WSIG--EBOC Unified Theory of Spacetime/ Time/ Space

WSIG–EBOC Unified Theory of Spacetime/Time/Space Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.45 Abstract Within the WSIG (Windowed Scattering & Information Geometry) and EBOC (Eternal-Block Observer-Computing) frameworks, we establish an operational spacetime theory based on windowed scattering: taking the phase derivative– spectral shift density–Wigner–Smith group delay triple equivalence as the metrological primitive; using the Kramers–Kronig causality–analyticity (restricted to stable LTI) and the light cone support of wave equation retarded Green’s function (time-domain support, applicable to LTV), we provide an upper bound in terms of front-propagating optical metric: causal front does not exceed c; and the equality holds if and only if the front-detectability condition ( § 5) is satisfied. Using the first-detection time of threshold mutual information to establish the information light cone bound: in vacuum or static media (LTI), cinfo := lim δ↓0sup D Tδ≤c, with equality cinfo =cif and only if the front-detectability ( § 5) holds; where D=Dfront (link normalization, th= 0) or D=Dsys := Dfront +c th(system normalization, th>0). For LTV, only Tδ≥tmin is obtained. We provide operational definitions for time and space, and characterize spacetime as a four-tuple (E,⪯, g, µφ): where ⪯is induced by light cone support, gis the (media-inclusive) optical Lorentzian metric, µφ= (φ′/π)dx is the phase density measure given by de Branges kernel diagonal. The core theorems prove the four-way equivalence of “phase slope = group delay = spectral shift density = SI realization”, and provide non-asymptotic detection bounds under the Nyquist–Poisson–Euler– Maclaurin (NPE) error ledger. The theory is compatible with the discrete light cone of the CHL theorem for reversible cellular automata (RCA), and establishes isomorphisms with density thresholds (Landau, Wexler–Raz, Balian–Low) for sampling/interpolation/frames. Keywords: Wigner-Smith group delay; Birman-Kre˘ın spectral shift; retarded Green’s function; optical metric; information light cone; NPE error ledger; de Branges phase density; reversible cellular automata; Landau density; Balian-Low theorem MSC 2020: 81U05; 47A40; 83C05; 94A12; 42C15; 37B15 1 Contents 1 Notation and Preliminaries  Scattering matrix S(E)∈U(N). Wigner–Smith delay matrix defined as Q(E) := −i S(E)†dS dE (E), with tr Q(E) having dimension 1/energy; physical group delay is τWS(E) := ℏtr Q(E), with dimension of time. This definition traces back to Smith’s characterization of the “lifetime matrix”. Standard literature defines Wigner–Smith delay as “derivative of phase with respect to energy” corresponding to tr Q, with τ=ℏtrQhaving time dimension.  Birman–Kre˘ın (BK) formula: If det S(E) = e−2πi ξ(E), then tr Q(E) = −2π ξ′(E).  KK–causality (LTI restricted): For stable LTI systems (impulse response hsys(t)), strict causality ⇒frequency response analytic in upper half-plane; and under additional conditions (hsys ∈L1(R), H(ω) polynomially bounded growth, no upper half-plane poles, etc.), real and imaginary parts satisfy Kramers–Kronig dispersion, and strict causality can be recovered (analyticity ⇒causality). LTV/nonstationary cases do not apply this frequency-domain equivalence; this paper uses only the timedomain support formulation of Gret (see § 5).  Retarded Green’s function: In 3D vacuum unbounded domain, the solution to the scalar wave equation □Gret =δis Gret(t, r) = δ(t− |r|/c)/(4π|r|), with support exactly at t=r/c;Maxwell’s time-domain dyadic kernel is obtained from this scalar kernel via tensor–differential operators (containing δand its derivatives), with support likewise only on the light cone.In bounded domains/media/dispersion,precursors/tails generally appear; see § 0 “front decomposition (definition)”.  Optical metric (Gordon): Front time satisfies tmin ≥dfront c, where dfront is defined by the 3D Fermat metric of high-frequency limit refractive index n∞. The equality tmin =dfront/c holds if and only if the frontdetectability condition ( § 5) is satisfied; 3D vacuum pure propagation is a special case. Many passive media satisfy “vacuumization” at high frequencies n(x, ω)→1 (thus n∞= 1), giving dfront =|A−B|; in this case, in experimental coordinates one can unconditionally only assert tmin ≥ |A−B|/c, and only when the above equality condition holds does one recover “front velocity = c”. (Group velocity can differ from c—Sommerfeld–Brillouin precursor). For modern reviews and extensions, see Leonhardt–Philbin and subsequent work.  Fast/slow light and information velocity: Experimental and informationtheoretic definitions show that detectable information velocity does not exceed c.  NPE error ledger: Nyquist sampling theorem and aliasing condition (Shannon/Nyquist); Poisson summation formula (NIST DLMF § 1.8(iv)); Euler–Maclaurin bounded remainder (DLMF § 2.10). 2  de Branges kernel diagonal: For the space H(E) of a Hermite–Biehler function E, we have K(x, x) = 1 πφ′(x)|E(x)|2, where φis the phase function. (Notation unification) This paper uses Eto denote energy (also as real frequency variable); in de Branges sections, xdenotes the same real axis variable and is identified with E, written as x≡E. Thus µφ= (φ′(E)/π)dE, consistent with the normalization 1 2πRwR(E)dE = 1 in § 2.  Weyl–Heisenberg representation and uniqueness: Stone–von Neumann theorem and foundations of coherent/time-frequency frameworks, see Folland.  Windowed convolution and averaging notation: Take energy window wR∈ L1(R) and unit integral kernel η∈L1(R). Normalization 1 2πZR wR(E)dE = 1,ZR η(ν)dν = 1. Define convolution and windowed average [η ⋆ f](E) := ZR η(ν)f(E−ν)dν, fw,η := 1 2πZR wR(E) [η ⋆ f](E)dE. For vacuum pure delay link SL(E) = eiEL/(ℏc), from tr QL≡L/(ℏc) and the above normalization, we obtain ℏ⟨tr QL⟩w,η =L/c (see § 4). (Symbol clarification) This paper uses ηfor unit integral kernel (for windowed readout), hsys(t) for system impulse response (LTI filter), with th:= inf{t:hsys(t)= 0}as its onset delay.  Distributions and generalized function notation:δ(·) denotes the Dirac δ distribution; Θ(t) is the Heaviside step function, Θ(t) = 0 (t < 0), Θ(t) = 1 (t > 0) (taking Θ(0) = 1 2does not affect results).  Front decomposition (definition): In vacuum or static media (LTI) and satisfying front assumption (F), the retarded Green’s function in the distributional sense can be written as Gret(t;x, y) = m X k=0 Kk(x, y)δ(k) t−dfront(x,y) c+ Θ t−dfront(x,y) cg(t;x, y), where m < ∞,Kk(x, y) are amplitude coefficients of front singularities, gis the tail kernel (locally integrable). We say “front-detectability (at the distributional level)” if and only if there exists some k≥0such that Kk(x, y)= 0; Maxwell case allows k≥1.  Scattering background and energy representation: Let Hbe a Hilbert space, H0, H be self-adjoint operators (free/full Hamiltonian). Assume wave operators W±:= slimt→±∞ eitHe−itH0exist and are complete, then scattering operator S:= W∗ +W−fiberizes in energy representation to S(E)∈U(N). Wigner–Smith delay matrix is defined as Q(E) := −i S(E)†∂ES(E), consistent with notation in § 0.  Trace notation: This paper uses tr for finite-dimensional matrix trace (e.g. S(E)∈ U(N)), reserving Tr for trace-class operators on Hilbert spaces (e.g. Twφin § 6). 3 2 Axioms (WSIG–EBOC Spacetime) Let S=E,⪯, g, µφ;H, H0, S(·); W,H. Hierarchy clarification: This paper calls (E,⪯, g, µφ) the spacetime four-tuple (ontological layer), while (H, H0, S(·); W,H) are realization data (realization layer, used to give windowed scattering readouts and proof chain); these two must not be confused. We call SaWSIG–EBOC spacetime if: (A1) Events and causal order:Eis the event set; if the wave equation’s Gret has nonzero support on the pair (e1, e2) (i.e. Gret(e2, e1)= 0), then set e1⪯e2. For discrete systems, take RCA neighborhood propagation bound (see § 9). (A2) Representation and observability:His a Hilbert space; (Uτ, Vσ) is a projective unitary representation of Weyl–Heisenberg, supporting windowed readouts. (A3) Phase–density–delay dictionary: There exists window Wsuch that windowed readouts of ∂Earg det S= tr Q=−2π ξ′(E) hold. (A4) Causality–analyticity consistency: In the stable LTI case, strict causality ⇒frequency response analytic in upper half-plane; and under additional conditions (hsys ∈L1(R), H(ω) polynomially bounded growth, no upper half-plane poles, etc.), real and imaginary parts satisfy Kramers–Kronig dispersion, and strict causality can be recovered (analyticity ⇒causality). LTV/nonstationary cases do not use this frequency-domain equivalence, only the time-domain support formulation of Gret(t, τ) (see § 5). In time-invariant/static media, we only assert tmin ≥Dfront c, Dfront =(L, vacuum, dfront(x, y),media/inhomogeneous/bounded domain. The equality tmin =dfront/c holds if and only if front-detectability ( § 5) is satisfied; 3D vacuum pure propagation is a special case. If cascaded with a strictly causal LTI filter (impulse response hsys(t), onset delay th:= inf{t:hsys(t)= 0}), then tmin ≥Dfront c+th, and only when front-detectability ( § 5) is satisfied and there is no systematic cancellation at the front, do we have tmin =Dfront c+th. (A5) Front–information consistency: In vacuum or static media (LTI), the first-detection time Tδof threshold mutual information satisfies cinfo := lim δ↓0sup D Tδ≤c, where Dis the front optical path in the normalization used. The equality cinfo =c holds if and only if front-detectability is satisfied; here “front-detectability” means: according to § 0 front decomposition Gret(t;x, y) = Pm k=0 Kk(x, y)δ(k) t−D c+Θ(·)g, there exists some k≥0such that Kk(x, y)= 0, or the measurement chain contains 4 aDirac pass-through component (impulse response contains δ(t)), or there exist tn↓D/c and unit-energy short pulse ψsuch that RGret(tn, τ;x, y)ψ(τ)dτ = 0. (i) Link normalization (th= 0):D=Dfront.(ii) System normalization (th>0): D=Dsys := Dfront +c th.(LTV supplement): For time-varying systems, only assert Tδ≥tmin (defined by time-domain support of Gret(t, τ)). (A6) Sampling–error closure: Readout error is given by NPE three-term decomposition with non-asymptotic upper bounds. (A7) SI alignment:ctakes SI fixed value; “length from delay” and “time from length” are mutually inverse realizations ( § 4, § 12). (A8) Phase density geometry: de Branges kernel diagonal K(x, x) = 1 πφ′(x)|E(x)|2 induces µφ:= (φ′/π)dx as windowed calibration. 3 Operational Definitions: Time and Space Definition 2.1 (Time) Take window wRand unit integral kernel η. For link scattering S(E), define windowed group delay readout T[wR, η] := ℏ1 2πZR wR(E)η ⋆ tr Q(E)dE, where Q:= −iS†dS dE , and tr Qis equivalent to ∂Earg det Sand −2πξ′(E). Normalization: Convention (2π)−1RRwR(E)dE = 1 and RRη(E)dE = 1 (unit integral kernel), giving T=L/c on vacuum pure delay link. Verification: for SL(E) = eiEL/(ℏc)we have constant tr Q=L/(ℏc), substituting into the above and using the normalization immediately yields T=L/c; consistent with Theorem 4.1. For a vacuum link of length L, the defined time coordinate difference satisfies ∆t=L/c. Definition 2.2 (Space) In vacuum, define spatial distance by radar distance: d(A, B) := c 2Troundtrip(A→B→A). In media/inhomogeneous/bounded domains, take high-frequency limit refractive index n∞(x) = limω→∞ n(x, ω). Define 3D Fermat (optical) metric dsfront =n∞(x)|dx|, and accordingly define front optical path dfront(A, B) := inf γ:A→BZγ n∞(x)ds, then earliest reachable time satisfies tmin(A, B)≥dfront(A, B) c. Equality holds if and only if front-detectability ( § 5) is satisfied. For isotropic case with n∞≡1, dfront =|A−B|. The above is equivalent to the null geodesic description of 4D Gordon optical metric in the high-frequency limit, but for timing purposes the 3D optical path formulation is more direct. Group refractive index: For isotropic passive media, define ng(x, ω) := n(x, ω) + ω ∂ωn(x, ω) = c vg(x, ω), 5 where vgis the group velocity. Thus group time tg=Rγngds/c, phase time tϕ=Rγn ds/c. In general, there is no guarantee of the ordering between tgor tϕand tmin (in anomalous dispersion and gain/loss situations, one can have tg< tmin or tϕ< tmin). What is universal and consistent with causality is only tmin ≥dfront c, Tinfo ≥tmin, and only when front-detectability ( § 5) is satisfied do we have tmin =dfront/c, where Tinfo is the first arrival time of any detectable information (see § 5). Round-trip time: Troundtrip(A→B→A;wR, η) := T[wR, η;A→B] + T[wR, η;B→A]. If link and readout protocol are reciprocal (bidirectional symmetric), then d(A, B) = c 2Troundtrip(A→B→A). Front assumption (F): Medium is passive linear, isotropic, and high-frequency refractive index has finite limit n∞(x) := lim ω→∞n(x, ω)∈[1,∞). Accordingly define front optical metric gfront and front optical path dfront. If only upper/lower bounds for n∞can be given 1 ≤n∞(x)≤n∞(x)≤n∞(x)<∞, then corresponding front optical paths satisfy dfront ≤dfront ≤dfront.In general, one can only unconditionally assert tmin(A, B)≥dfront(A, B) c, while tmin(A, B)≤dfront(A, B) c holds only under additional conditions:tmin ≤dfront/c only when front-detectability ( § 5) holds; if cascaded with a strictly causal LTI filter and onset delay thhas an upper bound, then tmin ≥dfront c+th, and only when front-detectability ( § 5) is satisfied and there is no systematic cancellation at the front do we have tmin =dfront c+th≤dfront c+th. Definition 2.3 (Simultaneity slice) Select a reference worldline and round-trip protocol, let Σt:= {e∈ E :Troundtrip = 2t}, whose three-dimensional metric is induced by radar distance or gfront. 4 Structured Definition of Spacetime Definition 3.1 (WSIG–EBOC Spacetime) If there exists windowed scattering readout such that: (1) Metric–readout consistency: On vacuum link Lwe have T=L/c; media front optical path is consistent with front, i.e. dfront is the geodesic optical path minimum of 3D Fermat (optical) metric dsfront =n∞|dx|; 6 (2) Triple dictionary:∂Earg det S= tr Q=−2πξ′(E); (3) NPE detectability: Error aliasing/Poisson/EM remainder have global upper bounds; (4) Information–causality consistency (vacuum or static media [LTI]): Firstdetection time Tδof mutual information satisfies cinfo := lim δ↓0sup Dfront Tδ≤c, with equality cinfo =conly when front-detectability is satisfied (see A5, § 5). (Cascade filter supplement): If measurement chain contains strictly causal LTI filter with onset delay th>0, then all Dfront above are replaced by Dsys := Dfront +cth.(LTV supplement): For time-varying systems, only assert Tδ≥tmin (defined by time-domain support of Gret(t, τ)). Then the four-tuple (E,⪯, g, µφ) is a spacetime. 5 Main Equivalence Theorem (Phase–Delay–Spectral Shift–SI) Theorem 4.1 (Four-way equivalence) Let vacuum link Lhave scattering SL(E) = exp iEL/(ℏc). Then T[wR, η;L] = ℏ∂Earg det SLw,η =ℏtr QLw,η =−ℏ2πξ′(E)w,η =L c, and in the Nyquist bandwidth limit, the obtained c= lim L/Tis independent of window/kernel and agrees with SI value. Proof. Assumption (single-mode pure delay link): Take single-channel SL(E) = exp iEL/(ℏc). Window wR∈L1(R), kernel η∈L1(R) satisfy normalization 1 2πZR wR(E)dE = 1,ZR η(E)dE = 1. Lemma 1 (logarithmic derivative = Wigner–Smith): For differentiable unitary matrix (scalar here) S, ∂Earg det S(E) = Im ∂Elog det S(E) = Im trS†(E)∂ES(E)=−itrS†(E)∂ES(E)= tr Q(E), where Q(E) := −i S†(E)∂ES(E). For scalar SL, tr QL(E) = −iS∗ L(E)∂ESL(E) = L/(ℏc) (constant). Lemma 2 (Birman–Kre˘ın): If det S(E) = e−2πi ξ(E), then tr Q(E) = −2π ξ′(E). Thus for SLwe get ξ′(E) = −1 2π L ℏc. Main proof: Define windowed readout T[wR, η;L] = ℏ1 2πZR wR(E)η ⋆ tr QL(E)dE. 7 Since tr QL≡L/(ℏc) is constant, convolution and integration commute and using both normalizations, T[wR, η;L] = ℏ·1 2π·L ℏcZwR(E)dE ·Zη(ν)dν =L c. Combining Lemmas 1–2, given the integrand is constant, T=ℏ∂Earg det SLw,η =ℏtr QLw,η =−ℏ·2πξ′(E)w,η =L c. Error closure note (NPE): Nyquist/Poisson/Euler–Maclaurin three terms are each 0 in the “constant integrand” case: alias term is 0, Poisson periodization term is 0, EM remainder is 0 because higher derivatives are 0. Thus limit is independent of window/kernel and agrees with SI value. 6 Causal Front and Information Light Cone Theorem 5.1 (Causal front does not exceed c; equality condition) For linear strictly causal, time-invariant (static media) channel, the earliest nonzero response time satisfies tmin ≥Dfront c, Dfront =(L, vacuum, dfront(x, y),media/inhomogeneous/bounded domain. For 3D vacuum free space pure propagation,tmin =L/c; for general media/bounded domains, if front-detectability ( § 5) holds, also tmin =dfront/c; if cascaded with strictly causal LTI filter, tmin ≥Dfront/c +th(equality requires front-detectability and no cancellation). (LTV remark): For linear time-varying systems, front is formulated only via timedomain support of Gret(t, τ); this paper does not represent time-varying media front by static dfront. Proof. Assume system is linear, strictly causal, time-invariant (static media). By causality, Gret(t, τ;x, y) = 0 when t < τ. Let source signal xbe supported on t≥0, then y(t) = ZR Gret(t, τ;x, y)x(τ)dτ, y(t) = 0 (t < 0). Vacuum, uniform, lossless 3D scalar wave equation: Gret(t, r) = δ(t−r/c) 4πr , support only at t=r/c, so for link length Lwe have tmin(L) = L/c.Maxwell case uses dyadic kernel applying tensor–differential operators (containing δand its derivatives) to the above, support likewise only on light cone, conclusion unchanged. 8 General media/inhomogeneous/bounded domain (infimum calibration): Under front assumption (F),n∞(x) exists and is finite, from high-frequency geometric optics front propagation bound supptGmed ret (t, τ;x, y)⊆[τ+dfront(x, y)/c, ∞). Thus when t < dfront(x, y)/c response is zero. If front-detectability ( § 5) holds, then at t=dfront(x, y)/c there exists distributional nonzero response (front singularity PkKkδ(k) or Dirac pass-through component), giving tmin =dfront(x,y) c; otherwise only tmin ≥dfront(x,y) c, typically strictly greater. If cascaded with strictly causal LTI filter with onset delay th>0, tmin ≥dfront(x,y) c+th, and only when front-detectability ( § 5) is satisfied and there is no systematic cancellation at the front can equality be taken. Thus in vacuum case, when t < L/c, for any τ≥0, t−τ < L/c ⇒Gret(t−τ, L) = 0, so y(t) = 0. Take unit-energy short pulse family xη=ψη(ψη→δapproximate identity kernel), then yη(t) = Gret ∗ψη(t) = ψη(t−L/c) 4πL , thus limη↓0yη(t) = δ(t−L/c)/(4πL), earliest nonzero response still at t=L/c. This argument is consistent with “detectable readout” calibration. Theorem 5.2 (Information light cone; bound and equality condition) In vacuum or static media (LTI), first-detection time Tδof threshold mutual information satisfies cinfo := lim δ↓0sup D Tδ≤c, where Dis the front optical path in the normalization used. Equality cinfo =cif and only if front-detectability holds. (i) Link normalization (th= 0):D=Dfront; if front-detectability (same definition as A5) holds, then cinfo =c. (ii) System normalization (cascade strictly causal LTI, th>0):D=Dsys := Dfront +c th; if front-detectability holds, then cinfo =c. (LTV supplement): For linear time-varying systems, only assert Tδ≥tmin. Proof. Let input process Xbe supported on t≥0, noise Nindependent of X, receiver observes Yt:= {Y(s):0≤s≤t}, Y (t) = ZR Gret(t, τ;x, y)X(τ)dτ +N(t). (1) Zero mutual information (before front): For case (i), if in vacuum t < L/c (or in media/inhomogeneous/bounded domain t<dfront(x, y)/c), then Y(s) = N(s), so I(X;Yt) = 0. For case (ii), front changes to Dsys/c. 9