Finite-Window Reversible Completion Entropy:\\ Axiomatization, Structural Properties, and Non-Asymptotic Error Theory Under EBOC--WSIG--RCA Framework

Finite-Window Reversible Completion Entropy: Axiomatization, Structural Properties, and Non-Asymptotic Error Theory Under EBOC–WSIG–RCA Framework Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.18 Abstract We propose and establish “finite-window reversible completion entropy” as a localized information measure for “record–interpretation” problems under reversible propagation and energy/regularity constraints: given finite window observation, the logarithm of minimal equivalent representative count (or capacity) of completion set consistent with reversible global dynamics. This measure solely characterizes “how many reversible worlds interpret the same record”, independent of prior probabilities and mixed-state entropy. We provide rigorous definitions at both discrete reversible cellular automaton (RCA) and continuous windowed scattering– information geometry (WSIG) ends, proving structural properties including monotonicity, reversible covariance, and splice subadditivity; under Toeplitz/Berezin compression and bandlimited/exponential window settings, construct non-asymptotic upper/lower bounds following Nyquist–Poisson–Euler–Maclaurin (NPE) three-fold decomposition with finite-order termination, providing quantitative law “boundary = information resource”, strictly compatible with unified scale of “phase derivative– relative state density–Wigner–Smith group delay trace”. Examples of bandlimited signals and RCA with theorem-level proofs conclude the paper. Keywords: Reversible completion entropy; EBOC; WSIG; RCA; Finite window; NPE error theory; Trinity scale Notation & Axioms / Conventions 1. Trinity Scale Identity (Trinity) holds almost everywhere on absolutely continuous spectrum: φ′(E) π=ρrel(E) = 1 2πtr Q(E),Q(E) = −i S(E)†∂ES(E), S(E)∈U(N), φ(E) := 1 2arg det S(E). (1) 1 Qis Wigner–Smith group delay matrix, its trace compatible with relative state density; det Sand Kre˘ın spectral shift function ξsatisfy Birman–Kre˘ın formula det S(E) = e2πi ξ(E), thus ξ′(E) = 1 2πi ∂Elog det S(E) = 1 2π∂Earg det S(E) = 1 2πtr Q(E), hence φ′(E)/π =ξ′(E) = ρrel(E). 2. Window–Readout–Object: Readout expressed as “operator–measure–function” objects. Given window wand kernel scale h, corresponding Toeplitz/Berezin compression or localization operator denoted Kw,h or Ta,w,h (symbol a). This paper writes ker(w;E) for non-negative weight kernel induced by window wand scale h on energy axis (can take Rker(w;E)dE = 1 normalization), used to define windowed readouts Φ[w],Ξ[w], etc. 3. Error Theory Discipline (NPE): All approximations decompose into Poisson aliasing term, Euler–Maclaurin (EM) boundary layer, and tail term, strictly terminating at finite order m; EM remainder and explicit upper bounds of Bernoulli polynomials/numbers used for non-asymptotic control. 4. Reversibility and Propagation Constraints: Discrete end characterized by RCA’s bidirectional reversible evolution (global map bijection, inverse evolution also CA), continuous end by unitary/scattering reversible propagation; Garden-ofEden theorem and Curtis–Hedlund–Lyndon characterization for equivalence and structure of reversible/surjective/injective. 5. Equivalence Classes: If two global completions related by structure-preserving reversible isomorphism (RCA conjugacy/lattice shift, scattering isospectral isomorphism), they belong to same interpretation equivalence class. 6. Measure Convention: Write vol(·) for counting measure (RCA end) or Lebesgue volume (WSIG end), multiplicative on product domains; boundary volumes in Theorem 5.1 etc. computed per this convention. 7. Complexity Unit Convention: Write K(y) for prefix Kolmogorov complexity, in nats:K(y) := (ln 2) Kbits(y). Accordingly, inequalities in § 6 and units of HW, Hw consistent. 8. Lattice Distance and Boundary Band (RCA): On Zd, take ℓ∞distance dist(x, W) := minz∈W|x−z|∞. Define ∂rW:= {x∈Zd: 0 <dist(x, W)≤r}, its volume computed by counting measure per (6). Boundary bands in § 5/ § 6 interpreted per this metric. 9. Notation Unification (Cross-End): In summary/conclusive statements ( § 12– § 13), write Has total symbol: RCA end H:= HW, WSIG end H:= Hw; if distinction needed, explicitly write HW, Hw. 10. Norm and Tolerance Ball (WSIG): Write |·|Hwfor Hilbert norm of Hw,Bε(y) := {z∈ Hw:|z−y|Hw≤ε}. All sets involving “tolerance/error” measured by this norm. 2 1 Problem Setup In perspective called EBOC (abstract framework of static-block–reversible-consistency completion, abbreviated herein), record is readout yof global invariant on finite window W. Under global reversible dynamics and energy/regularity constraints, define global reversible completion set consistent with y, characterize its scale by logarithm of “equivalent representative count/capacity”, obtaining finite-window reversible completion entropy. This measure describes “global reversible ambiguity of local record”, emphasizing structural consistency and propagation constraints, not statistical uncertainty. 2 Discrete End (RCA): Definition and Basic Properties Let Ω be global state space of oneor multi-dimensional CA, update radius r, global update G: Ω →Ω. If Gbijective and G−1also CA, call RCA. Let finite window domain W⊂Zd, observation y∈ YW. Define completion set consistent with yand reversible C(y;W) := {ω∈Ω : ω|W=y, ∃bidirectional trajectory consistent with global constraints}. (2) With structure-preserving reversible homeomorphism equivalence class C(y;W)/∼rev, define HW(y) := log min n#R:R⊂C(y;W),Rtransversally intersects all equivalence classeso. (3) Convention: Throughout take log as natural logarithm (unit: nats); if C(y;W)/∼rev infinite set, stipulate HW(y) := +∞. Convention (Window-Fixed Equivalence Relation) In definition of HW(y), ∼rev induced only by structure-preserving reversible isomorphisms identity on window W and preserving readout y: for any Φ, require Φ|W= id and (Φω)|W=ω|W=y; continuous end analogous, scattering/unitary isomorphisms need preserve yon image of Kw,h. Accordingly, C(y;W)/∼rev and “transversal” operation well-defined on same set. Convention (Feasibility, Revised) If C(y;W) = ∅, then HW(y) := −∞ (extended real). Propositions involving Irev(W1:W2) and splice subadditivity stated only when C(y;W1)=∅,C(y;W2)=∅,C(y;W1∪W2)=∅; (4) otherwise Irev undefined. Supplement (Irev Finiteness): All statements involving Irev(W1:W2) = HW1(y1) + HW2(y2)−HW1∪W2(y) (5) defined and discussed only when HW1(y1), HW2(y2), HW1∪W2(y)∈R(finite); otherwise Irev undefined. Theorem 2.1 (Monotonicity).If W1⊆W2and yi=y|Wi, then HW2(y2)≤HW1(y1). Proof. Constraint increase only filters completions, equivalence class count non-increasing, thus transversal representative count non-increasing, taking logarithm yields conclusion. 3 Theorem 2.2 (Reversible Covariance).For any structure-preserving reversible transformation Φ:Ω→Ω(translation, group action, RCA conjugacy), HΦ(W)(Φ(y)) = HW(y). Proof. Φ induces bijection on global states and equivalence classes, transversal count invariant. Theorem 2.3 (Splice–Reversible Mutual Information Identity; Subadditivity as Corollary).Let W=W1∪W2, two windows interact only through radius-rboundary band; and C(y;Wi),C(y;W)=∅with HW1(y1), HW2(y2), HW(y)∈R. Then have identity HW(y) = HW1(y1) + HW2(y2)−Irev(W1:W2),(6) where Irev(W1:W2) := HW1(y1) + HW2(y2)−HW1∪W2(y).(7) Corollary (Subadditivity): By § 5.2’s Irev(W1:W2)≥0, immediately get HW(y)≤HW1(y1) + HW2(y2).(8) Proof. Fiberize equivalence classes, boundary consistency condition produces pairing constraints, defining formula immediately yields above identity decomposition; then use § 5.2’s non-negativity and monotonicity to obtain corollary. Note: RCA’s reversible/surjective/pre-injective structure guaranteed by Garden-ofEden theorem and Curtis–Hedlund–Lyndon theorem: on Zd,pre-injective ⇔surjective; and reversibility ⇔global bijection with inverse rule also CA, supporting above covariance and splice structure. 3 Continuous End (WSIG): Toeplitz/Berezin Compression and Capacity Definition Let Hilbert space H, window wand scale hyield compression/localization operator Kw,h :H → Hw, or more specifically Toeplitz/Berezin operator Ta,w,h (symbol a). For observation y∈ Hw, consider feasible set Cw(y) := nΨ∈ Erev :Kw,hΨ = yo,(9) where Erev is unitary/scattering reversible propagation class, subject to energy shell and regularity constraints. With Fisher–Rao (FR) metric volume as capacity Cap(·) (one-time normalization constant κsee below), define Hw(y) := log Cap Cw(y)/∼rev .(10) Convention: As above, log takes natural logarithm (nats); if Cap(Cw(y)/∼rev) = 0 or +∞, then Hw(y) takes corresponding extended real value. Convention (Capacity and Normalization) Continuous end uniformly takes Cap as Fisher–Rao volume: if {Ψ(θ)}θ∈Θ⊂ Erev is smooth parametrization of reversible family, then Cap Cw(y)/∼rev := κZCw(y)/∼rev pdet I(θ)dθ, (11) 4 where I(θ) is Fisher information matrix induced by map θ7→ Kw,hΨ(θ), κ > 0 onetime normalization constant, chosen such that under § 4’s scale identity Φ[w] = Ξ[w] = Rker(w;E)ρrel(E)dE,Hw’s unit consistent with nats. Discrete end viewed as counting measure; capacity model not altered hereafter. Convention (Continuous End Feasibility and Finite Volume): Write Cw(y)/∼rev for reversible equivalence class quotient. All statements involving log Cap(·) (including § 3.2, § 4.1, § 8, § 9.1, § 11) hold only when 0<Cap Cw(y)/∼rev <∞(12) and this quotient set is C1regular measurable manifold (or covered by finite union of such pieces); if capacity is 0 or +∞, take Hw(y) = −∞ or +∞and no longer state above equations/inequalities. Theorem 3.1 (Covariance).For any unitary/scattering isomorphism Uand window covariance W,HWw(Wy) = Hw(y). Proof. By U’s unitarity and scattering conformality, and natural covariance under Berezin– Toeplitz quantization, capacity preserved. Theorem 3.2 (Revised: Non-Asymptotic Tolerance of NPE Three-Fold).If wbandlimited or exponential type and sampling satisfies Nyquist condition, when 0<Cap(Cw(y)/∼rev )<∞and feasible quotient set is C1regular manifold, exists finite order mand constant Cw,h, for any ε > 0have log Cap Ψ : |Kw,hΨ−y|Hw≤ε/∼rev −log Cap Cw(y)/∼rev ≤Cw,h ε+ ∆(≤m) NPE . (13) where ∆(≤m) NPE = ∆alias+∆(≤m) Bernoulli+∆tail. This formulation consistent with § 8’s Toeplitz/Berezin compression result. 4 Consistency with Trinity Scale Define windowed readout quantities Φ[w] := Zker(w;E)φ′(E) πdE, Ξ[w] := Zker(w;E)1 2πtr Q(E)dE, (14) and Rker(w;E)ρrel(E)dE. By trinity identity get Φ[w] = Ξ[w] = Rker(w;E)ρrel(E)dE. Accordingly obtain: Theorem 4.1 (Scale Compatibility and Local Stability).If capacity normalization consistent with above scale, then for smooth deformation δw of window and symbol perturbation δa, exists constant Csuch that |δHw(y)| ≤ C|δw|W1,1+|δa|W1,1+O∆(≤m) NPE .(15) Proof. Gateaux variation controlled by first-order response of windowed trace; scale identity unifies φ′(E)/π,ρrel(E), and (2π)−1tr Qnormalization; NPE finite-order termination yields non-asymptotic upper bound for residual. 5 5 Boundary = Information Resource: Propagation Radius and Mutual Information RCA’s finite propagation radius ror continuous system’s finite group delay/microsupport propagation bound means degrees of freedom outside window act through boundary band of thickness r. Definition 5.1 (RCA End Boundary Patch Count Comp).Let ∂rW:= {x∈Zd: 0 <dist(x, W)≤r},(16) where dist and volume vol take values per Notation(8)/(6). Define Comp(y;∂rW) := min n#B:B ⊂ S(∂rW),∀ω∈ C(y;W)∃b∈ B s.t. ω|∂rW=bo,(17) where S(∂rW)only refers to RCA alphabet configuration space. Definition 5.2 (Boundary Distinguishability, BD).For any ω, ω′∈ C(y;W), if ω∼rev ω′, then ω|∂rW=ω′|∂rW.(18) In other words, different equivalence classes must correspond to different boundary patches. Theorem 5.3 (RCA End Boundary-Dominated Upper Bound; Unified Version).Exists constant c > 0(depending only on propagation radius r, dimension, and local rule/alphabet size) such that log Comp(y;∂rW)≤c·vol(∂rW).(19) If further satisfying boundary distinguishability (Definition 5.1a), then HW(y)≤log Comp(y;∂rW)≤c·vol(∂rW).(20) where vol per “measure convention(6)” takes counting measure. This theorem does not involve WSIG end; continuous end’s corresponding upper bound and scale control see § 9 and § 4. Theorem 5.4 (Non-Negativity of Reversible Mutual Information).Let Irev(W1:W2) := HW1(y1) + HW2(y2)−HW1∪W2(y)≥0,(21) and monotonically non-decreasing with strengthening boundary consistency constraint; takes extremum when boundary fully closed. Proof. By equivalence class fiberization and matching number submodularity, combined with reversible consistency, obtain non-negativity and monotonicity. 6 6 Complexity Lower Bound and Random Robustness Write K(y) for computable upper bound proxy of Kolmogorov complexity. Definition (Boundary Length Budget): Let ∂rWand constant cas in Theorem 5.1, define bdry(W) := c·vol(∂rW).(22) Definition (Model Capacity): Using constant cfrom Theorem 5.1 and “measure convention(6)”, define ModelCap(W) := c·vol(∂rW).(23) Then in weak dependence and finite propagation scenarios, have HW(y)≳K(y)−bdry(W)+, HW(y)≲ModelCap(W) = c·vol(∂rW),(24) where [x]+:= max{x, 0}, both ends’ constants consistent with Theorem 5.1’s scale. Highly incompressible yneeds larger interpretation family, thus raising HW’s lower bound. 7 Relation to Shannon/von Neumann Entropy Hwmeasures “structurally consistent reversible interpretation family scale”, while Shannon/von Neumann entropy measures distribution or density matrix uncertainty. In i.i.d. and window scale tending to large limit, unit volume density of Hwcan couple with entropy rate; but in strong reversible constraint and boundary-dominated geometric scenarios they separate: Hwmore sensitive to “reversible consistency” and “propagation radius”. 8 Non-Asymptotic Bounds for Toeplitz/Berezin Compression Let Ta,w,h be Toeplitz/Berezin compression of symbol aand window w; write tolerance ball Bε(y). Exists constant Cw,a,h and finite order msuch that log Cap Ψ∈ Erev :|Ta,w,hΨ−y|Hw≤ε/∼rev −log Cap Cw(y)/∼rev ≤Cw,a,h ε+∆(≤m) NPE . (25) where ∆(≤m) NPE jointly controlled by Poisson aliasing, EM finite-order boundary layer, and tail term; if abandlimited/analytic symbol, wexponentially decaying and satisfying frame density condition, constant uniformly bounded by bandwidth and Bernoulli constants. 9 Nyquist Example for Bandlimited Signals Let one-dimensional bandlimited signal, bandwidth B, sampling rate fs≥2B, window wcompact-supported or exponential, energy shell |Ψ|H≤E. Theorem 9.1 (Boundary Dominance Under Nyquist Condition). Hw(y) = log Cap Cw(y)/∼rev ,(26) 7  log Cap Cw(y)/∼rev −log Cap boundary phase/amplitude degrees of freedom/∼rev  ≤∆(≤m) NPE . (27) When window scale R→ ∞ with B, E fixed, unit length Hw/R →0. Proof. By Paley–Wiener space’s Landau density necessary condition and Poisson summation controlling aliasing error, EM finite-order convergence controls boundary layer; boundary dominance given by finite propagation/kernel decay and frame stability. Note (Frame and Uncertainty) Window family taking tight frame can optimize error constant; Balian–Low forbids simultaneous optimal double-localization at critical density, suggesting capacity lower bound cannot be excessively compressed. 10 RCA Examples and Theorems Consider one-dimensional RCA, radius r, window length R. Theorem 10.1 (Capacity Upper Bound for Finite-Radius Propagation).If length-2r boundary patch uniquely determines extension, then HR(y)≤log #{boundary patches} ≤ c·r, (28) and when R→ ∞ with rfixed, unit length HR/R →0. Proof. By Markov-type propagation restriction and boundary determinacy, transversal count bounded by boundary patch count. Theorem 10.2 (Splice and Reversible Mutual Information).For adjacent windows W1, W2 with common boundary band, Irev(W1:W2)monotonically non-decreasing with strengthening boundary consistency constraint, maximized when closed. (29) Proof. Analogous to Theorem 2.3; RCA reversibility and Garden-of-Eden/CHL structure ensure reversible consistency and non-negativity of pairing. 11 Variation and Second-Order Structure Write F(a, w) := log Cap Cw(y)/∼rev . Within regular classes of symbol/window (bandlimited/analytic/exponential window), have δF=⟨Ga, δa⟩+⟨Gw, δw⟩+O∆(≤m) NPE ,(30) where sensitivity functionals Ga,Gwderived from windowed trace and trinity scale; their second-order symmetric part establishes H1/2-type stable Hessian lower bound, reflecting capacity function’s strong convex/concave complementary structure on feasible domain (depending on capacity model and normalization). 8 12 Axiomatization Summary  A1 Reversible Consistency: Count only completions consistent with observation and reversible;  A2 Covariance: Invariant under structure-preserving reversible transformations;  A3 Monotonicity: Non-increasing with window expansion;  A4 Subadditivity: Corrected by reversible mutual information in splice;  A5 NPE–EM Non-Asymptotic Closure: All errors terminate at finite order with explicit upper bounds;  A6 Singularity Non-Increasing/Poles = Principal Scales: Windowing introduces no stronger singularities, poles determine principal scales;  A7 Scale Identity Compatibility: Strictly consistent with φ′/π =ρrel = (2π)−1tr Q. 13 Conclusive Theorems Theorem 13.1 (Well-Definedness and Robustness).Under above axioms and regularity conditions, finite-window reversible completion entropy H(RCA end HW, WSIG end Hw) well-defined, invariant under reversible isomorphisms, Lipschitz stable under window/symbol perturbations, obtains non-asymptotic upper/lower bounds and variational estimates under NPE–EM discipline; its scale dominated by boundary band and propagation radius, embodying quantitative law “boundary = information resource”; strictly compatible with trinity scale. Proof. Synthesis of 3.1, 3.2, 4.1, 5.1, 5.2, and NPE–EM finite-order control. Theorem 13.2 (Local Information Law).In systems with RCA finite radius or continuous bounded propagation, as window scale increases, unit volume density of H(RCA end HW, WSIG end Hw) decays to zero; for bandlimited/exponential windows, decay rate of H/R uniformly controlled by constant set of bandwidth, window decay, and EM order. Proof. By 5.1’s boundary-dominated upper bound and 9.1’s Nyquist framework, combined with EM finite-order termination, obtain unit-scale decay. Appendix A: Explicit Constants of NPE Three-Fold  Poisson Aliasing Term: Let frequency-domain bandwidth Band sampling interval ∆ satisfy 1/∆≥2B(Nyquist), then aliasing error estimated by window spectrum decay of high-frequency leakage; Poisson summation yields main term of discrete–continuous difference.  EM Boundary Layer: For p≤morder termination, remainder given by L∞ bound of periodized Bernoulli function Ppand ζ(p) constant as Rp≤2ζ(p) (2π)pR|f(p)|.  Tail Term: Given by exponential/sub-exponential decay of window and energy shell regularity (finite-order bounded derivatives), yielding geometric or power-law convergence. 9