat arc length coordinate ℓunder energy E, substituting ω=E/ℏ; if medium is anisotropic or non-isotropic, ngshould be understood as projected group refractive index (equivalent slowness) along ray direction. Fourier convention:b f(τ) := ZR f(E)e−iτE dE, f(E) = 1 2πZRb f(τ)eiτE dτ. Window and kernel normalization (with dimension): Take dimensionless prototypes W, K ∈L1(R) satisfying RW=RK= 1, define wR(E) := 2π ∆w WE ∆w, κ(E) := 1 ∆κ KE ∆κ, then 1 2πRwR(E)dE = 1, Rκ(E)dE = 1, and wR, κ both have dimension E−1. From normalization, for constant cwe have [κ⋆c]≡c,⟨c⟩w,κ;E0=c. Regularity assumptions and approximate identity:wR∈W1,1(R) (or compactly supported, or band-limited; these two cannot coexist), and wR, w′ R∈L1, wR(±∞) = 0; κ∈L1,Rκ= 1, and in distributional sense [κ ⋆ ∂EΦ] = ∂E[κ ⋆ Φ]. Approximate identity condition: (wR)Rsatisfies supR∥wR∥L1<∞,∥wR∥L∞→0, and bwR S′ −−→ 2πδ0(R→ ∞). Note: Strict band-limitedness and compact support cannot hold simultaneously (Paley–Wiener); this paper uses effective support in frequency domain to provide “alias-free” verifiable sufficient criterion, with remaining leakage uniformly counted in Etail. 2 Convolution definition: [κ⋆f](E) := RRκ(E−E′)f(E′)dE′. Weighted average notation (unified): For any integrable function fand central energy E0, define ⟨f⟩w,κ;E0:= 1 2πZR wR(E−E0) [κ⋆f](E)dE . If unspecified, default is E0= 0. For constant cand Rκ= 1, we have ⟨c⟩w,κ;E0≡c. Notation:wR,E0(E) := wR(E−E0). Target spectral density:ρ⋆∈ {ρ, ρrel} (absolute spectral density and relative state density respectively). Sampling and Nyquist condition: Let f:= wR,E0·(κ ⋆ ρ⋆), En:= E0+n∆E, b f(τ) := Rf(E)e−iτEdE.Poisson summation (with offset, standard form): X n∈Z f(E0+n∆E) = 1 ∆EX m∈Z ei2πm ∆EE0b f2πm ∆E. Alias-free condition remains: supp b f⊂(−π/∆E, π/∆E). Effective support width (revised): Let b f(τ) = bwR(τ)∗bκ(τ)·bρ⋆(τ), define Ωeff := inf nΩ>0 : supp b f⊂[−Ω,Ω]o≤Ωw+ min Ωκ,Ωρ. The verifiable sufficient condition for “alias-free” is unified as ∆E<π Ωeff ,common upper bound substitute: ∆E<π Ωw+ min(Ωκ,Ωρ). NPE error ledger: Any windowed readout decomposes as Ealias +EEM +Etail. Under band-limitedness and Nyquist, Ealias = 0. Only finite-order Euler–Maclaurin (EM) exchange is allowed to ensure singularity set and pole order do not increase. The above “three-term error” is purely mathematical analytic remainder and upper bounds (including Poisson summation alias term, finite-order EM remainder, and bandwidth tail term), not involving experimental noise or instrument error. 2 Axiom Family A1 (Causal spacetime): (M, g) is spacetime manifold, causal front is calibrated by ds2= 0 and constant c. A2 (Leaf decomposition, local/global conditions): In the considered spacetime domain satisfying global hyperbolicity (existence of Cauchy leaf) or within causally convex domain, can select time function tsuch that spacelike hypersurfaces Σtform smooth leaf decomposition, inducing 3-metric h. When lacking global hyperbolicity, statements about “leaf/radar” below apply only within that causally convex domain. A3 (Energy scale): On absolutely continuous spectrum dµφ(E) = φ′(E) πdE = ρrel(E)dE =1 2πtr Q(E)dE. A4 (Realizability): Any instrument readout is equivalent to “window Ö kernel” weighting of spectral density, subject to NPE three-term error closure. A5 (Probability and pointer): Born probability is equivalent to minimum relative entropy (I-projection); given window operator Wself-adjoint and W ≥ 0, pointer 3 basis takes Ky-Fan maximum principle: given window operator Wand dimension k, let P be projection of rank k, then max rankP=kTr(PW) = k X i=1 λ↓ i(W); pointer subspace is spanned by top keigenvectors of W. A6 (Exchange discipline): All discrete–continuous exchanges use only finite-order EM; singularity set is preserved. A7 (Operational definition of time and space): Time is joint calibration of causal partial order and earliest detectable mutual information; space is intrinsic distance within spacelike leaf Σtby metric h;radar calibration is used only for correspondence with cin A1, not as original definition of length. 3 Trinity Theorem Theorem 2.1 (Phase–density–delay trinity) For self-adjoint scattering pair (H, H0) satisfying Birman–Kre˘ın formula applicability conditions, on absolutely continuous spectrum almost everywhere φ′(E) = π ρrel(E) = 1 2tr Q(E). Proof. Weyl–Titchmarsh–Herglotz theorem gives spectral density representation ρ(E) = π−1ℑm(E+i0), where m(z) is corresponding channel’s Weyl–Titchmarsh–Herglotz function, e.g. m(z) = ⟨ψ, (H−z)−1ψ⟩boundary value (ψappropriate boundary vector/channel choice); its non-negative imaginary part satisfies ρ(E) = π−1ℑm(E+i0) with Radon– Nikodym density of absolutely continuous spectrum. Birman–Kre˘ın formula gives det S(E) = exp(−2πi ξ(E)), ρrel(E) = −ξ′(E). From Q(E) = −i S†dS dE and Jacobi identity get ∂EArg det S(E) = tr Q(E). Combining yields conclusion. Corollary 2.2 (Threshold and singularity preservation) Under band-limitedness+Nyquist and finite-order EM, windowing does not introduce new singularities or pole order increase; thresholds and resonances are determined by master scale, can distinguish physical structure from numerical artifact via “window-invariant singularities”. 4 Windowed Readout and NPE Error Closure Theorem 3.1 (Windowed readout identity) For band-limited window wRand kernel κ, windowed readout of any observable is represented as Obs = 1 2πZR wR(E−E0) [κ⋆ρ⋆](E)dE +Ealias +EEM +Etail =⟨ρ⋆⟩w,κ;E0+Ealias +EEM +Etail. where ρ⋆takes ρ(absolute spectral density) or ρrel (relative state density). Proof. First introduce weighted average notation ⟨f⟩w,κ;E0:= 1 2πZR wR(E−E0) [κ ⋆ f](E)dE. Case A (ρrel): From trinity ρrel =1 2πtr Q=1 πφ′and φ=1 2Φ, ⟨ρrel⟩w,κ;E0=1 2πZwR(E−E0) [κ⋆1 πφ′](E)dE =−1 2π2Zw′ R(E−E0) [κ⋆φ](E)dE =−1 4π2Zw′ R(E−E0) [κ⋆Φ](E)dE, 4 using [κ⋆∂EΦ] = ∂E[κ ⋆ Φ] and integration by parts (understood distributionally), with Φ(E) := Arg det S(E) taking continuous phase branch from § 0. Case B (ρ): From Weyl–Titchmarsh–Herglotz representation ρ(E) = π−1ℑm(E+i0), directly ⟨ρ⟩w,κ;E0=1 2πZwR(E−E0) [κ⋆ρ](E)dE. Subsequently via Poisson summation (giving Ealias), finite-order Euler–Maclaurin (giving EEM) and bandwidth truncation (giving Etail) obtain three-term error closure. Non-asymptotic upper bounds: There exist constants CEM(k) and Ctail(R) such that |EEM| ≤ CEM(k) sup |∂2k E(·)|,|Etail| ≤ Ctail(R)|·|L1(E /∈[−Ω,Ω]), monotonically convergent with kand bandwidth R. 5 Windowed Group Delay Readout of Speed of Light and Fourfold Alignment 5.1 Metrological Protocol (Readout) Windowed group delay readout is defined as T[wR, κ;L;E0] := ℏ 2πZR wR(E−E0)hκ ⋆ 1 Ntr QLi(E)dE =ℏD1 Ntr QLEw,κ;E0 , where Lis intrinsic distance within spacelike leaf Σtbetween endpoints by metric hij,1 Ntr QLis per-channel average group delay of link. Windowed group delay readout of speed of light constant (metrological protocol) is denoted cread := lim energy window width ↑ L T[wR, κ;L;E0], where “energy window width↑” is equivalent to “τdomain bandwidth Ωw(R)↓0”, i.e. bwR⇒2πδ, while (2π)−1RwR= 1 always holds. This limit is independent of E0(see § 4.2). And is equivalent to constant cin A1; does not constitute redefinition of c. 5.2 Existence, Uniqueness, and Window/Kernel Independence Proposition 4.2(a) (Vacuum link–identity): If SL(E) = eiEL/(ℏc)IN, then q(E) := 1 Ntr QL(E)≡L ℏcis constant, thus T[wR, κ;L;E0] = ℏ 2πZwR(E−E0) [κ⋆q](E)dE =ℏ 2π L ℏcZwR(E−E0)dE =L c, independent of (wR, κ), bandwidth Rand E0. Proposition 4.2(b) (General link–limit form, rigorous version): Let (wR)R be approximate identity from § 0, satisfying 1 2πRwR= 1, supR|wR|L1<∞,|wR|L∞→0 and bwR S′ −−→ 2πδ0; take κ∈L1and Rκ= 1. Let q(E) := 1 Ntr QL(E) = q0+δq(E), δq ∈L1(R). 5 Then limit exists and lim R→∞ T[wR, κ;L;E0] = ℏq0+1 2πZR δq(E)dE, independent of (wR, κ, E0). 5.3 Equivalence with Trinity Scale From 1 2πtr Q=ρrel =φ′/π get T[wR, κ;L;E0] = ℏD1 Ntr QEw,κ;E0 =2ℏ N⟨φ′(E)⟩w,κ;E0=ℏ N⟨Φ′(E)⟩w,κ;E0. For vacuum link Φ′(E) = NL ℏcis constant, so T=L/c independent of E0. 5.4 Equivalence with Causal Front Under Kramers–Kronig analyticity and retarded Green’s function support conditions, earliest nonzero response front velocity is c.For vacuum pure delay link (SL(E) = eiEL/(ℏc)IN), from 4.1–4.3 know T=L/c; if measured T=L/c, then either violates NPE closure or violates front causality. For general media/geometry links, front velocity still c. Under conditions of pure transmission without significant reflection/standing waves and satisfying geometric optics (WKB) approximation, window–kernel weighted group delay can be written as T[wR, κ;L;E0] = 1 cZγng(ℓ, E)w,κ;E0dℓ . In general only strict identity T[wR, κ;L;E0] = ℏD1 Ntr QLEw,κ;E0 . 5.5 Equivalence with Information Light Cone Under conditions that receiver contains only independent noise before threshold and no pre-shared information-bearing variable, supremum of speed of first-detectable mutual information equals c. This supremum is jointly constrained by front velocity and trinity scale. 5.6 Mutual Inverse with Metrological Realization Length from time: Take cas constant, use ℓ=c∆tto define length unit; length from delay: Use Tto back-calculate L. Strict mutual inverse for vacuum link, consistent within NPE bounds under weak dispersion. 6 6 In-System Definition of Four Bridge Constants 6.1 ℏ: Weyl–Heisenberg Central Charge (Physical Scale of phase– time 2-cocycle) WSIG definition (operational): Let U(∆t) be physical time translation, V(∆E) be energy modulation unitary representation. Their projective commutation phase satisfies V(∆E)U(∆t) = expi ℏ∆E∆tU(∆t)V(∆E). ℏis defined as the unique scale lifting geometric 2-cocycle exp(iτσ) to physical quantity expi∆E∆t/ℏ; and anchored by critical lattice ∆t∆ω= 2πto E=ℏω. Stone–von Neumann uniqueness theorem guarantees this calibration is unique in canonical class. EBOC definition (structural): Under EBOC’s time–frequency translation action, ℏis central charge of Weyl–Heisenberg group, mapping product of phase increment on static block and time leaf parameter to energy scale: ∆Φ = ∆E∆t/ℏ. Its value is uniquely fixed by WSIG calibration, thus covariant in block–leaf reading. RCA definition (operational–metrological): Let one-step evolution U:= F= exp(−iHCA∆t/ℏ), spatial translation σa= exp(iKa). Floquet eigenvalue Uψ =e−iω∆tψ gives quasi-energy frequency ω, accordingly readout E:= ℏω.ℏis uniquely determined by scale from “step phase ω∆t” to energy scale. Meaning (three-side comparison):ℏis central charge of “phase–time 2-cocycle”: in WSIG lifts geometric phase ∆E∆tto measurable phase; in EBOC guarantees covariance of leaf advancement to E=ℏω; in RCA converts eigenphase of discrete stepping to energy scale, achieving unified calibration E↔ωon three sides. 6.2 e: Minimal Coupling Quantum of U(1) Gauge Holonomy WSIG definition (operational): For closed loop Crealized by minimal elementary U(1) carrier (e.g. single electron, not Cooper pair), interference phase ∆ϕ(C) = q ℏΦB(C) (mod 2π). Let Φ0,B be minimal positive magnetic flux of elementary carrier loop family realizing first phase recurrence ∆ϕ= 2π, define e:= 2πℏ Φ0,B . 6.3 kB: Scale from Information Temperature to Thermal Temperature WSIG definition (operational): Under maximum entropy (minimum KL/I-projection) with energy constraint, optimal distribution p⋆∝exp(−βE). Align natural parameter βwith thermal temperature Tby T:= 1 kBβ 7 defining kBas scale mapping “nats per energy” to “per kelvin”. Spectral slope readout satisfies ∂ωloghI(ω) ω3i=−ℏ kBT·eℏω/(kBT) eℏω/(kBT)−1, in Wien limit ℏω≫kBTapproximates to −ℏ/(kBT). Accordingly jointly anchor kB by “exact formula + Wien limit approximation” with Carnot temperature, guaranteeing uniqueness and acyclicity. 6.4 G: Geometric Coupling Coefficient of Curvature–Energy Density WSIG definition (operational): Use windowed group delay and phase tomography to invert geometric curvature (or weak field potential), use energy window to read energy– momentum flux of Tµν. Define Gto be unique scaling coefficient making Gµν =8πG c4Tµν hold in measured domain; Newtonian limit ∇2ϕ= 4πGρE/c2as consistency constraint. Band-limitedness+Nyquist and finite-order EM ensure window/kernel independence of both “delay/deflection” and “energy flow” sides. 6.5 Acyclicity and Independence Proposition 5.1 (DAG): Unit–calibration dependency graph is directed acyclic graph of L0→L1→L2→L3: L0: (M, g), c and phase/count readouts; L1: Observable ratio classes (Φ′,tr Q, ρrel etc.); L2:ℏ(time–frequency anchor), e(holonomy anchor), kB(temperature and spectral slope double anchor), G(curvature–energy flow correspondence anchor); L3: All physical quantities in absolute scale. Calibration anchors of four constants are mutually independent, and none depend on undetermined constants themselves; any hypothetical loop would contradict trinity scale, NPE error closure, or corresponding double-anchor verification, hence excluded. 7 Derived Physical Quantities and Semantic Equivalence in EBOC and RCA We set EBOC as 2D subshift X⊂ AZ2, coordinates denoted (i, t)∈Z×Zor via suspension flow (i, t)∈Z×R; each time leaf Σt:= {(i, t) : i∈Z}equipped with induced metric h. We set reversible cellular automaton (RCA) as 1D subshift Y⊂ BZwith bijective sliding block code F:Y→Y, local radius r, lattice spacing a, reference time slot ∆t. Define maximal propagation cone (light cone) Λ := {(i, n)∈Z2:|i| ≤ rn}, and via suspension flow construct operator spectrum U:= exp(−iHCA∆t/ℏ) and shift spectrum σa:= exp(iKa). This section provides semantic equivalence dictionary and readout formulas for physical quantities in EBOC ↔RCA, all following unified master scale φ′(E)⇐⇒ 1 2πtr Q(E)⇐⇒ ρrel(E). 8 7.1 Time–Space–Velocity–Acceleration Time  WSIG: Time scale given by windowed group delay T[wR, κ;L;E0].  EBOC: Parameter tof causal partial order.  RCA: Step index nand t=n∆t. Space and length  WSIG: Length unit given by cT(radar method).  EBOC:ℓ(γ) = Rphij ˙γi˙γjds.  RCA: Graph metric dCA(i, j) := |i−j|a, radar method ℓCA(γ) := Ngate(γ)a 2(minimum round-trip gate number Ngate(γ)); calibrated via cCA =cconsistent with EBOC. Velocity and upper bound  WSIG: Velocity upper bound ccalibrated by T[wR, κ;L;E0].  EBOC:v=dℓ/dt,|v| ≤ c.  RCA:vCA := limn→∞ |i(n)−i(0)|a n∆t≤ra ∆t=c. 7.2 Wave–Phase–Dispersion and Group Parameters Plane mode  WSIG:φ′(E) = 1 2tr Q(E), dispersion given by windowed phase readout.  EBOC: Wave mode expi(k·x−ωt), dispersion relation ω=ω(k).  RCA: Koopman mode ψk,ω(j, n) = expi(kaj −ωn∆t),k∈[−π/a, π/a] (first Brillouin zone); local rule linearization gives discrete ω=ω(k). 7.3 Energy–Momentum–Mass–Action Energy and momentum  WSIG:E=ℏω,p=ℏkgiven by windowed phase readout.  EBOC:E=ℏω,p=ℏk.  RCA: Suspension generators U= exp(−iHCA∆t/ℏ), shift generator σa= exp(iKa) give E=ℏω,p=ℏk. 9