Hierarchical ``Halting''--``View-Relative Completeness'' Unified Theory\\ (Formal Construction Under EBOC--WSIG--RBIT Framework)

Hierarchical “Halting”–“View-Relative Completeness” Unified Theory (Formal Construction Under EBOC–WSIG–RBIT Framework) Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: 1.29 Abstract Under the frameworks of Windowed Scattering–Information Geometry (WSIG/CCS), Eternal-Static-Block Computation (EBOC), and Resource-Bounded Incompleteness Theory (RBIT), we establish viewand resource-dependent hierarchical “halting” criteria and provide equivalence theorems consistent with sampling–frame thresholds, Poisson–Euler–Maclaurin (EM) error theory, and “trinity” scale (phase derivative–relative state density–group delay). We unify the sign and branch conventions of the trinity equality chain, clarify the interchangeable use of Φf= arg det S’s absolutely continuous branch and Birman–Kre˘ın (BK) gauge; characterize “readout/pointer” and Ky–Fan minimum at operator level; tighten equivalence conditions of “sampling-period closed-loop” and “readout halting” under strict bandlimiting and tight frame regularity premises; and restate RBIT assertions in G¨odel–Chaitin type and extension chain incompleteness. Keywords: Hierarchical halting; View-relative completeness; WSIG; EBOC; RBIT; Trinity scale; Sampling theory; Frame theory 1 Notation, Gauge, and Regularity Discipline 1.1 View Quadruple and Windowed Readout Let the view quadruple be View = (H0, wR, h, sample),(1) and introduce window center parameter c∈R. Define shifted-scaled window wR,c(E) := R−1wE−c R.(2) 1 where H0is reference end (free end of scattering pair (H, H0)), wRis energy window (scale R), his bandlimited frontend kernel, sample is sampling/frame scheme (step ∆, point set Λ, etc.). Windowed readout takes Obs(R, ∆; c) = ZR wR,c(E)h∗ρ⋆(E)dE +εalias +εEM +εtail,(3) where ρ⋆can take relative state density ρrel or Herglotz density ρm. Throughout we uniformly adopt standard convolution (h∗ρ)(E) := ZR h(E−E′)ρ(E′)dE′.(4) When necessary, use wR,c as weight measure without affecting regularity and error theory conclusions. Window Normalization and Baseline (Zero-Sum Condition): Adopt one of two gauges: (A) Zero-Mean Window: RRwR(E)dE = 0. (B’) External Baseline (Non-Trivial Zero-Sum): Take cext Rindependent of ρrel (obtainable from reference end H0’s density ρm0or independent low-pass calibration), define ˜ρrel := ρrel −cext R. Then ZR wR˜ρrel = 0 ⇐⇒ ZR wRρrel =ZR wRcext R.(5) Equivalence Applicability: Theorem A.1 in § 2 is uniformly stated under gauge (A) or (B’) to avoid tautology from endogenous baseline. Translation Invariance: For any c∈R, ZR wR,c(E)dE =ZR wR(E)dE, (6) thus zero-sum/baseline relations under gauge (A)/(B’) are strictly consistent with subsequent conditions formulated via wR,c. (Unified Notation) To cover both gauges (A)/(B’), write ¯ρrel := (ρrel,gauge (A); ˜ρrel,gauge (B).(7) Hereafter all mentions of “zero-sum/balanced flux” expressed via ¯ρrel; if smoothing adopted, uniformly compute via h∗¯ρrel. Regularity and Bandlimiting Discipline (Poisson–EM Non-Introducing Singularities): Fix an EM truncation order M≥1. To ensure Poisson–EM error closure without introducing new singularities, require h∈C2M(R)∩L1,wR∈C2M c(R) or strictly bandlimited, and (h∗ρ⋆)’s derivatives up to order 2Mintegrable at window endpoints; then EM truncation remainder consists only of endpoint derivatives and Bernoulli polynomials, Poisson term auditable or zero under bandlimiting or sufficient decay. Above regularity and Poisson–EM closure confined to σac(H); for δ-mass in σpp(H), adopt wRsupport avoidance or suppress its contribution via h’s smooth convolution, thus maintaining “non-introducing singularities”. 2 1.2 Windowed Scattering “Trinity” Scale (Normalization, Branch, and BK Sign) Almost everywhere on absolutely continuous spectrum, take scattering matrix S(E)∈ U(N), Wigner–Smith matrix Q(E) := −i S(E)†∂ES(E),(8) global phase Φf(E) := arg det S(E), φ(E) := 1 2Φf(E).(9) Define relative state density (Krein–Friedel sense) ρrel(E) := 1 2π∂EΦf(E).(10) On σac, fix absolutely continuous branch of Φf, thus ∂EΦfunique almost everywhere; equivalently, using BK gauge det S(E) = exp(−2πi ξ(E)), ρrel(E) = −ξ′(E). One-Line Derivation: tr Q(E) = −itrS†∂ES=−i ∂Elog det S(E) = ∂EΦf(E).(11) Accordingly, strict equality chain holds almost everywhere on σac: φ′(E) = 1 2tr Q(E) = π ρrel(E).(12) 1.3 Sampling–Frame Thresholds and Impossibilities With phase density dν =φ′ πdE as geometric scale: Landau provides necessary density lower bound for sampling/interpolation; Wexler–Raz dual characterizes Gabor/multiwindow frame consistency; Balian–Low theorem gives global localization obstruction at critical lattice. These jointly delimit applicability domain of stable readout and fixedpoint criteria. Strict Definition of Nyquist Threshold: Write windowed-smoothed density gR,c(E) := wR,c(E) (h∗ρ⋆)(E), its Fourier support half-width Ωeff(R) := inf{Ω>0 : supp dgR,c ⊆[−Ω,Ω]}.(13) Define aliasing threshold ∆c(R) := π Ωeff(R).(14) Aliasing is zero if and only if supp dgR,c ⊆[−π/∆, π/∆]; sufficient condition is ∆ ≤∆c(R) (translation doesn’t change frequency domain support). 1.4 “Readout/Pointer” Operator-Level Definition and Ky–Fan Minimum; Born = I-Projection Induce positive operator WR(Toeplitz/Gram form) from wRand h.One verifiable construction (positive-definite, trace-class, robust): Let uR,c(E) := wR,c(E)2(or take uR,c(E) = |wR,c(E)|),(15) 3 take measurable matrix kernel Kh:R→CN×N, define WR:= ZR uR,c(E)Kh(E)†Kh(E)dE. (16) Then WR≥0 and tr WR<∞;add non-triviality constraint tr WR>0, thus p(j) = λj tr WR , N X j=1 p(j) = 1 (17) well-defined. This construction only applies to “pointer/Ky–Fan” criterion (iii), not changing (i)(ii)(iv)’s window weight and zero-sum criteria (still computed via wR,c). Action Space and Dimension: Below stipulate WRacts on channel space CN (induced by window/kernel energy weighting on each channel), so its spectral decomposition written as {(λj, ψj)}N j=1, with this Nas dimensional reference for subsequent rank conditions like rank P⊥=N−k. Correspondingly, index jin p(j) = ⟨ψj, WRψj⟩/tr WR takes 1 ≤j≤N. Let WR=PjλjPψjbe spectral decomposition. Eigenvalue Gauge: Take {ψj}N j=1 as orthonormal eigensystem, let Pψj=|ψj⟩⟨ψj|, then ⟨ψj, WRψj⟩=λj, tr Pψj= 1. Accordingly define p(j) := ⟨ψj, WRψj⟩ tr WR =λj tr WR , N X j=1 p(j) = 1.(18) Take admissible family as coaxial mixed family M:= nX j gjPψj:gj≥0,X j gj= 1o.(19) Tighten admissible family to coaxial block-coarsened family: Take partition of index set P={Bα}, and block weights µα≥0,Pαµα= 1, define ΠP,µ := X α µα 1 |Bα|X j∈Bα Pψj∈ M.(20) Corresponding block-coarsening channel ΓPis intra-class uniformization on {ψj}basis: (ΓPp)(j) = 1 |Bα|X k∈Bα p(k), j ∈Bα.(21) (Notation Clarification) ΓPis channel (Markov linear operator), ΠP,µ is mixed state; they are different objects. This paper does not use notation ΓΠP,µ . Non-Degenerate Coarsening Assumption: Unless otherwise stated, partition P is not single-point refinement, i.e., exists αwith |Bα| ≥ 2. This constraint excludes trivial case ΓP= Id, making conclusion “Born=I-projection fixed point holds only with intra-block spectral degeneracy” consistent with feasible set. Given rank k, define “pointer projection” P⋆ k∈arg min P:P2=P, P †=P, rank P=k tr(PWR),(22) 4 whose optimal value equals sum of smallest keigenvalues of WR(Ky–Fan principal spectral sum minimization). Dual Explanation: Let complementary projection P⊥:= I−P(also orthogonal projection), then min P⊥: (P⊥)2=P⊥,(P⊥)†=P⊥, rank P⊥=N−k tr(P⊥WR) (23) equivalent to maximization for P, choosing largest keigenvalues; this paper’s gauge fixes adoption of “Pointer=Ky–Fan Minimum”. On probability side, perform Csisz´ar Iprojection with channel family (two-block threshold feasible set): Γ⋆∈arg min P∈T2 DKLp∥ΓPp,T2:= P={B↓, B↑}induced by spectral threshold of WR. (24) Under coaxial partitioning and frame regularity, pointer Ky–Fan minimum and this two-block threshold I-projection optimal channel ΓP∗are compatible but generally non-equivalent. Only when simultaneously satisfying (a) intra-block spectral degeneracy and (b) strict bandlimiting + perfect reconstruction, holds (iii) ⇐⇒ (i) ⇐⇒ (ii) ⇐⇒ (iv).(25) If only satisfying (a) without (b), only have (iii) ⇐⇒ (i) ⇐⇒ (ii),(26) while (iv) is only near-closed-loop within error budget (not included in equivalence). See § 2’s (iii) and its two “moreover” clauses for details. Thus this paper fixes adoption of “Pointer=Ky–Fan Minimum” as necessary criterion for stability; if intra-block degeneracy exists, then “Born=I-projection fixed point” holds simultaneously. 2 Hierarchical “Halting” Predicates Let system S, resource R= (L;m, N, ε), view View = (H0, wR, h, sample). Define H0⇒H1⇒H2⇒H3⇒H4⇒H5,(27) converse generally doesn’t hold:  H0: Syntactic halting (no successor state).  H1: Dynamical halting (function graph enters loop); loop length p= 1 is self-loop fixed point,non-equivalent to “no successor” H0;p > 1 is oscillatory halting.  H2: Readout halting. Fix View, write windowed-smoothed density gR,c(E) := wR,c(E) (h∗ρ⋆)(E). If its Fourier support satisfies supp dgR,c ⊆[−π/∆, π/∆],(28) then εalias = 0. A set of sufficient conditions is supp cwR⊆[−Ωw,Ωw],suppbh⊆[−Ωh,Ωh],supp bρ⋆⊆[−Ωρ,Ωρ],(29) 5 with Ωw+ Ωh+ Ωρ≤π/∆. Simultaneously require d dlog RObs(R, ∆; c)→0, εEM, εtail controlled,(30) and balanced flux satisfies ZR wR,c(E) ¯ρrel(E)dE = 0.(31) where d dlog Runiformly refers to logarithmic scale derivative under wR,c(E) = R−1w(E− c)/Rwith fixed c.  H3: Trinity halting. Satisfies φ′=1 2tr Q=πρrel and “Pointer=Ky–Fan minimum stable” (if WRintra-block degenerate, then Born=I-projection fixed point also holds).  H4: Multi-view halting (reaching H3uniformly over view family V).  H5: Theoretical halting/resource completeness (globally unreachable at RBIT level). 3 Main Theorem A (Under Fixed View: “Relative Halting ⇔Sampling-Period Closed-Loop”) Theorem 3.1 (Tightened Equivalence).Let View = (H0, wR, h, sample)satisfy: effective bandwidth Ωeff (R)of gR,c finite and ∆≤∆c(R) = π/Ωeff(R)(thus εalias = 0), windows/bases used constitute tight frame (Parseval; or frame bounds uniform and normalized), and satisfy § 0.4’s structural coaxial assumption and coaxial blockcoarsening channel (M,ΓP) hold. (Measure Unification Convention) Uniformly write ¯ρrel := (ρrel,gauge (A); ˜ρrel,gauge (B).(32) If (i)’s observation main term adopts smoothing h∗ρrel, then in (ii) synchronously express zero-sum via h∗¯ρrel to ensure (i)’s observation main term and (ii)’s zero-sum criterion are in same measure. (Phase Monotonicity) On window support require φmonotone non-decreasing to ensure generalized inverse E(·)in (iv) well-defined; if not satisfied, (iv) doesn’t participate in equivalence chain. Under § 0.1 gauge (A) or (B’) and threshold/frame regularity premises: (Strict Bandlimiting) If wR, h, ρ⋆strictly bandlimited and ∆≤∆c(R)(no aliasing), and moreover adopt Poisson–Shannon perfect reconstruction of Parseval tight frame (or equivalent exact summation) to compute readout, making εEM =εtail = 0, then (i) ⇔(ii) ⇔(iv); (General Regularity) Under only Poisson–EM regularity and bounded tail, (i) ⇔ (ii); (iv) is only near-closed-loop within error budget (gives (i)⇒(iv); while (iv) only implies approximate zero-sum, i.e., (ii) holds approximately in sense of εEM +εtail,does not imply strict equality of (ii)). 6 Moreover: (Strict Bandlimiting+Perfect Reconstruction) Under strict bandlimiting and perfect reconstruction premise, if and only if WRintra-block degenerate, exists channel ΓPsuch that (iii) ⇐⇒ (i) ⇐⇒ (ii) ⇐⇒ (iv); (General Regularity) If and only if WRintra-block degenerate, then (iii) ⇐⇒ (i) ⇐⇒ (ii), while (iv) is only near-closed-loop within error budget and not equivalent. In general, “Pointer=Ky–Fan minimum” (denoted (iii)) is compatible but not necessarily equivalent to above criteria. (i) Readout halting (H2; definition see § 1, includes Nyquist no-aliasing, Poisson–EM error controlled, and balanced flux zero-sum); (ii) ZR wR,c(E) ¯ρrel(E)dE = 0; (iii) Pointer basis is Ky–Fan minimum; (iii) Two-block threshold Born = I-projection fixed point (coaxial partition): Exists two-block channel ΓP∈ {ΓP:P ∈ T2}induced by spectral threshold of WR, such that p= ΓPp. (33) This fixed point holds non-trivially if and only if WRhas spectral degeneracy within each non-trivial block Bα(i.e., λjconstant on Bα); then (iii)’s equivalence conclusion with (i)(ii) (and under “strict bandlimiting+perfect reconstruction” with (iv)) consistent with above “moreover” two paragraphs. (iv) One-step closed-loop of phase-coordinate uniform sampling (generalized inverse version): Take right-continuous generalized inverse E(θ) := inf{E:φ(E)≥θ}.(34) Given window center cn, set cn+1 := Eφ(cn) + π.(35) When ∆≤∆c(R)and tight frame regularity (verifiable via Wexler–Raz dual kernel): (Strict Bandlimiting) If wR, h, ρ⋆strictly bandlimited and ∆≤∆c(R), then εalias = 0; if moreover adopt Parseval tight frame and Poisson–Shannon perfect reconstruction (or equivalent exact summation formula) to compute windowed readout, can make εEM =εtail = 0; under this premise Obs(R, ∆; cn+1) = Obs(R, ∆; cn).(36) Otherwise, only obtain near-closed-loop within error budget (consistent with next “general regularity”). (General Regularity) Under only § 0.1’s Poisson–EM regularity and bounded tail, closed-loop is near-closed-loop within error budget: Obs(R, ∆; cn+1)≈Obs(R, ∆; cn),(37) with deviation controlled by εEM +εtail. Proof Sketch.  (ii)⇒(i): Main term zero and Poisson–EM controlled;  (Strict Bandlimiting+Perfect Reconstruction) (iv)⇒(ii): Under strict bandlimiting and Poisson–Shannon perfect reconstruction, via Parseval and kernel diagonal identity, reduce one-step closed-loop to density zero-sum; (General Regularity) only implies approximate zero-sum, with deviation controlled by εEM +εtail; 7  (i)⇒(iv): Under Nyquist and tight frame premise, strict bandlimiting+perfect reconstruction gives equality closed-loop; otherwise near-closed-loop within error budget (deviation controlled by εEM +εtail).  Side Note: From (i) can construct non-increasing Ky–Fan objective, but not guaranteed to reach minimum, so (iii) is compatible but not equivalent with (i)(ii)(iv); (iii)’s equivalence scope see case-by-case statements in above summary. Thresholds and obstructions jointly guaranteed by Landau necessary density, Wexler– Raz duality, and Balian–Low theorem. 4 Main Theorem B (Changing View ≡Adding Theory/Extending Dictionary) Definition 4.1 (View Extension).View 7→ View′includes H07→ H′ 0(relabeling ρrel), (wR, h, sample)7→ (wR′, h′,sample′) and scale (Mellin/logarithmic) switching. Theorem 4.2 (Invariants and Re-Illumination).Under Poisson–EM regularity and tight frame premise, view extension generates no new singularities, but can change visibility of windowed phase density: H3under one view can be broken by another view, thus revealing new oscillation–period. 5 RBIT Interface: Completeness Growth = Halting Boundary Extrapolation Theorem 5.1 (Chaitin-Type Incompleteness).For any consistent and computably enumerable theory Tinterpreting PA, exists constant cTsuch that when n > cT, concrete proposition “K(x)≥n” is true but unprovable in T. Theorem 5.2 (Extension Chain Non-Termination).For any consistent and computable extension chain Tt+1 =Tt+ ∆t, each stage thas G(t)undecidable in Tt. Corollary 5.3. In resource–statistics unified coordinate R= (L;m, N, ε)and view family V, “pursuing completeness” i.e., continuously expanding resources and extending views, is structurally equivalent to “pursuing non-halting”. 6 EBOC–Consensus Chain Discrete Mirror and Computability Under causal net–window constraint–uniform choice function, obtain unique successor function graph f:V→V. Its connected components decompose into directed loops and attached in-trees (line/half-line as limit cases); “halting” is self-loop (length 1 loop), general p > 1 is oscillatory non-halting. Loop detection achievable in linear time, constant space. 8 7 Operationalization of “Heat Death” and Verifiable Scheme Definition 7.1 (View-Relative Heat Death).Given View, if ZwR,c(E) ¯ρrel(E)dE ≈0,ZwR,c(E)1 2tr Q(E)dE ≈0,(38) and Poisson–EM three-fold error closes within budget, then reaches view-relative heat death. Protocol (Verifiable Closed-Loop): Reference calibration (inverting (H, H0)’s ρm, ρm0); window/kernel KKT optimum; record (∆, M, L) and Landau threshold, multiwindow frame verified via Wexler–Raz dual kernel; jointly solve Ky–Fan minimum and minimum-KL criterion; scan views over H0, window/kernel/scale to detect “re-illumination”. 8 Engineering Thresholds and Cross-Domain Consistency 1. Nyquist Alias Elimination: Strict bandlimiting and ∆ ≤∆cmake εalias = 0, Poisson for alias audit. 2. Frame Stability and Obstruction: Count via dν =φ′ πdE, Landau lower bound, Wexler–Raz duality, and Balian–Low obstruction jointly control stability domain and critical degeneracy. 3. EM Truncation: Finite-order EM only introduces endpoint Bernoulli correction, generates no new singularities. 4. Group Delay Cross-Domain Unification: Q=−i S†∂EShas unified expression and statistical structure in quantum, acoustic, and electromagnetic scattering. 9 Conclusion Under trinity scale φ′(E) = 1 2tr Q(E) = πρrel(E) and Poisson–EM regularity discipline, under fixed view: (Strict Bandlimiting+Perfect Reconstruction) If strictly bandlimited and adopting Poisson–Shannon perfect reconstruction of Parseval tight frame (or equivalent exact summation), making εEM =εtail = 0, then (when φmonotone non-decreasing on window support) (i) Readout halting ⇐⇒ (ii) Window-weighted zero-sum ⇐⇒ (iv) Phase–period closed-loop (39) (General Regularity) Only have (i) ⇔(ii); (iv) gives near-closed-loop within error budget, compatible but not equivalent to (i)(ii). “Pointer=Ky–Fan minimum” (denoted (iii)) is compatible with above criteria, but generally non-equivalent (serves as necessary criterion for stability, not claiming equivalence with (i)(ii) or (iv)). Regarding (iii): 9