Phase--Frequency Unified Metrology and Experimental Testbeds\\ in Computational Universe:\\ Unified Time Scale Implementation\\ from FRB Vacuum Windowing Upper Limit\\ to \delta-Ring Scattering Identifiability

Phase–Frequency Unified Metrology and Experimental Testbeds in Computational Universe: Unified Time Scale Implementation from FRB Vacuum Windowing Upper Limit to δ-Ring Scattering Identifiability Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous “computational universe” framework, universe axiomatized as discrete object Ucomp = (X, T,C,I), upon which constructed discrete complexity geometry, discrete information geometry, control manifold (M, G) induced by unified time scale, task information manifold (SQ, gQ), and time–information–complexity joint variational principle. Unified time scale given by scattering master scale κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) unifying phase derivative, spectral shift density, and Wigner–Smith group delay trace as single scale. However, this framework still remains mainly at “theoretical geometry” level, has not systematically given how to metrologically measure and calibrate unified time scale and computational universe structure in actual experiments. This paper, on basis of computational universe–unified time scale–spectral windowing readout, constructs cross-platform metrology paradigm using “phase–frequency” as sole observable, implementing it on two representative testbeds: cosmologicaldistance Fast Radio Burst propagation (FRB) and laboratory-scale δ-ring–Aharonov– Bohm (AB) flux scattering. Core idea: from computational universe perspective, all observables realized through phase–frequency readout under unified time scale; FRB and δ-ring scattering respectively provide cosmic-scale and laboratory-scale “homologous readouts”, viewable as implementations of same metrology paradigm at different scales under unified time scale and complexity geometry. Main results of this paper: 1. Under framework of categorical equivalence between computational–physical universes, introduce “phase–frequency readout functor” PhFr, sending any physically realizable computational universe object to metrology object containing only phase–frequency data. Prove PhFr compatible with unified time 1 scale master scale: under traceable perturbation and wave operator completeness, PhFr output completely determined by κ(ω) and finite spectral– scattering invariants. 2. For FRB, construct “vacuum polarization windowing upper limit” model: window FRB frequency-domain phase using PSWF/DPSS type window functions, prove under fixed complexity budget and cosmological distance constraints, any unified time scale variation δκ(ω) contribution to FRB phase residual can be bounded by strict upper bound; if observed residual below this bound, obtain unified time scale type upper limit on vacuum polarization or other new physics. 3. For δ-ring–AB flux scattering, restate equivalence between spectral quantization equation f(k, αδ, θ) = cos(kL)+(αδ/k) sin(kL)−cos θ= 0 and “amplitude-corrected phase closure” cos γ(k) = |t(k)|cos θ and prove under computational universe–control manifold perspective: under spectral observation {kn(θ)}at fixed (L, θ), δ–coupling strength αδand AB flux θare identifiable in non-pathological domain (Jacobian full rank), usable as “laboratory ruler” for unified time scale–phase metrology. 4. Under unified time scale–spectral windowing readout framework, embed FRB and δ-ring scattering in same “phase–frequency metrology universe”, prove existence of “cross-platform scale unification condition”: when FRB phase residual and δ-ring scattering spectral shift both explained by same κ(ω) model, their windowed readouts belong to same equivalence class on appropriate PSWF/DPSS space, thus can calibrate and consistency-test unified time scale through joint fitting. 5. Embed above phase–frequency metrology structure into time–information– complexity variational principle, formalize “choosing FRB/δ-ring window functions and control parameters” as variational problem on joint manifold, give variational conditions for “simultaneously using cosmic-scale and laboratoryscale phase–frequency readouts to maximize unified time scale identifiability under finite complexity budget”. This paper thus completes experimental implementation design of “phase–frequency unified metrology” within computational universe framework: FRB and δ-ring scattering become two-end testbeds of unified time scale and complexity geometry, PSWF/DPSS window functions become natural tools for error control, both jointly constructing cross-scale, cross-platform, yet completely unified phase–frequency metrology system under computational universe perspective. Keywords: Computational universe; Phase-frequency metrology; FRB; δ-ring scattering; Unified time scale; PSWF/DPSS; Experimental testbed 1 Introduction In previous series works, we have completed constructions at following levels: 2 1. At discrete level, abstract universe as axiomatic computational universe object Ucomp = (X, T,C,I), upon which construct complexity graph Gcomp = (X, E, C), complexity distance dcomp, complexity dimension and discrete Ricci curvature. 2. At unified time scale–scattering theory level, introduce κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω), as “trinity master scale”, unifying scattering phase derivative, spectral shift density, and group delay trace. 3. Through unified time scale and complexity geometry construct control manifold (M, G), prove discrete complexity distance converges to geodesic distance dGin refinement limit. 4. Through observer family and relative entropy second-order structure construct task information manifold (SQ, gQ), and give joint variational principle of time– information–complexity on joint manifold EQ=M×SQ. 5. In spectral windowing error control work, introduce PSWF/DPSS window functions, show in unified time scale–frequency domain, they are optimal readout windows under finite time–bandwidth–complexity budget. These results lay foundation for constructing “unified time scale–computational universe” theoretical system, but do not directly answer key question: how to metrologically **measure and calibrate** this unified time scale master scale through concrete physical experiments? Mathematical existence of unified time scale insufficient to demonstrate its physical measurability; we need to connect scattering master scale with actually observable phase–frequency data, and design cross-platform metrology strategy so phase– frequency readouts from cosmic scale and laboratory scale can be jointly used to test and calibrate unified time scale. In this context, Fast Radio Bursts (FRB) and δ-ring–AB flux scattering become two very natural testbeds:  FRB are short-duration broadband radio signals traversing cosmological distances, whose propagation phase, group delay and dispersion structure contain integrated information about cosmological medium and vacuum properties; under unified time scale–scattering perspective, FRB essentially “cosmic-level scattering experiment”.  δ-ring–AB flux scattering is precise measurement of spectral–scattering structure of one-dimensional ring geometry, point potential and AB flux at laboratory scale; its spectral quantization equation and phase closure provide highly controllable phase– frequency testbed, usable for “reverse calibration” of unified time scale model under known geometric parameters and coupling constants. Goal of this paper: unify embedding of FRB and δ-ring scattering into computational universe–unified time scale framework, establish metrology paradigm using phase– frequency as sole readout, so two types of experiments can mutually calibrate and consistencytest on same unified time scale master scale. 3 2 Phase–Frequency Readout Functor in Computational Universe This section, under background of computational–physical universe categorical equivalence, introduces “phase–frequency readout functor” PhFr, whose output contains only phase–frequency data, directly connected with unified time scale master scale. 2.1 Review of Physical–Computational Universe Equivalence In previous categorical equivalence work, we constructed physical universe category PhysUnivQCA and computational universe category CompUnivphys, giving mutually inverse functors F:PhysUnivQCA →CompUnivphys, G:CompUnivphys →PhysUnivQCA. Physical universe object abstractable as Uphys = (M, g, F, κ, S), computational universe object as Ucomp = (X, T,C,I). Functors F, G preserve structure of unified time scale density κ(ω) and scattering data S(ω): from physical side to computational side, unified time scale discretized as singlestep cost; from computational side to physical side, complexity geometry continuized as control–scattering manifold. 2.2 Phase–Frequency Data Objects Define “phase–frequency data object” as UPhFr = (Ω,Θ(ω), κ(ω)), where Ω ⊂Ris effective frequency band, Θ(ω) is total scattering phase (or its normalization), κ(ω) is unified time scale density. According to unified time scale master scale, under traceable perturbation condition κ(ω) = Θ′(ω)/π holds up to additive constant. 2.3 Phase–Frequency Readout Functor Definition 2.1 (Phase–Frequency Readout Functor).Define functor PhFr :PhysUnivQCA →PhFrUniv, where PhFrUniv objects are UPhFr, morphisms are frequency-domain transformations preserving ωand Θ(ω), κ(ω) structure. 4 For physical universe object Uphys,PhFr(Uphys) = (Ω,Θ(ω), κ(ω)) determined by its scattering data and unified time scale master scale. Through categorical equivalence, given computational universe object Ucomp, first use G(Ucomp) = Uphys to obtain physical universe, then apply PhFr to obtain phase–frequency data object. Thus obtain composite functor PhFr ◦G:CompUnivphys →PhFrUniv. Proposition 2.2 (Consistency of Unified Time Scale and Phase–Frequency Readout). Under traceable perturbation and wave operator completeness conditions, phase–frequency readout object UPhFr = (Ω,Θ(ω), κ(ω)) completely determined by unified time scale density κ(ω)and constant phase shift, i.e., Θ(ω) = πZω κ(ω′) dω′+ Θ0. In particular, any two pairs (Θ, κ),(Θ′, κ′)with κ≡κ′and Θ−Θ′constant correspond to same unified time scale structure. Proof omitted. 3 FRB Vacuum Polarization Windowing Upper Limit: Computational Universe Perspective This section views FRB propagation as cosmic-scale scattering–propagation process, constructs “vacuum polarization windowing upper limit” under unified time scale–spectral windowing. 3.1 Scattering–Propagation Model of FRB Propagation For simplicity, consider FRB signal complex amplitude A(ω) in frequency domain, whose phase part writable as A(ω) = |A(ω)|exp(iΦFRB(ω)). If propagation includes only known dispersion and reionized medium contributions, then ΦFRB(ω) = Φknown(ω)+Φnew(ω), where Φknown from conventional dispersion measure and medium model, Φnew represents possible contributions from vacuum polarization, new particles, or unified time scale perturbations. Under unified time scale–scattering perspective, ΦFRB(ω) understandable as effective scattering phase ΘFRB(ω), whose derivative gives effective time scale density perturbation δκFRB(ω) = 1 π∂ωΦnew(ω). 5 3.2 Windowed FRB Phase and Error Upper Bound Observationally, we can only measure phase–frequency data in finite frequency band ΩFRB = [ωmin, ωmax] with finite resolution. Introduce window function WFRB(ω) (e.g., generated by PSWF/DPSS spectrum) and define windowed residual RFRB =ZΩFRB WFRB(ω)ΦFRB(ω)−Φknown(ω)dω. If unified time scale perturbation δκFRB(ω) of FRB signal has constraint |δκFRB| ≤ Λ under some spectral norm, then through integration and Cauchy–Schwarz inequality obtain |RFRB| ≤ Λ|WFRB|L2(ΩFRB )CFRB, where CFRB determined by cosmological propagation kernel and geometric factors. Conversely, if observationally residual |RFRB| ≤ εobs, obtain upper bound on unified time scale perturbation Λ≥Λmin ≥|RFRB| |WFRB|CFRB . Writing optimal window function choice problem as constrained minimization of |WFRB|, classical results show PSWF/DPSS type windows minimize error upper bound under given time–frequency–complexity budget, thus giving “FRB vacuum polarization windowing limiter”. 4δ-Ring-AB Flux Scattering Spectral-Phase-Scattering Equivalence and Identifiability This section, from computational universe–control manifold perspective, restates spectral– phase–scattering structure of δ-ring–AB flux scattering, gives identifiability theorem. 4.1 δ-Ring-AB Flux Model Consider one-dimensional ring, circumference L, coordinate x∈[0, L) with periodicity x∼x+L. Introduce point δ–potential on ring, strength αδ, and AB flux θ∈[0,2π). Corresponding Hamiltonian (unit mass, ignoring constants) writable as H=−∂2 x+αδδ(x), boundary condition includes AB phase: ψ(L−) = eiθψ(0+), ψ′(L−) = eiθψ′(0+). Solving eigenequation Hψ =k2ψyields spectral quantization condition f(k, αδ, θ) = cos(kL)+(αδ/k) sin(kL)−cos θ= 0. Corresponding scattering amplitude t(k) and phase γ(k) satisfy certain phase closure, typical form 6 cos γ(k) = |t(k)|cos θ, where functional relationship between γ(k) and kL,αδgiven by scattering theory. 4.2 δ-Ring Control Manifold in Computational Universe View δ-ring scattering as computational universe on low-dimensional control manifold: control parameter space Mδ-ring ={(L, αδ, θ)}, equipped with metric G, e.g., G=gLLdL2+gααdα2 δ+gθθdθ2. Spectral observation {kn(θ)}corresponds to data points on information manifold, while unified time scale density given by scattering phase derivative and group delay. δring thus becomes highly controllable “computational sub-universe” on three-dimensional control manifold, usable for unified time scale and phase–frequency metrology. 4.3 Spectral–Phase–Scattering Equivalence Theorem Theorem 4.1 (Spectral Quantization and Phase Closure Equivalence).In δ-ring–AB flux model, spectral quantization condition f(k, αδ, θ)=0 and phase–amplitude closure cos γ(k) = |t(k)|cos θ equivalent under usual scattering regularity conditions; in particular, when |t(k)| → 1 (weak scattering or transmission resonance), reduces to pure phase closure cos γ(k) = cos θ. Proof sketch. From boundary conditions and δ–potential jump conditions derive transfer matrix, require wave function to match itself after one loop around ring, obtain spectral quantization equation; on other hand, compute scattering matrix elements t(k), r(k) and phase shift γ(k), rewrite spectral condition as phase–amplitude closure. Algebraic equivalence between them verified through direct substitution and simplification. See Appendix B.1 for details. 4.4 Parameter Identifiability Theorem Theorem 4.2 (Local Identifiability of δ-Ring).Given Land several AB flux values θj, if observe sufficiently many eigenwavenumbers {kn(θj)}, and Jacobian matrix at these points J= (∂f/∂k, ∂f/∂αδ, ∂f/∂θ) has full rank at (k, αδ, θ), then (αδ, θ)are locally identifiable parameters of spectral data near this point, i.e., local inverse function exists writing (αδ, θ)as function of {kn(θj)}. 7 Proof sketch. Apply implicit function theorem: if ∂f/∂k = 0 and partial derivative submatrix with respect to (αδ, θ) full rank, can solve k=k(αδ, θ) locally, then construct composite map for multiple θj, obtain local invertibility. For multiple eigenvalues, stack components; if combined Jacobian full rank, overall identifiability holds. See Appendix B.2 for details. 4.5 Pathological Domains and Condition Numbers Through explicit computation ∂k ∂αδ =−fαδ fk , where fαδ= (sin(kL))/k,fk=−Lsin(kL)+αδ(. . . ), can define pathological condition number region fk≈0, corresponding to spectral quantization curve highly sensitive to parameters or non-invertible. Under computational universe–complexity geometry perspective, these pathological regions correspond to regions on control manifold with large curvature, spectral–phase information’s “geodesic sensitivity” to parameters dramatically amplified, requiring window functions and experimental design to avoid or specially handle within complexity budget. 5 FRB and δ-Ring Unified Phase-Frequency Metrology This section embeds FRB and δ-ring scattering in same phase–frequency metrology framework, gives geometric conditions for “cross-platform scale unification”. 5.1 Juxtaposition of Phase–Frequency Readout Objects For FRB and δ-ring, we respectively obtain phase–frequency data objects UFRB PhFr = (ΩFRB,ΘFRB(ω), κFRB(ω)), Uδ PhFr = (Ωδ,Θδ(ω), κδ(ω)). Under unified time scale hypothesis, exists “master scale density” κuniv(ω) such that effective time scales corresponding to FRB and δ-ring respectively κFRB(ω) = gFRB(ω)κuniv(ω), κδ(ω) = gδ(ω)κuniv(ω), where gFRB, gδare weight functions determined by geometry and propagation kernels. 8 5.2 Cross-Platform Scale Unification Condition Definition 5.1 (Cross-Platform Scale Unification).FRB and δ-ring scattering called scale-unified on unified time scale if there exist master scale density κuniv and weights gFRB, gδsuch that windowed phase residuals satisfy consistent interpretation: RFRB(WFRB)≈ZWFRB(ω)δgFRB(ω)κuniv(ω) dω, Rδ(Wδ)≈ZWδ(ω)δgδ(ω)κuniv(ω) dω, for window function family WFRB, Wδ. Theorem 5.2 (Cross-Platform Consistency Test of Unified Time Scale).If there exist master scale density κuniv and weights gFRB, gδsuch that for PSWF/DPSS type window function family {Wj}, windowed residuals of FRB and δ-ring satisfy RFRB(Wj) = λjRδ(Wj) + O(εj), where λjare ratios precomputable from geometric factors, when εjacceptable within experimental error, then phase–frequency data of FRB and δ-ring scattering consistent with unified time scale model. Conversely, if for some window functions Wjsystematic deviation exists exceeding error tolerance, can determine unified time scale model has inconsistency in this frequency band and scale, need to correct κuniv or weight model. Proof sketch. Using completeness of PSWF/DPSS, expand κuniv in window function space, write FRB and δ-ring residuals as coefficient vectors for same basis, test whether they satisfy prespecified linear relationship. See Appendix B.3 for details. 6 Joint Variation of Window Functions and Control Parameters: Optimal Cross-Platform Metrology Strategy This section incorporates FRB and δ-ring window function and control parameter choices into time–information–complexity joint variational principle, gives variational form of “optimal cross-platform metrology under finite complexity budget”. 6.1 Extended Joint Manifold Previous joint manifold EQ=M × SQ. Now introduce two types of additional degrees of freedom: 1. FRB window function parameter space WFRB, e.g., spanned by linear coefficients of several PSWF modes; 2. δ-ring control parameter space Mδ={(L, αδ, θ)}and corresponding window function space Wδ. 9