Formal Construction of WSIG--EBOC Holographic Reconstruction Theory

Formal Construction of WSIG–EBOC Holographic Reconstruction Theory Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.2 Abstract We establish a holographic reconstruction theory with Windowed Scattering– Information Geometry (WSIG) as the observational layer and Eternal Block Observation– Computation (EBOC) as the ontological layer. The core proposition is: under an overlapping window family satisfying frame stability and topological consistency, the “trinity” of local readouts consisting of phase derivative–relative state density– Wigner–Smith group delay can be stably glued into a global scattering field within a non-asymptotic Nyquist–Poisson–Euler–Maclaurin three-term closed error budget, thereby providing a global encoding of the EBOC static block. We present unified notation, conventions, and gluing consistency conditions, prove three main theorems on existence–stability–uniqueness (under gauge choice), and propose implementable inversion–projection algorithms along with semantic correspondence to reversible cellular automata (RCA). Keywords: Holographic reconstruction; WSIG; EBOC; Wigner-Smith delay; Frame theory; ˇ Cech cohomology; Birman-Kre˘ın formula; Reversible cellular automata MSC 2020: 81U05; 47A40; 94A15; 42C15; 55N05 Contents 1 Notation, Conventions, and Axioms Energy domain and multi-port scattering. Let the energy domain be a measurable set Ω ⊂R, with channel number N∈N. At almost every Lebesgue point of the absolutely continuous spectrum, the Wigner–Smith delay matrix and total phase for scattering matrix S(E)∈U(N) are defined as Q(E) := −i S(E)†∂ES(E),Φ(E) := Arg det S(E). Let the half-phase be φ(E) := 1 2Φ(E). Then ∂EΦ(E) = tr Q(E), ∂Eφ(E) = 1 2tr Q(E). 1 These three quantities satisfy the gauge identity with the relative state density ρrel(E): φ′(E) π=ρrel(E) = 1 2πtr Q(E), where ρrel equals the derivative of the Kre˘ın spectral shift function (this paper adopts scattering phase sign conventions and channel orientation that make the above equation hold). These equalities respectively stem from ∂Elog det S= tr(S−1S′) and the Birman– Kre˘ın criterion. Windows and frames. Take window family {wj}j∈J⊂L1∩L2(Ω) with corresponding analysis operators Wjf:= f∗wj. Suppose there exist constants 0 < A ≤B < ∞ such that A|f|2 L2(Ω) ≤X j∈J |Wjf|2 L2(Ω) ≤B|f|2 L2(Ω),∀f∈L2(Ω), then {Wj}is called a frame; denote the frame operator S:= PjW∗ jWj, which is bounded, invertible, and |S−1| ≤ A−1. Three-term error closure (NPE). Let sampling step h > 0, effective bandwidth W > 0, and suppose the window has m≥2 orders of integrable derivatives. The total error from discretization–convolution–truncation decomposes as ηNPE := ηalias +ηBern +ηtail, where the aliasing term is controlled by Poisson summation, the Bernoulli correction layer comes from finite-order Euler–Maclaurin, and the tail term characterizes truncation leakage; corresponding bounds are given in § 5. Gauge and topology. We adopt the Birman–Kre˘ın gauge: fix an absolutely continuous branch of Φ = Arg det Sand let φ=1 2Φ. Assume the window covering is overlapping and has no topological gaps (see § 2), to exclude phase multivaluedness from ˇ Cech 1-cocycles. 2 Local Observation Model 2.1 Trinity Local Readout Define the window-wise local readout Rj(E) := φ′ j(E),tr Qj(E), ρrel,j(E), satisfying the window-wise trinity constraint φ′ j=1 2tr Qj=πρrel,j. The observation model assumes the existence of a global field R(E) := φ′(E),tr Q(E), ρrel(E) such that Rj=WjR+εj,|εj|L2≤CNPE bounded by § 5. 2 2.2 Unitarity Constraint and Generating Equation From the definition Q=−iS†S′we derive the equivalent matrix ordinary differential equation S′(E) = i S(E)Q(E), S(E0)∈U(N). When Q†=Q∈L1 loc(Ω), the solution is given by the path-ordered exponential, and S†S≡I. Therefore, given Q(requiring only that its trace satisfies the gauge identity), one can always construct Ssatisfying unitarity. 3 Gluing Consistency and ˇ Cech Energy Let the overlap region be Ujk := {E∈Ω : wj(E)wk(E)= 0}. Define the phase derivative difference and ˇ Cech energy as ∆jk(E) := φ′ j(E)−φ′ k(E),Eˇ C:= X j∼kZUjk |∆jk(E)|2dE. Consistency requirement: There exists a constant κ0>0 such that Eˇ C≤κ0|ηNPE|2 L2(Ω). In particular, for any closed 1-cycle Con the nerve of the covering, we have ZC ∆jk(E) dE≤ctop |ηNPE|L1, excluding branch cut inconsistencies induced by covering gaps. 4 Variational Model for Global Reconstruction Define the constraint sets Ctri := nR:φ′=1 2tr Q=πρrelo,Cunit := nS:S†S=I, −iS†S′=Qo. Coupling constraint. The Q(E) appearing in both places is the same function, uniquely determined by S:Q(E) := −i S(E)†∂ES(E). Therefore tr Qin Ctri equals −itr S†S′, synchronized with Cunit. In weighted space |·|W, minimize min R,S J(R, S) := X j∈J |WjR−Rj|2 Ws.t. R∈Ctri, S ∈ Cunit. To suppress overlap inconsistencies, introduce ˇ Cech regularization Jλ:= J+λEˇ C, λ > 0. 3 5 Main Theorems and Proofs 5.1 Theorem A (Existence) Theorem 5.1 (Existence).If {Wj}is a frame with overlapping covering (existence of lower bound A > 0), and Eˇ C≤κ0|ηNPE|2 L2, then there exists a minimizing pair (R⋆, S⋆) such that Jλattains its minimum, with R⋆∈ Ctri and S⋆∈ Cunit. Proof. First ignoring constraints, the normal equation SR=X j W∗ jRj,S=X j W∗ jWj, has Sinvertible with |S−1| ≤ A−1by the frame lower bound, so the unconstrained minimizing solution R(0) =S−1X j W∗ jRj exists uniquely. Choose any Hermitian Q(0) such that tr Q(0) = 2φ(0)′(e.g., Q(0) =2φ(0)′ NI), and construct S(0) ∈ Cunit by § 1.2. Apply alternating projection and minimization to the closed set Ctri×Cunit ( § 6); using non-expansiveness of projections and lower semicontinuity of Jλ, combined with the ˇ Cech term absorbing error sources, we obtain existence of a minimizing pair. Frame theory ensures coercivity and boundedness, completing existence via the direct method. 5.2 Theorem B (Stability) Theorem 5.2 (Stability).Under the conditions of Theorem ??, let b Rbe the Rcomponent of the minimizing solution. Then |b R − R|L2(Ω) ≤κ(A, B)εmeas +|ηNPE|L2(Ω), κ(A, B)≤pB/A, where εmeas := maxj|WjR−Rj|L2. Proof. By frame inequalities and triangle inequality, A|b R − R|2 L2≤X j |Wj(b R−R)|2 L2≤2X j |Wjb R−Rj|2 L2+ 2 X j |WjR−Rj|2 L2. Minimality and the ˇ Cech penalty term give boundedness on the right-hand side; normalizing with upper bound Byields the stated estimate. 5.3 Theorem C (Uniqueness Under Gauge) Theorem 5.3 (Uniqueness Under Gauge).If the nerve of the covering has no 1-cycles (all closed loops Csatisfy RC∆jk dE= 0), and the BK gauge is fixed, then (R⋆, S⋆)is unique up to left multiplication by constant unitary equivalence (U(N)). Proof. The condition implies that φ′after covering gluing has a global primitive φconsistent with the BK gauge; tr Qis rigidly determined by tr Q= 2φ′. The solution set of equation S′=iSQfor given Qdiffers only by left multiplication by a constant unitary element, equivalent to constant unitary equivalence classes (U(N)). 4 6 Non-Asymptotic NPE Error Budget Suppose window wjhas morders of integrable derivatives, step size h, bandwidth W. Then there exist constants C1, C2, C3(depending on window decay and regularity) such that |ηalias|L2≤C1e−2πW h,|ηBern|L2≤C2hm|∂m ER|L2,|ηtail|L2≤C3|R · 1Ωc|L2. The first term is given by the exponential-type aliasing bound from frequency spectrum periodization in Poisson summation; the second term derives the O(hm) estimate from the Euler–Maclaurin bounded remainder (Fourier series bound on Bernoulli polynomials); the third term is controlled by the L2mass of the truncation residual. Thus |b R − R|L2≤κ(A, B)εmeas +C1e−2πW h+C2hm|∂m ER|L2+C3|R · 1Ωc|L2. 7 Inversion–Projection Algorithm 7.1 Frame Inversion Initialization From the normal equation, solve R(0) =S−1X j W∗ jRj,|S−1| ≤ A−1. 7.2 Alternating Projection and Unitarization Use alternating projection to solve constrained feasibility minimization: R(n+1 2):= ΠCtri [R(n)], S(n+1) := ΠCunit [S(n);R(n+1 2)],R(n+1) := arg min RX j |WjR−Rj|2 W. Here ΠCtri can be implemented by least squares fitting of (φ′,1 2tr Q, πρrel); ΠCunit is defined as: first at each Efind Q⋆(E) := arg min Q†=Q,tr Q=2φ′(E)Q+i S†S′2 F, whose closed-form solution is Q⋆= sym −iS†S′−1 Ntr sym −iS†S′−2φ′I; then integrate e S′=ie SQ⋆and use the unitary factor of polar decomposition as retraction. Classical global convergence of alternating projection applies only to two closed convex sets; given that Cunit is nonconvex and U(N) is a unitary manifold, this paper treats polar decomposition only as retraction and claims local convergence under local angle conditions/transversality; polar decomposition still provides the nearest unitary matrix in Frobenius norm. 7.3 ˇ Cech Energy Regularization In overlap regions, add λEˇ Cto localize and suppress phase inconsistency sources. λcan be adaptively chosen according to C1, C2, C3in § 5 and measurement noise εmeas to balance bias–variance. 5 8 Multi-Port Spectral Decomposition and Channel Selection Spectral decomposition S(E) = PN ℓ=1 eiθℓ(E)Pℓ(E) gives tr Q(E) = N X ℓ=1 θ′ ℓ(E), φ′(E) = 1 2 N X ℓ=1 θ′ ℓ(E). For sparse channel selection, one can select a set I(E)⊂ {1, . . . , N}on the spectral decomposition S(E) = PN ℓ=1 eiθℓ(E)Pℓ(E) and define composite projection PI(E) := Pℓ∈I(E)Pℓ(E); then impose rank constraint rank PI(E)≤kor penalize with Ky–Fan k-norm to favor low-rank coupling while maintaining trinity and unitary coupling. 9 Semantic Correspondence with EBOC and RCA EBOC views (Ω, S) as the global encoding of a static block; window selection only changes the locally accessible cross-section without altering the ontological encoding. On the RCA side, sliding block codes provide local–global stitching: assign consistent transitions and “recorded entropy” readouts to each overlapping local state configuration, then Kolmogorov consistency constructs a global reversible flow. The frame stability and ˇ Cech energy in this paper correspond to RCA’s conflict-free local transitions and codebook redundancy threshold; the BK gauge corresponds to the unification of global phase (gauge) choice. 10 Sufficient Conditions and Failure Mechanisms Sufficient conditions. Frame covering (existence of lower bound A > 0); BK gauge fixed; nerve of covering has no 1-cycles; three-term errors within § 5 bounds; ˇ Cech energy bounded. Failure mechanisms. Covering gaps (some energy region without overlap); topological gaps (nonzero 1-cycles on nerve inducing multivaluedness); frame degeneration (A→0); aliasing dominance (W/h too small); tail leakage (energy mass outside window not negligible). 11 Explicit Constants for Kaiser–Bessel Window For Kaiser–Bessel window (parameter β > 0), one can take C1≲C(β), C2≲C(β)W−m, C3≲C(β)e−β, thus |b R − R|L2≤κ(A, B)εmeas +C(β)e−2πW h+C(β)W−mhm|∂m ER|L2+C(β)e−β. The mainlobe–sidelobe tradeoff of Kaiser–Bessel window and the role of parameter β are detailed in the literature. 6 12 Verifiable Predictions 1. At fixed bandwidth, increasing overlap (improving frame lower bound A) should monotonically decrease reconstruction error; 2. Injecting phase perturbations in overlap regions will manifest as nonzero Eˇ Cand be localized by the regularization term; 3. When W/h crosses the Nyquist threshold, the error curve exhibits a phase transition point from “aliasing-dominated  Bernoulli-dominated”, consistent with sampling density criteria. 13 Conclusion Within an axiomatic framework of “gauge identity” and NPE three-term closure, this paper provides stable invertible gluing from local trinity readouts to global scattering field, achieving uniqueness under BK gauge; algorithmically implemented via frame inversion and alternating projection–polar decomposition; semantically in strict correspondence with the holographic–stitching structure of EBOC/RCA. This theory establishes verifiable qualitative predictions and non-asymptotic error budgets among window design, sampling density, topological unification, and channel selection. References [1] E. P. Wigner. Lower limit for the energy derivative of the scattering phase shift. [2] F. T. Smith. Lifetime matrix in collision theory. Physical Review, 118:349–356, 1960. [3] U. R. Patel and E. Michielssen. Wigner–Smith Time Delay Matrix. arXiv:2003.06985, 2020. [4] M. Sh. Birman and M. G. Kre˘ın. On the theory of wave operators and scattering matrix. [5] A. Strohmaier and A. Waters. The Birman–Kre˘ın formula. [6] O. Christensen. An Introduction to Frames and Riesz Bases. Springer, 2nd edition, 2016. [7] H. H. Bauschke and J. M. Borwein. On Projection Algorithms for Convex Feasibility. SIAM Review, 38(3):367–426, 1996. [8] N. J. Higham. Computing the Polar Decomposition. SIAM Journal on Scientific and Statistical Computing, 11(6):1160–1174, 1990. [9] F. J. Harris. On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. Proceedings of the IEEE, 66(1):51–83, 1978. [10] H. J. Landau. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Mathematica, 117:37–52, 1967. 7