Representation and Inversion of ``Time'' in EBOC

Representation and Inversion of “Time” in EBOC — Characterizing “Sequence” and “Choice” in a Static Block Universe, and Deriving Consciousness Self-Linearization from Recursive Unfolding of Observation Windows Anonymous Author Version: 2.7 Abstract In a “timeless” static block universe (EBOC), all facts are given as once-for-all structure– measure objects; so-called “time” should be a secondary calibration endogenously invertible from that object, not a primitive coordinate. This paper provides a rigorous route under the unified semantics of EBOC: first, via the window–consensus paradigm, we define “sequence” as a bidirectionally infinite path (consensus chain) on the function graph driven by a unified selector, ensuring “unique successor” through preference aggregation and well-order disambiguation; then, via the identity windowed trace = phase–density calibration (phase derivative = relative density of states = Wigner–Smith group delay trace), we establish “time readout” as window-weight density integral and close it under finite-order Nyquist–Poisson–Euler–Maclaurin error discipline; finally, in KL/Bregman information geometry, we characterize the recursive unfolding of observation windows as an I-projection (minimal KL) sequence, thereby obtaining the consciousness selflinearization theorem and inversion parameters in dual (expectation) coordinates. The core conclusion is: the one-dimensionality of “narrative time” in EBOC can be endogenously inverted via the combined force of “structural selection + metric readout”. 1 Notation & Axioms / Conventions (Calibration Card I: Trinity) The calibration identity holding almost everywhere in the absolutely continuous spectrum φ′(E) π=ρrel(E) = 1 2πtr Q(E),Q(E) = −i S(E)†dS dE (E). where S(E) is the scattering matrix, φ′(E) is the total scattering phase derivative, ρrel is the relative density of states; the identity arises on one hand from the Birman–Kre˘ın formula det S(E) = e2πiξ(E)⇒ξ′(E) = ρrel(E) = 1 2πi d dE log det S(E). Define total scattering phase φ(E) := 1 2ilog det S(E) (continuous branch), then φ′(E) π=1 2πi d dE log det S(E) = ξ′(E) = ρrel(E), fully consistent. On the other hand from Wigner–Smith time-delay matrix and Kre˘ın–Friedel relation ρrel(E) = 1 2πtr Q(E) [?]. 1 (Calibration Card II: NPE Finite-Order Discipline) All windowed computations only allow finite-order Euler–Maclaurin (EM) and Poisson summation; error strictly decomposes as ε=εalias +εEM +εtail, where under Nyquist sampling (band-limited signal, sampling rate >2B)εalias = 0; EM remainder controlled by Bernoulli polynomials and higher-order derivatives of integrand; tail controlled by fast decay and band limitation. This discipline guarantees non-increasing singularity and “pole = primary scale” [?]. Windows and kernels. On energy axis RE, given even window wR≥0 and front-end kernel h≥0 (band-limited, regular, and RRh(E)dE = 1), convolution denoted (h⋆ρ)(E). Working energy band. Denote B:= ess suppXkwRk⊂RE, the essential support of pointwise weight sum of the window family. All assertions in this paper about coverage,bounded overlap (strong/weak), readout and inversion are stated on B. Integrability. Assume ρrel ∈L1 loc(B); accordingly all RE(t) E0ρrel appearing in this paper are well-defined on B. Window family coverage. Let window family {wRk}satisfy X k wRk(E)>0 a.e. on B. Window family bounded overlap. Strong form: there exists C < ∞such that X k wRk(E)≤Ca.e.; Weak form: there exist M < ∞and Wmax <∞such that for any E, #{k:wRk(E)>0} ≤ Mand supk∥wRk∥∞≤Wmax. Under either condition and h≥0,Rh= 1, we have X k wRk(E)h⋆ρrel(E)∈L1 loc;accordingly, Fdefined in § ?? is a locally bounded variation (absolutely continuous) function on B; if further assuming ZBX k wRk(E)h ⋆ ρrel(E) dE < ∞(e.g., finite window family, or PkwRk∈L1(B) and ρrel ∈L1(B)), then Fis globally bounded variation on B. Window family normalization (PUC) and approximate identity kernel. Let PkwRk(E)≡1 a.e. on B, take nonnegative kernel family {hε}ε>0satisfying Rhε= 1 and for all f∈L1 loc(B) we have hε⋆ f →fin L1 loc(B) (ε→0). Under PUC and NPE finite-order discipline, the band-limited quantity hε⋆ ρrel satisfies Fε(E) := X kZE −∞ wRk(E′)hε⋆ ρrel(E′)dE′=ZE −∞ ρrel(E′)dE′+Cε+O(εEM +εtail), where constant Cεtogether with EM/tail terms give a uniform upper bound, and Cε→C0as ε→0. Frames and band limitation. Multi-window Gabor/frame Parseval/Tight construction and Wexler–Raz biorthogonality provide stability and density criteria for windowed reconstruction and multi-channel cooperation; critical sampling constrained by Balian–Low phenomenon [?]. Information geometry. Adopt Legendre potential Λ and Bregman/KL construction: ∇Λ gives expectation coordinates, I-projection is minimal KL under linear moment constraints; KKT conditions characterize unique optimal point and give sensitivity [?]. 2 2 Timeless Characterization of “Sequence” and “Choice” 2.1 Window Graph, Causal Compatibility, and Feasible Paths Take window radius rand allowed fragment set C. Construct De Bruijn-type window graph Γ: vertices are local fragments of length 2r, edges are one-step slides; impose causal compatibility (advancing along edges does not violate underlying dependency preorder). Thus feasible sequences X:Z→ C on the block correspond one-to-one with bidirectional paths on Γ [?]. 2.2 Unified Selector and Function Graph Decomposition For each vertex, aggregate multi-agent preferences as weighted extremum, and disambiguate with well-order, obtaining unified selector Sel and deterministic successor; this yields function graph ΓSel (each point out-degree = 1). Any finite out-degree-1 directed graph decomposes into several directed cycles plus their in-trees; periodic points form cycles, others are transient nodes. This paper defines bidirectionally infinite consensus chain as bidirectionally extended paths on cycles [?]. Proposition 2.1 (Function Graph Structure–Finite Fragment Case).Let allowed fragment set Cbe finite (equivalently: alphabet finite and window radius finite), then each connected component of ΓSel contains exactly one directed cycle, with other vertices flowing into that cycle via directed trees; the cycle admits bidirectional infinite paths, called consensus chains. General infinite case: each connected component contains at most one directed cycle. Proof. Function graphs are standard functional digraphs; their decomposition properties are as stated in the literature (cycles + in-trees) [?]. 2.3 Linear Extension and Threshold Stability For dependency preorder ⪯, by Szpilrajn’s theorem any partial order extends to a total order; on the consensus chain image set take this consistent linear extension as the index coordinate t∈Z. When weights and disambiguation have minimal gap, unique successor remains invariant under small perturbations (threshold stable) [?]. Definition 2.2 (Sequence and Choice).Choice: Given window state v,Sel(v)selects unique successor edge; Sequence: Bidirectional path (vt)t∈Zon ΓSel satisfying vt→vt+1. 3 Representation of “Time”: Phase–Density–Windowed Trace 3.1 Phase Derivative = Relative Density of States = Group Delay Trace On the absolutely continuous spectrum, the Birman–Kre˘ın formula connects spectral shift function ξand S(E): det S(E) = e2πiξ(E)⇒ξ′(E) = ρrel(E) = 1 2πi d dE log det S(E). On the other hand, Wigner–Smith defines Q(E) = −iS†S′, Kre˘ın–Friedel relation gives ρrel(E) = 1 2πtr Q(E). Together yield the identity in Calibration Card I [?]. 3 3.2 Windowed Readout and Non-Asymptotic Closure Define windowed trace readout Obs(R;ρrel) := ZR wR(E)h⋆ρrel(E)dE. Discrete implementation obeys NPE three-way decomposition: aliasing term (Poisson side), boundary Bernoulli layer (EM side), and tail (band-limited decay). When sampling satisfies Nyquist, εalias = 0; EM remainder controlled by Bernoulli coefficients and higher-order derivative bounds; tail controlled by band limitation and window regularity, thus no new singularities introduced [?]. Definition 3.1 (EBOC Time Readout Functional).Given window family {wRk}and kernel h, define T[ρrel] := X kZR wRk(E)h⋆ρrel(E)dE, and under additional integrability assumption ZB |h⋆ρrel|dE < ∞,X k wRk∈L∞(B)∩L1(B) (or finite window family), T[ρrel]is finite and can be given a uniform upper bound via NPE finite-order error; under Nyquist εalias = 0 [?]. 4 Inversion of “Time”: Recovering Linear Order from Window Data 4.1 Phase Integral Index Let consensus chain C={vt}t∈Z. In B,E(t) is taken from the monotone preimage of the readout functional: denote F(E) := X kZE −∞ wRk(E′)h⋆ρrel(E′)dE′. Under band limitation, Nyquist sampling, wRk≥0, h ≥0and window family bounded overlap (strong or weak form),Fis locally bounded variation (absolutely continuous) on B; if further satisfying the global integrability condition given in the previous section, then Fis globally bounded variation on B. On the absolutely continuous part of B, ρrel =ξ′a.e. holds, hence ρrel ≥0a.e. ⇔ξis non-decreasing there. Under this premise Fis monotone non-decreasing; if further adding window family coverage and ρrel >0 (a.e.), then Fis strictly monotone. Select step calibration ∆ >0, define effective index set TF:= {t∈Z|t∆∈ranF|B}. For t∈ TF, take right-continuous generalized inverse E(t) := F−1(t∆), F −1(y) := inf{E:F(E)≥y}. To eliminate additive constant, take baseline index t⋆∈ TFand set E0:= E(t⋆), accordingly define τ(t) := ZE(t) E0 ρrel(E)dE =1 2πZE(t) E0 tr Q(E)dE, by ρrel ∈L1 loc(B) in Notation, the above integral is well-defined on B. Thus τ(t⋆) = 0; if ρrel ≥0(a.e.) then τis monotone non-decreasing, and when window family coverage and ρrel >0 (a.e.) hold, τis strictly increasing [?]. 4 Under the general condition of only satisfying “window family coverage + bounded overlap”, Fprovides strictly monotone energy parameter E(t) and order equivalence; phase coordinate τstill needs to be constructed through Rρrel. If further satisfying PUC + approximate identity kernel, then there exists constant Csuch that τ(t) = F(E(t)) −F(E0) + O(εEM +εtail), thus “time readout” can be directly given by prefix windowed readout (up to constant) with error uniformly controlled by NPE discipline. 4.2 Inversion Theorem Theorem 4.1 (Time Inversion).Under band-limited windows, Nyquist sampling, and finiteorder EM conditions: (1) (General condition) The linear order of any consensus chain Ccan be inverted from the prefix windowed readout Fvia the generalized inverse F−1to a strictly monotone energy parameter E(t), and accordingly obtain a bounded variation parameter equivalent to the chain index t; if ρrel ≥0(a.e.), this parameter is monotone non-decreasing, and when window family coverage and ρrel >0(a.e.) hold, it is strictly increasing. (2) (Additional PUC + approximate identity kernel) Further we have τ(t) = F(E(t)) − F(E0) + O(εEM +εtail), thus phase coordinates can be directly recovered from F(or its Nyquist sampling {F(Ej)}) within a uniform error bound. Proof sketch. (i) By Calibration Card I and Kre˘ın–Friedel relation, reduce windowed trace to Rρrel; (ii) Nyquist closes aliasing to zero, EM remainder and tail controlled; (iii) By integrability assumption ρrel ∈L1 loc(B) we know τis well-defined on B; when ρrel ≥0(a.e.),τis monotone non-decreasing; under window family coverage and ρrel >0 (a.e.), τis strictly monotone and invertible [?]. 5 Recursive Unfolding of Observation Windows ⇒Consciousness Self-Linearization 5.1 Submission = I-Projection (Minimal KL) Let internal state be represented by natural parameter θ, potential function Λ of Legendre type, expectation coordinate X=∇Λ(θ). Each observation step updates target moment Fnto Fn+1, submission/collapse equivalent to θn+1 = arg min θKL(Pθ∥Pθn) s.t. Eθ[T] = Fn+1, i.e., I-projection on linear constraints; unique solution exists and satisfies KKT. Bregman– Euclidean Pythagorean property gives optimal decomposition of projection [?]. 5.2 Linear Response and Quasi-Linear Trajectory When window change is “mild” and NPE order fixed, then Xn+1 −Xn=∇2Λ(θn) (θn+1 −θn) + o∥θn+1 −θn∥, KKT and strong convexity give first-order linear response; reparametrizing the iteration with τfrom § ??, can be viewed as approximate equal-step advance along some fixed vector v∗in expectation coordinates. 5 Theorem 5.1 (Consciousness Self-Linearization).Let Λbe essentially smooth strictly convex potential; windows wRand kernel hband-limited and satisfy Wexler–Raz/Parseval frame stability; sampling Nyquist. Then the submission states {Xn}driven by recursive windows admit a strictly increasing reparametrization map σ:Z→Zand constant vector v∗in expectation coordinates, and there exists function εmicro(R, ∆) = oR→∞,∆→0(1), such that for any baseline nand all integers m ∥Xσ(n+m)−Xσ(n)−m v∗∥ ≤ |m|hCεEM +εtail+εmicro(R, ∆)i. Convention: εmicro(R, ∆) depends only on window/kernel and NPE order, satisfying εmicro(R, ∆) → 0(as R→ ∞,∆→0with NPE order fixed), the bound reflects linear accumulation of error with step count |m|. Accordingly, consciousness exhibits quasi-linear dominant trajectory in its own dual coordinates. Proof sketch. Wexler–Raz and Parseval/Tight guarantee readout mapping and reconstruction stability; KKT and Bregman geometry’s Pythagorean identity give first-order linearization of each I-projection step; NPE constraint ensures noise terms are entirely dominated by finite-order remainder [?]. 6 Unified Trinity of “Sequence–Choice–Time”  From block to sequence: Unified selector generates consensus chain (bidirectionally infinite path) on function graph, and via Szpilrajn assigns consistent linear extension;  From sequence to time: Phase–density–group delay calibration identity makes “time readout” become integral of window-weighted density; Nyquist–EM guarantee non-asymptotic closure;  From time to consciousness: I-projection on recursive windows enables dual coordinates to acquire quasi-linear principal axis, with τas the endogenous parameter of that axis. 7 Thresholds, Singularities, and Implementation Notes 1. Threshold/resonance: Singularities (such as poles, branch points) of φ′(E) (equivalently ρrel(E)) correspond to continuous spectrum thresholds and resonances; zeros do not constitute general criteria. Windowing and finite-order EM do not increase singularity, maintaining “pole = primary scale” [?]. 2. Frames and density: Multi-window Parseval/Tight and Wexler–Raz biorthogonality guarantee robust reconstruction; critical sampling constrained by Balian–Low, redundant sampling intervals recommended [?]. 3. Sampling and aliasing: Band limitation and Nyquist sampling are sufficient conditions for closing aliasing; in engineering implementation, modulation–downsampling strategy can achieve in-band Nyquist [?]. 8 Conclusion From the EBOC static block perspective, “time” is not a primitive axis but generated in three steps: (i) window–consensus condenses choice into function graph’s consensus chain;(ii) phase–density enables the chain to acquire invertible time calibration (windowed trace readout); (iii) KL/Bregman makes submission of recursive windows exhibit self-linearization 6 in dual coordinates. This route is entirely anchored on verifiable criteria: function graph and linear extension, phase–density identity and NPE finite-order error discipline, Legendre–Bregman and KKT optimization structure, thereby reducing the one-dimensionality of “narrative time” to the result of structural choice + metric readout. References [1] Yafaev, D.R. 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