Cross-Platform Metrological Paradigm for Unified Phase--Frequency Readout: Windowed Upper Limits on FRB Vacuum Polarization, Spectral--Scattering Equivalence and Identifiability of \delta-Ring--AB Flux, and Critical Coupling Metrology for Scattering Topological Invariants

Cross-Platform Metrological Paradigm for Unied PhaseFrequency Readout: Windowed Upper Limits on FRB Vacuum Polarization, SpectralScattering Equivalence and Identiability of δ -RingAB Flux, and Critical Coupling Metrology for Scattering Topological Invariants Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Propose unied metrological paradigm using "phasefrequency" as sole readout, penetrating three classes of systems with disparate scales and physical backgrounds: fast radio burst (FRB) propagation at cosmological distances, point interactions ( δ potential) and AharonovBohm (AB) ux coupling in cold atom/electronics one-dimensional ring systems, and scattering topological invariants for condensed matter class D endpoints. Under unied kernelwindowing generalized least squares/Fisher syntax provide: First, self-consistent estimation of refractive index modication order via curved spacetime QED one-loop eective action, constructing reproducible windowed upper limiter ; on FRW background scale upper limit phase residual scale below observational threshold, yielding only rigorous upper bounds. Second, rigorously prove equivalence between spectral quantization trigonometric equation on ring cos θ= cos(kL) + αδ ksin(kL) and " amplitude-corrected phase closure formula" cos γ(k) = |t(k)|cos θ, γ =kL −arctanαδ k,|t(k)|=k pk2+α2 δ fully equivalent, degenerating to pure phase closure only in limit |t| → 1 ; provide structural identiability theorem that {kn(θ)} alone can uniquely invert αδ≡mg/ℏ2 , with closedform sensitivity, ill-conditioning avoidance criteria, and rst-order equivalentization of narrow potential width. Third, on class D endpoints use Q= sgn det r(0) as topological criterion, propose under near-unitary premise unied nite-size (exponential) and contact/boundary (algebraic) bias extrapolation regression Jc(L) = Jc+βe−L/ξ +γ/L, provide linear response mapping to cQED readout (cavity frequency shift and additional loss). Proofs and engineering protocols for above conclusions provided with paper, directly reproducible and transplantable to similar platforms. Keywords : Phasefrequency metrology; fast radio burst; curved spacetime QED; windowed upper limits; δ potential; AharonovBohm eect; self-adjoint extension; scattering topological invariant; Majorana; cQED 1 1 Introduction & Historical Context Metrological schemes using "phasefrequency" as single observable naturally span multiple frontiers: in cosmological electromagnetic propagation, phase frequency perturbations record geometry and medium; in one-dimensional quantum rings, phase shifts quantify scattering and geometric coherence; in condensed matter scattering metrology, low-energy reection subblock phase and determinant characterize topological number. Curved spacetime QED one-loop eective action indicates vacuum polarization introduces only minute refractive index correction on weak curvature backgrounds, satisfying high-frequency causality and analyticity; this makes FRB carrier phase naturally a channel for setting upper limits , not realistic detection. δ -ring problem under selfadjoint extension U(2) framework yields rigorous spectralscattering equivalence unifying with AB ux; topological endpoints implementable via single-end reection matrix zero-energy criterion for near-eld metrology of non-local topology. Above backgrounds increasingly clear in recent literature; this paper's goal is condensing into unied metrological syntax and reproducible recipes . 2 Model & Assumptions 2.1 Unied Observation Equation, Weights, and Windowing Across three platforms, observables abstractable as phase-type readout m(ω) on frequency. Unied expression m(ω) = ZP K(ω, χ)x(χ) dχ+ P−1 X p=0 apΠp(ω) + ϵ(ω), where K is observation kernel, χ denotes propagation parameter (e.g., conformal coordinate), geometric phase, or energy scale; x(χ) is target function (or windowed parameter); Πp are systematic/foreground basis functions, ap their amplitudes; ϵ is noise process, covariance Cϕ(ω, ω′) . On discrete frequency axis adopt weighted inner product ⟨f, g⟩ ≡ X ij f(ωi) [C−1 ϕ]ij g(ωj). Dene weighted GramSchmidt orthonormalized frequency windows Wj(ω) , construct design matrix with Ajµ =⟨Wj,K(·, χµ)⟩ ; generalized least squares (GLS) and Fisher information ˆ θ= (A⊤A)−1A⊤y, F =A⊤A, where yj=⟨Wj, m⟩ , θ are windowed parameters (or coecients of x(χ) in piecewise constant basis). Classical CRB for single-frequency phase in white noise limit degenerates to σϕ≥(2Nρ)−1/2 , signal processing reference for this paper's upper limit derivation. 2.2 Platform-Specic Assumptions and Kernels  FRB propagation : Along conformal coordinate χ ( dℓ=a(η)dχ ) integration, local frequency ωloc = (1 + z)ωobs cancels with scale factor, thus KFRB(ωobs, χ) = ωobs/c , 2 taking x(χ)≡δn(χ) (refractive index correction) as windowed target kernel. This form guarantees high-frequency causality n(ω→ ∞)→1 and worldlinePenrose limit analyticity constraint.  δ -ring + AB ux : Observable is eigenwavevector k(θ) satisfying quantization condition, αδ≡mg/ℏ2 represents point interaction strength, geometric length L is ring circumference, θ= 2πΦ/Φ0 is ux phase. Kernel function manifests in dispersion relation and boundary conditions, see Section 4.  Topological scattering : Observable is characteristic phase of zero-energy reection matrix r(0) or sign of its determinant; kernel K from scatteringtopology mapping and inputoutput linear response, see Section 6. 3 Main Results (Theorems and Alignments) Theorem 1 (A: FRBWindowed Upper Limit and Order of Magnitude for One-Loop Vacuum Polarization) . Curved spacetime QED under weak curvature background provides one-loop correction order of magnitude for refractive index δn ∼αem πλ2 eR, taking FRW upper bound |R| ∼ H2 0/c2 , phase accumulation upper limit for 1 GHz, 1 Gpc ∆ϕ∼ωobsL cδn ≈1.2×10−53 rad. Therefore under any realistic baseband data volume and noise, can only provide windowed upper limit for x(χ) = δn(χ) , not eective detection. Proof in Appendix A. Theorem 2 (B: δ -RingEquivalence of Trigonometric Equation and Amplitude-Corrected Phase Formula) . For one-dimensional ring δ potential (strength g ) and AB ux θ , eigenwavevector k satises cos θ= cos(kL) + αδ ksin(kL), which is fully equivalent to cos γ(k) = |t(k)|cos θ, γ(k) = kL −arctanαδ k,|t(k)|=k qk2+α2 δ and degenerates to pure phase closure cos γ= cos θ only in limit |t(k)| → 1 (weak scattering or transmission resonance). Proof in Appendix B. Theorem 3 (C: δ -RingStructural Identiability from {kn(θ)} Alone) . Under spectral observation {kn(θ)} at xed (L, θ) , can uniquely invert αδ=mg/ℏ2 , but cannot separate individual values of m and g . If introducing second independent readout type (e.g., energy spectrum En=ℏ2k2 n/(2m) or absolute potential strength geff =RVdx calibration), then at generic points (m, g) Jacobian full rank and decoupling possible. Proof in Appendix B. 3 Theorem 4 (D: δ -RingImplicit Function Sensitivity and Ill-Conditioning) . Let f(k, αδ, θ) = cos(kL) + αδ ksin(kL)−cos θ, then ∂k ∂αδ = sin(kL) k Lsin(kL)−αδ kLcos(kL) + αδ k2sin(kL). Ill-conditioned domain given by ∂f/∂k = 0 ; when |αδ|/k ≪1 approximately sin(kL)≃(αδ/k) cos(kL) . Proof in Appendix B. Theorem 5 (E: First-Order Equivalentization of δ -Potential Finite Width) . For narrow potential V(x) (width w≪L ) with RVdx=g xed, in (kw)≪1 region, eective strength on ring αeff δ(k) = m ℏ2g1 + η2[V] (kw)2+O(kw)4, where shape factor η2[V] determined by second moment of V ; rectangular and Gaussian potentials respectively have η2=1 3,1 2 . Proof and numerical comparison in Appendix B. Theorem 6 (F: Topological Scattering InvariantNear-Unitary Extrapolation) . Under near-unitary premise ( |r†r−⊮| ≤ ε⋆ ), nite length L , contact imperfection, and weak dissipation introduce bias to critical coupling Jc , unied by Jc(L) = Jc+βe−L/ξ +γ/L extrapolation regression absorption, where exponential term corresponds to endpoint state coupling overlap, algebraic term corresponds to contact/boundary eects. Q= sgn det r(0) ip provides Jc(L) observation trajectory. Proof in Appendix C. Theorem 7 (G: cQED Redundant Readout and Scattering Invariant Consistency) . Single-port weak coupling, inputoutput relation gives r(ω) = 1−Z0Y(ω) 1 + Z0Y(ω)≈1−2Z0Y(ω), δωc∝g2 cav Re χ(ωc), κadd ∝g2 cav Im χ(ωc), where χ is endpoint subsystem linear response. For appropriate detuning |ωc−ωcont|≳3κ , zero crossings of {δωc, κadd} consistent with Q ip. Proof in Appendix C. 4 Proofs 4.1 Proof of Theorem A (FRB) QED eective action on curved background writable as Leff =−1 4FµνFµν +1 m2 eaRFµνFµν +bRµνFµαFν α+cRµναβFµνFαβ+· · · , linearized dispersion relation provides refractive index correction δn ∝(αem/π)λ2 eR . On FRW, taking |R| ∼ H2 0/c2 , integrating along χ with kernel Eq. (2.4), 4 ∆ϕ=Zχs 0 ωobs cδn(χ) dχ≤ωobsL cmax χδn(χ), substituting constants (αem/π)≃2.32×10−3 , λe= 3.8616×10−13 m , |R| ≃ 5.4×10−53 m−2 and ωobsL/c ≃6.46 ×1026 (1 GHz/1 Gpc), obtain ∆ϕ≈1.2×10−53 rad. HollowoodShore worldline Penrose limit guarantees n(ω→ ∞)→1 ; this paper's constructed windowed kernel satises this causality/analyticity constraint. 4.2 Proof of Theorem B ( δ -Ring) From straight-line δ potential scattering amplitude t=k k+iαδ and |t|=1 p1+(αδ/k)2 , expanding transfer matrix trace condition tr T= 2 cos θ yields cos θ= cos(kL) + αδ ksin(kL). On the other hand cos(kL) + αδ ksin(kL) = ℜ h(cos kL +isin kL)1−iαδ ki=p1+(αδ/k)2cos kL −arctan αδ k, i.e., cos γ=|t|cos θ , proved. 4.3 Proof of Theorem C (Identiability) Spectral equation (3.3) contains only combined parameter αδ=mg/ℏ2 ; if (m1, g1)= (m2, g2) exists but m1g1=m2g2 , then for any (θ, n) yields identical {kn} . Therefore {kn(θ)} alone uniquely determines αδ , while m , g non-separable. If adding energy spectrum En=ℏ2k2 n/(2m) , then J=∂f/∂m ∂f/∂g ∂E/∂m ∂E/∂g= ∂f ∂αδ ∂αδ ∂m ∂f ∂αδ ∂αδ ∂g −ℏ2k2 2m20!, at generic points det J= 0 (except degenerate subsets like k= 0 ), thus (m, g) decoupling possible, proved. 4.4 Proof of Theorem D (Sensitivity) By implicit function theorem ∂k ∂αδ =−fαδ fk , fαδ=1 ksin(kL), fk=−Lsin(kL) + αδL kcos(kL)−1 k2sin(kL), obtain Eq. (3.6). Ill-conditioned domain satises fk= 0 , when |αδ|/k ≪1 approximately sin(kL)≃(αδ/k) cos(kL) , proved. 5 4.5 Proof of Theorem E (Finite Width) Let narrow potential V(x) = V0u(x/w) ( Ru= 1 ) with g=V0w xed, transfer matrix low-energy expansion at kw ≪1 yields t−1(k) = 1 + iαδ k+η2[V] (kw)2+O(kw)4, equivalentizing to αδ→αeff δ(k) replacement. Calculate second moment for rectangular/Gaussian u comparing with solvable models, obtain η2=1 3,1 2 . Numerical verication in Appendix B Figure B2. 4.6 Proof of Theorems F and G (Topological Scattering and cQED) Class D system topological number via single-end reection matrix zero-energy criterion Q= sgn det r(0) . Finite length coupling leads to zero mode energy splitting ∝e−L/ξ , contact/boundary introduces 1/L scaling; under near-unitary assumption, det r(0) sign ip trajectory Jc(L) thus exhibits Eq. (3.8) shape. On the other hand, inputoutput gives r≈1−2Z0Y , while Y∝g2 cavχ ; when cavity mode suciently detuned from continuum, χ(0) sign change produces zero crossing in {δωc, κadd} , consistent with Q ip, proved. 5 Model Apply 5.1 FRB "Windowed Upper Limiter"  Kernel and weights : Adopt KFRB =ωobs/c , Πp(ω)∈ {1, ω, ω−1,log ω} . Cϕ by threechannel bootstrap estimation: o-source, o-band, sidelobe.  Orthogonal windows : On discrete frequency axis {log ωi} , use C−1 ϕ weighted GramSchmidt obtaining Wj , whitening test ⟨Wj, Wℓ⟩ ≃ δjℓ .  Upper limits : For windowed parameters θ adopt GLS and prole-likelihood parallel computing 95% upper limit band, providing scaling curve with bandwidth/event number and systematic basis envelope; use "whether includes ω−1 " as robustness switch taking envelope. 5.2 δ -Ring Inversion and Ill-Conditioning Avoidance  Inversion : With Eq. (3.3) f= 0 as target, use Eq. (3.6) as Newton step iteration inverting ˆαδ .  Ill-conditioned domain : Plot gray region |fk|< τ in (kL, αδ/k) plane ( τ∼10−2 ), experimental point selection avoids domain.  Finite width : For given w and potential shape, use Eq. (3.7) correcting ˆαδ , propagate η2 uncertainty to covariance. 5.3 Topological Extrapolation and cQED Redundancy  Extrapolation regression : Measure Q ip Jc(Li) on {Li} , regress (Jc, β, γ, ξ) via Eq. (3.8), compare "exponential/algebraic/composite" three models using AIC/BIC.  Near-unitary quality control : Weight by εi=|r† iri−⊮| or enter regression as covariate, report Jc sensitivity to ε . 6  cQED consistency : Under detuning |ωc−ωcont|≳3κ , use {δωc, κadd} zero crossings as redundant evidence, cross-validate with Q ip. 6 Engineering Proposals  FRB : Release minimized script loading public baseband examples, complete coherent dedispersion, structure-maximized DM, three-channel bootstrap covariance, weighted window orthonormalization and 95% upper limit plotting, attach "whether includes ω−1 basis" robustness comparison.  δ -ring : Provide {kn(θ)} inversion αδ executable script, plot ill-conditioned domain gray map and ∂k/∂αδ contours; provide nite width correction and error propagation example.  Topological + cQED : Release scripts for extracting Q from r(ω) , regressing Jc(L) with model comparison, and {δωc, κadd} consistency test with Q . 7 Discussion (Risks, Boundaries, Past Work) FRB : At FRW background order of magnitude, one-loop vacuum polarization phase eect unmeasurable; this paper positions as "windowed upper limiter", emphasizing kernelmetric selection consistency and systematic envelope, avoiding bias setting. δ -ring : Strictly limit "pure phase closure" to limit |t| → 1 , avoiding misuse at generic θ ; nite width equivalentization requires use within kw ≪1 region. Topological : Near-unitary threshold and model selection key to extrapolation reliability, recommend using residual structure and cross-validation suppressing overtting. 8 Conclusion Using unied phasefrequency readout and "kernelwindowingGLS/Fisher" syntax, established reproducible metrological recipes for three platforms: FRB side provides rigorous windowed upper limiter; δ -ring side proves spectralscattering amplitude-corrected phase equivalence, structural identiability and closed-form sensitivity with nite width correction; topological scattering side implements critical coupling metrology via near-unitary extrapolation and cQED redundancy. Supporting engineering protocols and minimal scripts make paradigm easily transferable to similar systems. Acknowledgements, Code Availability Thank related public data and literature providing reproducible benchmarks. Supporting code and minimal data examples released under general license, including environment lockles, unit checks, and injectionrecovery tests. FRB scripts with δ -ring, topological extrapolation scripts adopt unied I/O conventions and random seeds ensuring reproducibility. Data usage and citation follow respective dataset policies. 7 References 1. Drummond, I. T., Hathrell, S. J., "QED vacuum polarization in a background gravitational eld," Phys. Rev. D 22 , 343 (1980). 2. Hollowood, T. J., Shore, G. M., "The causal structure of QED in curved spacetime: Analyticity and the refractive index," JHEP 12 , 091 (2008); related works (20082009). 3. CHIME/FRB Collaboration, "Updating the First CHIME/FRB Catalog of Fast Radio Bursts with Baseband Data," ApJ 969 , 145 (2024). 4. Hessels, J. W. T., et al. , "FRB 121102 Bursts Show Complex TimeFrequency Structure," ApJL 876 , L23 (2019). 5. Castillo-Sánchez, M., Gutiérrez-Vega, J. C., "Quantum solutions for the delta ring and delta shell," Am. J. Phys. 93 , 557565 (2025). 6. Fülöp, T., Tsutsui, I., "A Free Particle on a Circle with Point Interaction," Phys. Lett. A 264 , 366374 (2000). 7. Fulga, I., Hassler, F., Akhmerov, A. R., Beenakker, C. W. J., "Scattering formula for the topological quantum number of a disordered multi-mode wire," Phys. Rev. B 83 , 155429 (2011). 8. Araya Day, J., et al. , "Identifying biases of the Majorana scattering invariant," SciPost Phys. Core 8 , 047 (2025). 9. Dmytruk, O., Trif, M., "Microwave detection of gliding Majorana zero modes," Phys. Rev. B 107 , 115418 (2023). 10. Rife, D. C., Boorstyn, R. R., "Single-tone parameter estimation from discrete-time observations," IEEE Trans. Inf. Theory 20 , 591598 (1974). A FRB: One-Loop Vacuum Polarization Scale, KernelMetric Consistency, and Windowed Upper Limit A.1 One-Loop Eective Action and Scale From Eq. (4.1) linearize Maxwell equations, under weak curvature and geometric optics approximation, dispersion relation k2=n2ω2/c2 yields n(ω, χ) = 1 + δn(χ) + O(R2), δn(χ)≈αem πλ2 eR(χ). FRW background Rµναβ low-order contractions provide |R| ∼ H2 0/c2 . A.2 KernelMetric Consistency (Redshift Cancellation) Phase increment along geodesic dϕ=ωloc cδn(χ) dℓ=(1 + z)ωobs cδn(χ)a(η) dχ=ωobs cδn(χ) dχ, i.e., when integrating in conformal coordinates (1 + z) and a exactly cancel, obtaining kernel Eq. (2.4). This form automatically satises causality requirement n(ω→ ∞)→1 . A.3 Fisher/GLS Windowed Upper Limit Expand δn(χ) on piecewise constant basis, dene Ajµ =⟨Wj, ωobs/c⟩∆χµ , then F=A⊤A, σUL = 1.96 pdiag(F−1), 8 take upper limit envelope over dierent systematic basis choices (whether includes ω−1 , log ω ). B δ -Ring: Boundary Conditions, Equivalence, Identiability, Sensitivity, and Finite Width B.1 Boundary Condition and Trigonometric Equation Straight-line δ potential boundary condition ψ′(0+)−ψ′(0−)=2αδψ(0), αδ=mg/ℏ2, combined with free propagation segment transfer matrix yields single-lap trace condition tr T= 2 cos θ , simplifying to Eq. (3.3). B.2 Equivalence Three-Line Derivation Using ℜheikL1−iαδ ki= cos(kL) + αδ ksin(kL) = p1+(αδ/k)2cos kL −arctan αδ k, obtain Eq. (3.4), where |t|= (1 + (αδ/k)2)−1/2 . B.3 Identiability (Rank) Construct observation equations {f= 0, E =ℏ2k2/(2m)} Jacobian for (m, g) , avoiding degenerate strips like k= 0 and strong scattering ensures full rank (Eq. 4.5). B.4 Sensitivity and Ill-Conditioned Domain From Eq. (4.6) obtain implicit function derivative and ill-conditioning condition fk= 0 . Experimental design uses criterion |fk|> τ for point selection, τ∼10−2  10−3 . B.5 Finite Width Equivalentization and Numerical Comparison For V(x) = V0u(x/w) , under low-energy expansion at kw ≪1 , t−1(k) = 1 + iαδ k+η2[V] (kw)2+· · · , where η2[V] = 1 w2Rx2V(x) dx RV(x) dx (after normalization and comparison calibration) . Numerical scan verication for rectangular ( η2= 1/3 ) and Gaussian ( η2= 1/2 ) potentials, relative error ≤1% in kw ≤0.2 region. Solvable model results consistent with expansion in overlap region. 9