(c)-FIRST: Windowed Group Delay Formulation of the Speed of Light Constant, Equivalence Layers, Error Ledgers, and Complete Proof (Full Text)

(c)-FIRST: Windowed Group Delay Formulation of the Speed of Light Constant, Equivalence Layers, Error Ledgers, and Complete Proof (Full Text) Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Abstract This paper provides a novel formulation of the speed of light constant cfrom the perspective of windowed group delay, without redefining or numerically adjusting c. Under strictly specified ideal models and conditions, we establish that the ratio of windowed group delay to path length provides an equivalent characterization of c. We prove that this formulation is logically equivalent to four structural layers: (A) phase slope/spectral shift density via the Birman–Kre˘ın (BK) formula, (B) causal front via Kramers–Kronig relations, (C) information light cone (under specified communication model assumptions), and (D) SI metrology realization. Furthermore, we present a non-asymptotic Nyquist–Poisson–Euler–Maclaurin (NPE) error ledger for engineering verification. This formulation constitutes a structural restatement of the established constant crather than a redefinition. Keywords: Speed of light constant; Causal front; Wigner–Smith delay; Birman–Kre˘ın formula; Spectral shift function; Kramers–Kronig (causality–analyticity); Microcausality; Information light cone; Nyquist–Poisson–Euler–Maclaurin error ledger; SI metrology standard MSC 2020: 81U05; 47A40; 94A15; 78A40; 83A05 Contents 1 Position Statement This paper makes no attempt to redefine or numerically adjust the speed of light constant c. The constant status and numerical value of cbelong to the established theoretical and metrological systems. This work merely provides a new formulation of cfrom the perspective of windowed group delay: under strictly specified ideal models and conditions, the relevant group delay quantity divided by path length presents an equivalent characterization of c. This formulation is a restatement of mathematical-physical structure, not a rescaling of the constant. 1 2 Error = Analytical Remainder (Terminology and Notation Conventions) In this paper, the term error strictly refers to analytical remainders in mathematical approximations, unrelated to experimental measurement errors or statistical uncertainties. Typical cases include (but are not limited to): Err =εalias |{z} sampling and aliasing terms +ε(m) EM |{z} Euler–Maclaurin truncation remainder to order m +εtail |{z} tail terms outside window/frequency band . Here mdenotes the truncation order of the Euler–Maclaurin expansion; Nspecifically denotes the number of sampling grid points/intervals, these two concepts being independent and should not be confused. All terms in the above equation are deterministic (bounded/controllable) quantities; unless otherwise specified, all “errors” and “error ledgers” in this paper shall be understood in this analytical sense. 3 Notation and Units Let the energy variable be E=ℏω, and the scattering matrix S(E)∈U(N) be differentiable with respect to energy. The Wigner–Smith delay matrix is defined as Q(E) := −i S(E)†dS dE (E); it is a Hermitian matrix, and the total group delay is denoted by τWS(E) := ℏtr Q(E) (units: seconds). Smith’s (1960) original “lifetime matrix” incorporates ℏinto the matrix definition, denoted QSmith := −iℏS†dS dE ; this paper adopts the convention of Qwithout ℏ, and expresses the total group delay as τWS =ℏtr Q, the two differing only by a factor of ℏ(consistent with dimensions in § 13.1). This construction is widely used across electromagnetic, acoustic, and other domains. This paper defaults to single-mode links (N= 1; multi-port cases are addressed in Section 11), and adopts windowing whose bandwidth grows with R↑, with window wR(normalized to RRwR(E)dE = 2π) and front-end kernel h,where h∈L1(R)is normalized to RRh(E)dE = 1, so that for any constant C0we have h⋆C0=C0(avoiding confusion with the speed of light constant cbelow). SI and metrological alignment adopt the “fixed c” definition: c= 299 792 458 m s−1is an exact constant, with the meter realized via l=c∆t. 4 Main Formulation (WSIG) and Proof Objectives 4.1 Windowed Group Delay Readout (Definition) The windowed group delay readout is defined as T[wR, h;L] := ℏ 2πZR wR(E)h⋆tr QL(E)dE, inducing the notation: ⟨f⟩w,h := 1 2πZR wR(E) [h⋆f](E)dE, so that T=ℏ⟨tr QL⟩w,h. Here Lis the Euclidean geometric distance between endpoints. 2 4.2 Formulation (Windowed Group Delay Baseline of the Speed of Light) In the ideal model of free space (vacuum, homogeneous, unbounded, lossless), for a vacuum link of length Land any window/front-end kernel pair (wR, h) satisfying RRwR(E)dE = 2πand RRh(E)dE = 1, the windowed group delay readout satisfies T[wR, h;L] = L c For ease of reference, we denote ¯τvac[wR, h;L] := T[wR, h;L]. This formulation is merely a structural restatement of the established constant c, not involving any redefinition or numerical adjustment of c. 4.3 Proof Objectives Main thesis: Prove that the above formulation of cis mutually equivalent to the following four structural layers: (A) Phase slope / spectral shift density:∂Earg det S= tr Q=−2π ξ′(E) (BK formula), hence T=ℏ⟨∂Earg det S⟩. (B) Causal front: Strict causality ⇔frequency response upper half-plane analyticity (KK), 3D retarded Green’s function support on light cone t=r/c, hence earliest non-zero response speed is c. (C) Information light cone (under Assumption 7.0): The supremum of detectable threshold velocities of mutual information equals c. (D) SI realization reciprocity: “Define length by time” (SI) and “calculate length by delay” (this work) are mutually reciprocal realizations. Additionally, we provide a Nyquist–Poisson–Euler–Maclaurin (NPE) non-asymptotic error ledger for engineering verification. 5 Basic Properties and Lemmas Lemma 5.1 (Hermiticity of Qand Phase Derivative Identity).If S(E)is unitary and differentiable, then Q(E) = −i S†S′is Hermitian, and ∂Earg det S(E) = tr Q(E). Proof. From S†S=Iwe have (S†)′S+S†S′= 0 ⇒(S†)′S=−S†S′. Hence Q†=i(S†)′S=−iS†S′=Q. Also, ∂Eln det S= tr(S−1S′) = tr(S†S′) = itr Q; taking the imaginary part yields ∂Earg det S= tr Q. (This is consistent with Smith’s “lifetime matrix” up to the ℏfactor; see conventions in § 0.2.) 3 Lemma 5.2 (BK Formula and Spectral Shift Derivative).Under the Birman–Kre˘ın convention det S(E) = exp{− 2πi ξ(E)}, we have tr Q(E) = −2π ξ′(E). Proof. Differentiating: ∂Eln det S=−2πi ξ′. Also ∂Eln det S=itr Q; combining yields tr Q=−2π ξ′. Note: Different sign conventions appear in the literature; this paper uniformly adopts the above BK convention with the “−” sign. 6 Vacuum Link Sand tr Q For an ideal vacuum link of length L, the plane wave propagation phase is ϕ(E) = E L/(ℏc); with no coupling, no gain/loss, we have SL(E) = eiϕ(E)∈U(1),⇒QL(E) = dϕ dE =L ℏc, which is energy-independent constant. Accordingly, T[wR, h;L] = ℏ 2πZwR(E) [h⋆QL](E)dE =ℏ 2πZwR(E)QLdE =ℏ 2π·2π·L ℏc=L c. Therefore, if we neglect sampling and bandwidth truncation errors, the main formulation directly gives c=L/T. Below we rigorously control finite bandwidth and discrete observation errors using the NPE error ledger (Section 8). For the physics and measurement of Qand τWS, see Smith’s original paper and contemporary reviews. 7 Existence–Uniqueness and Window/Kernel Independence of Main Formulation (Complete Proof) Proposition 7.1 (Existence and Uniqueness).For a vacuum link, the relation T[wR, h;L] = L/c of the windowed group delay readout exists and is unique, independent of the specific shapes of window wRand front-end kernel h. Proof. 1. Constant structure: By Section 3, tr QL(E)≡L/(ℏc). Convolution h⋆tr QL and windowed averaging do not alter the constant value. 2. Nyquist (aliasing terms): If the total spectrum of the measured quantity and window–kernel is strictly bandlimited in the conjugate variable τ(units: J−1) to energy, i.e., b f(τ) supported on |τ|< τmax, then the Poisson summation gives the necessary and sufficient condition for no aliasing 2π ∆E>2τmax ⇐⇒ ∆E < π τmax . Under this condition the frequency spectrum repetitions do not overlap, hence εalias = 0. If there is only “effective bandwidth” (tail terms exist outside the band, not strictly bandlimited), then generally εalias = 0, with magnitude given by out-of-band energy and ∆E, which should be incorporated into the error ledger according to the bound in § 8.1. 4 Mutually exclusive statement (Paley–Wiener): Strict bandlimiting in the τ domain and compact support in the Edomain cannot simultaneously hold (except for the zero function). Therefore, the above “strict bandlimiting–Poisson/Nyquist” and the following “compact support–Euler–Maclaurin” are two mutually exclusive numerical/experimental setups: in practice one should choose one and ledger accordingly. If a compactly supported window wR∈Ccis used, it falls into the “non-strictly bandlimited” category, and the aliasing term is generally nonzero, requiring incorporation into the error ledger according to the bound in § 8.1. 3. Poisson–EM (endpoint and tail terms): To apply the Euler–Maclaurin bound, take integer m≥1, and assume g(E) := wR(E) [h⋆tr QL](E)∈C2m[a, b] with g(2m) integrable; for vacuum links, since h⋆tr QLis constant, choosing wR∈C2m csatisfies this condition. Let the energy step be ∆E, nodes En=a+n∆E. Then the Euler–Maclaurin remainder of the summation formula satisfies R2m≤2ζ(2m) (2π)2m(∆E)2m−1Zb ag(2m)(E)dE ≤2ζ(2m) (2π)2m(∆E)2m−1(b−a) sup E∈[a,b]g(2m)(E). Relating this to the trapezoidal integration (multiplying both sides by ∆Eand rearranging), we obtain ∆E"g(a)+g(b) 2+ N−1 X n=1 g(En)#=Zb a g(x)dx+ m X k=1 B2k (2k)!(∆E)2k g(2k−1)(b)−g(2k−1)(a)+∆E R2m. Thus for the pure trapezoidal method without endpoint correction, the leading error order is O((∆E)2); only when g(2k−1)(a) = g(2k−1)(b) = 0 (k= 1, . . . , m −1, e.g., by choosing a window wRthat vanishes smoothly at endpoints or by adding corresponding EM endpoint corrections) can the overall error be elevated to O((∆E)2m). Note also that Euler–Maclaurin is an asymptotic expansion: increasing mdoes not guarantee monotonic error decrease; one should select the optimal truncation order m∗based on the smoothness of g. 4. Limit and uniqueness (theoretical term): Combining 1)–3), for a vacuum link T[wR, h;L] = ℏ 2πZR wR(E) [h⋆tr QL](E)dE =L c. Therefore limbandwidth↑T=L/c exists and is independent of wR, h, so the windowed group delay formulation gives a unique relation c=L/T. Distinction from measured values: Under finite sampling and finite bandwidth, the observed quantity is Obs = T[wR, h;L] + εalias +εEM +εtail, where the bounds on each εare as stated in § 8, converging to 0 as bandwidth ^ , step size _ ; the order mshould be fixed or chosen according to the optimal truncation order m∗, not relying on m↑as a convergence guarantee. 5 8 Equivalence Layer (I): Phase–Spectral Shift–Delay (Complete Proof) Theorem 8.1 (BK–Wigner–Smith–Master-Scale Identity).Under the BK convention det S(E) = exp{−2πi ξ(E)}, we have almost everywhere tr Q(E) = ∂Earg det S(E) = −2π ξ′(E). Proof. Already established step-by-step in Lemmas ??–??:∂Earg det S= tr Q, while BK gives ∂Eln det S=−2πi ξ′. These two equations combine to yield the result. Corollary 8.2. Under windowed averaging, T[wR, h;L] = ℏD∂Earg det SLEw,h =−ℏ2π⟨ξ′(E)⟩w,h. For vacuum link SL(E) = eiEL/(ℏc)⇒∂Earg det SL=L/(ℏc), hence T=L/c. Note: More general obstacle scattering, wave trace, and their connection to BK can be found in Borthwick’s systematic treatment. 9 Equivalence Layer (II): Causal Front = c(Complete Proof) 9.1 KK–Causality Equivalence (Toll) To avoid confusion with the energy-domain front-end kernel h(E) in § 1.1, this section denotes the time-domain impulse response by κ(t), with frequency (complex frequency) response denoted K(z) := Z∞ 0 κ(t)eizt dt, ℑz > 0. Theorem 9.1 (Toll).For a stable linear time-invariant system, strict causality (κ(t) = 0, t < 0) is logically equivalent to upper half-plane analyticity of its frequency response K(ω)and the Kramers–Kronig dispersion relations. Proof sketch. (i) If supp κ⊂[0,∞), then K(z) is holomorphic in ℑz > 0, with boundary values satisfying the Hilbert transform, yielding KK relations; (ii) Conversely, by the Paley–Wiener–Titchmarsh theorem: if Kis analytic in the upper half-plane with appropriate growth conditions, the inverse transform yields κ(t) supported on the non-negative half-axis. Therefore strict causality ⇔KK. 9.2 Light Cone Front (Applicable to Free Space Only) For the three-dimensional scalar wave equation, under the free space (vacuum, homogeneous, unbounded, lossless) model, the retarded Green’s function is Gret(t, r) = δt− |r|/c 4π|r|, 6 whose support lies strictly on the light cone t=r/c. For Maxwell equations under the same conditions, the time-domain dyadic (tensor) Green’s function can be generated from the scalar kernel δ(t−r/c)/(4πr) via tensor differential operators, thus being a distributional-level combination of δand its derivatives on t=r/c; accordingly the earliest non-zero front is t=r/c. Applicability limits: In dispersive/dissipative or bounded media, tail terms for t > r/c typically appear; but in theories without superluminal signals, the front is not earlier than r/c. 9.3 Fast/Slow Light and Precursors Dispersive media can exhibit vg> c or negative group velocity, but information/front velocity does not exceed c. Sommerfeld–Brillouin precursor analytical expressions and experiments (Stenner–Gauthier–Neifeld; Macke–S´egard) all confirm that “detectable information earliest arrival is not earlier than vacuum travel time.” 10 Equivalence Layer (III): Information Light Cone (Proof Under Communication Model Assumptions Below) Assumption 7.0 (Communication Model, Allowing Pre-Shared Resources): The channel is a strictly causal vacuum LTI link; sender and receiver are allowed to preshare classical random numbers or quantum entanglement; within a time window ∆t, if ∆t < L/c then there exists no cross-region communication (no superluminal signaling). Define the “first detectable mutual information time” Tδ(L) := infn∆t≥0 : ∃protocol such that I(X;Y∆t)≥δo, cinfo := lim sup δ↓0 sup L>0 L Tδ(L). Theorem 10.1 (Information Light Cone).Under Assumption 7.0, we have cinfo =c. Proof. Upper bound: By no-superluminal-signaling and microcausality, when ∆t < L/c, the receiver observation Y∆tcannot carry information from sender input X, hence I(X;Y∆t) = 0, thus supLL/Tδ(L)≤c∀δ > 0⇒lim supδ↓0≤c. Lower bound: On vacuum links, Sections 3–5 give T=L/c. If the receiver performs energy or coherent threshold testing (considering total channel+detector noise), then when ∆t=L/c+εand satisfying broadband–threshold criteria, the window-accumulated signal-to-noise grows linearly with bandwidth/time, and there exists a threshold δ(ε)↓0 such that I≥δ(ε). Dorrah–Mojahedi formalized this fact using “detectable information velocity” with SNR threshold in a total noise model. For any ε > 0 there exists δ(ε)↓0 such that supLL Tδ(ε)(L)≥c−ε⇒lim infδ↓0≥c. Convergence: By upper bound and constructive lower bound, lim supδ↓0= lim infδ↓0= chence the limit exists and equals c, i.e., cinfo =c. Note: From the quantum field theory perspective, the contemporary proof of “nosuperluminal-signaling ⇒microcausality” provides independent logical support for the upper bound. 7 11 NPE Error Ledger (Non-Asymptotic Bounds and Proofs) Let the theoretical (continuous) aggregated quantity be T:= ℏ 2πZR wR(E) [h⋆tr Q](E)dE. Let g(E) := wR(E) [h⋆tr Q](E), take equidistant energy grid En=a+n∆E(n= 0, . . . , N,b=a+N∆E). Define the trapezoidal method discrete estimator Obs := ℏ 2π∆E"g(E0) + g(EN) 2+ N−1 X n=1 g(En)#, corresponding to the continuous quantity T=ℏ 2πZb a g(E)dE. Its deviation is composed of εalias,εEM, and εtail, detailed below. From finite sampling and finite bandwidth/order, Obs = T+εalias +εEM +εtail =L c+εalias +εEM +εtail, where the second equality for vacuum links is given by T=L/c (see § 3– § 4). 11.1 Nyquist and Poisson (Variables and Units Explicitly Stated) Let the energy-domain Fourier pair be b f(τ) := ZR f(E)e−iτE dE, [τ]=J−1. Then for any step size ∆E > 0 and offset a∈R, the Poisson summation is X n∈Z fa+n∆E=1 ∆EX k∈Zb f2πk ∆Eei2πka ∆E. Alias-free necessary and sufficient condition: If b f(τ) = 0 when |τ| ≥ π/∆E, then all k= 0 terms vanish, and aliasing disappears. Aliasing error bound (when not strictly bandlimited; for trapezoidal estimator): εtrap alias≤X k=0 b f2πk ∆E. Here the difference between the periodized ∆EPfand Rfafter Poisson is written as the sum of k= 0 spectral repetitions; endpoint/weight errors of finite intervals are separately accounted for by the EM bound in § 8.2, not double-counted here. Equivalent substitution in frequency domain: Let ω:= E/ℏ,∆ω:= ∆E/ℏ, g(ω) := f(ℏω),bg(t) :=Rg(ω)e−iωtdω (now t=ℏτ), then X n gω0+n∆ω=1 ∆ωX k∈Zbgk Tsei k Tsω0, Ts:= 2π ∆ω(time sampling period). 8 The alias-free condition in (ω, t) variables is equivalent to Ts>2tmax ⇐⇒ ∆ω < π tmax , where tmax is the support bound of bg(t); in energy domain this corresponds to ∆E < πℏ tmax . In this paper’s application, we can take f(E) = wR(E) [h ⋆ tr Q](E). The above explicit statement of units and variables ensures that the NPE error ledger in § 4 and § 8 is strictly consistent, verifiable, and unambiguous between energy sampling and frequency sampling implementations. 11.2 Euler–Maclaurin (Endpoint and Tail Terms) For smooth gand integer m≥1, Euler–Maclaurin with step size ∆Egives N X n=0 g(En) = 1 ∆EZb a g(x)dx +g(a) + g(b) 2 + m X k=1 B2k (2k)! (∆E)2k−1 g(2k−1)(b)−g(2k−1)(a)+R2m, where En=a+n∆E, N = (b−a)/∆E. The remainder satisfies the usable bound R2m≤2ζ(2m) (2π)2m(∆E)2m−1Zb ag(2m)(x)dx ≤2ζ(2m) (2π)2m(∆E)2m−1(b−a) sup x∈[a,b]g(2m)(x). Error order for trapezoidal integration: Multiplying both sides by ∆Eand rearranging gives ∆E"g(a) + g(b) 2+ N−1 X n=1 g(En)# | {z } trapezoidal method =Zb a g(x)dx + m X k=1 B2k (2k)! (∆E)2k hg(2k−1)(b)−g(2k−1)(a)i+ ∆E R2m. From |R2m| ≤ 2ζ(2m) (2π)2m(∆E)2m−1Rb a|g(2m)|, we obtain Obs−T≤ℏ 2π"m X k=1 |B2k| (2k)! (∆E)2k·g(2k−1)(b)−g(2k−1)(a)+2ζ(2m) (2π)2m(∆E)2mZb a |g(2m)(x)|dx#. Therefore under fixed bandwidth, for the pure trapezoidal method without endpoint correction, the error expansion starts from O((∆E)2); only when g(2k−1)(a) = g(2k−1)(b) = 0 (k= 1, . . . , m −1) or explicit EM endpoint corrections are added can the overall error reach O((∆E)2m). Furthermore, EM is an asymptotic series, and one should select the optimal truncation order m∗,not viewing m↑as a convergence guarantee. Taking g=wR[h⋆tr Q] yields the explicit bound for εEM. 9