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
In previous ``computational universe'' framework, universe axiomatized as discrete object U_{comp} = (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 (S_Q,g_Q), and time--information--complexity joint variational principle. Unified time scale given by scattering master scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr Q(\omega) 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: cosmological-distance Fast Radio Burst propagation (FRB) and laboratory-scale \delta-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 \delta-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: enumerate \item 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 scale master scale: under traceable perturbation and wave operator completeness, PhFr output completely determined by \kappa(\omega) and finite spectral--scattering invariants. \item 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 \delta \kappa(\omega) 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. \item For \delta-ring--AB flux scattering, restate equivalence between spectral quantization equation f(k,\alpha_\delta,\theta) = \cos(kL) + (\alpha_\delta/k)\sin(kL) - \cos\theta = 0 and ``amplitude-corrected phase closure'' \cos\gamma(k) = |t(k)|\cos\theta and prove under computational universe--control manifold perspective: under spectral observation \{k_n(\theta)\} at fixed (L,\theta), \delta--coupling strength \alpha_\delta and AB flux \theta are identifiable in non-pathological domain (Jacobian full rank), usable as ``laboratory ruler'' for unified time scale--phase metrology. \item Under unified time scale--spectral windowing readout framework, embed FRB and \delta-ring scattering in same ``phase--frequency metrology universe'', prove existence of ``cross-platform scale unification condition'': when FRB phase residual and \delta-ring scattering spectral shift both explained by same \kappa(\omega) 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. \item Embed above phase--frequency metrology structure into time--information--complexity variational principle, formalize ``choosing FRB/\delta-ring window functions and control parameters'' as variational problem on joint manifold, give variational conditions for ``simultaneously using cosmic-scale and laboratory-scale phase--frequency readouts to maximize unified time scale identifiability under finite complexity budget''. enumerate This paper thus completes experimental implementation design of ``phase--frequency unified metrology'' within computational universe framework: FRB and \delta$-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.
In previous computational universe'' framework, universe axiomatized as discrete object U_{comp} = (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 (S_Q,g_Q), and time--information--complexity joint variational principle. Unified time scale given by scattering master scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr Q(\omega) 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: cosmological-distance Fast Radio Burst propagation (FRB) and laboratory-scale \delta-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 \delta-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: enumerate \item 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 scale master scale: under traceable perturbation and wave operator completeness, PhFr output completely determined by \kappa(\omega) and finite spectral--scattering invariants. \item 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 \delta \kappa(\omega) 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. \item For \delta-ring--AB flux scattering, restate equivalence between spectral quantization equation f(k,\alpha_\delta,\theta) = \cos(kL) + (\alpha_\delta/k)\sin(kL) - \cos\theta = 0 and amplitude-corrected phase closure'' \cos\gamma(k) = |t(k)|\cos\theta and prove under computational universe--control manifold perspective: under spectral observation \{k_n(\theta)\} at fixed (L,\theta), \delta--coupling strength \alpha_\delta and AB flux \theta are identifiable in non-pathological domain (Jacobian full rank), usable as laboratory ruler'' for unified time scale--phase metrology. \item Under unified time scale--spectral windowing readout framework, embed FRB and \delta-ring scattering in same phase--frequency metrology universe'', prove existence of cross-platform scale unification condition'': when FRB phase residual and \delta-ring scattering spectral shift both explained by same \kappa(\omega) 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. \item Embed above phase--frequency metrology structure into time--information--complexity variational principle, formalize choosing FRB/\delta-ring window functions and control parameters'' as variational problem on joint manifold, give variational conditions for simultaneously using cosmic-scale and laboratory-scale phase--frequency readouts to maximize unified time scale identifiability under finite complexity budget''. enumerate This paper thus completes experimental implementation design of ``phase--frequency unified metrology'' within computational universe framework: FRB and \delta$-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.
In previous series works on computational universe U_{comp} = (X,T,C,I), we have established discrete complexity geometry (complexity distance, volume growth, and discrete Ricci curvature), discrete information geometry (task information manifold (S_Q,g_Q) and embedding \Phi_Q), control manifold (M,G) induced by unified time scale, as well as time--information--complexity joint variational principle. Therein unified time scale given by scattering master scale $ \kappa(\omega) = \varphi'(\omega){\pi} = \rho_{rel}(\omega) = 1{2\pi}\tr Q(\omega), unifying physical time density, spectral shift function derivative, and Wigner--Smith group delay trace. However, above structures are still ``ideal limits'': radius T of complexity ball B_T(x_0) can be arbitrarily large, geodesics on control manifold can extend arbitrarily, Fisher structure on information manifold can be perfectly identified under infinite data. In actual computational universe, all readouts and decisions proceed under finite time, finite complexity budget, and finite frequency band constraints, thus necessarily carrying errors. To rigorously control errors within unified time scale--complexity geometry--information geometry framework, requires systematic ``spectral windowing readout'' theory. This paper introduces readout operators and error models within computational universe framework, unifying them as window function problem in unified time scale frequency domain. We prove: under unified time scale, writing readout operator as integral over frequency domain objects R(f) = \int_{\Omega} W(\omega)\,f(\omega)\,d\omega, where W(\omega) is window function, f(\omega) is frequency domain quantity related to universe evolution (e.g., \kappa(\omega) or its weighting), can naturally introduce class of time--band-limited--complexity-limited joint extremal problems. In continuous case, we prove: under constraints of given time truncation interval [-T,T] and frequency band [-W,W], Prolate Spheroidal Wave Functions (PSWF) are optimal window function family: they maximize energy concentration under dual restrictions of [-T,T] and [-W,W], thereby minimizing worst-case error of ``energy leakage outside complexity ball'' when unified time scale--complexity budget given. In discrete case, we introduce corresponding Discrete Prolate Spheroidal Sequences (DPSS), defining window sequences on finite-length complexity chains, proving they maximize energy concentration under discrete time--frequency restrictions, thereby giving optimal error control structure for ``finite-order readout'' under constraints of finite complexity steps N and finite bandwidth W. Under language of unified time scale--complexity geometry, we obtain following conclusions: enumerate \item For computational universe readouts with band-limited unified time scale frequency (e.g., scattering delay spectrum), if complexity budget only allows 2TW/\pi level degrees of freedom, then PSWF window functions give optimal error--complexity tradeoff under this budget; \item On discrete complexity graph G_{comp}, DPSS provides optimal readout sequence under finite step length N and finite bandwidth W, whose error decay and spectral concentration constants controlled by DPSS eigenvalues; \item These results can be embedded into time--information--complexity joint variational principle, viewing ``choosing readout window function'' as adding ``spectral windowing control dimension'' layer on joint manifold E_Q, thereby giving variational characterization of ``optimal observation strategy under finite resources''. enumerate This paper as ``error control'' chapter in computational universe series, at interface of unified time scale--frequency domain--complexity geometry, elevates classical time--frequency concentration results of PSWF/DPSS to error control and observability theory in computational universe, providing theoretical foundation for subsequent construction of unified readout design on specific physical--engineering testbeds such as FRB/\delta$-ring.
Within the unified context of ``universe = Generalized Light Structure (GLS),'' ``observer = filter chain (windowed compression → CP channel → POVM → threshold counting),'' and ``causality = lightlike cone preorder,'' we establish an axiomatic theory with the master scale πφ′(E)=ρrel(E)=2π1trQ(E),Q(E)=−iS(E)†dEdS(E) (phase derivative---relative density of states---Wigner--Smith group delay trace). \textbf{Causal red line:} Windowed group delay readout Tγ[wR,h] serves solely as operational time scale (can be negative in narrowband/resonance, having no universal size comparison with wavefront time t∗); causality determined exclusively by t∗ and lightlike cone partial order. Core results: (i) windowed group delay readout provides operational time scale, proven series-parallel additive and gauge covariant/relatively invariant (invariant when U,V energy-independent or more generally ∂E(argdetU+argdetV)=0); in vacuum links satisfying Nyquist discipline, high-frequency/alias-free limit aligns with wavefront velocity calibrating c; (ii) redshift characterized by spectral scaling with reciprocal time scaling law; (iii) light speed c normed by vacuum wavefront, yielding wavefront lower bound and no-supercone propagation for arbitrary physical channels; (iv) ``GLS ↔ causal manifold'' mutual construction outline with skeleton and consistency conditions---rigorous proof and naturality verification in separate work; (v) wave-particle duality and double-slit windowed complementarity D2+V2≤1 unified in same ledger; (vi) spectral resolution enhancement (bandwidth narrowing) and redshift amplification are dual; conversely, temporal/image resolution enhancement (σT decrease) and blueshift (bandwidth increase) are dual, consistent with σT≳Cw,h/B in \S4.4, yielding Nyquist--Poisson--Euler--Maclaurin (NPE) finite-order error closure and engineering prescription; (vii) high-frequency platform bridge: under \S4 premises, blueshift limit of any fixed-shape normalized window family---single channel limE0→∞Tγ=t∗, multichannel limE0→∞TC=Nefft∗ and per-mode average limE0→∞TC=t∗, treating causal wavefront t∗ as operational limit scale of Tγ and TC. Theory employs operator-measure-function language throughout (Toeplitz/Berezin compression Kw,h, readout = linear functional on spectral measure), unifying all time/density readouts under global unitarity postulate by Q=−iS†S′; experimental non-unitarity traced back to master scale via unitary dilation S.
Within abstract framework of de Branges--Kreĭn canonical systems and multi-channel scattering, this paper establishes unified system in pure theoretical language of ``operator--measure--function'', independent of experimental narrative: welding ``phase derivative--relative state density--Wigner--Smith group delay trace'' as universal measure coordinate of same parent scale, characterizing finite resources and observational choices via windowed readouts of Toeplitz/Berezin compression. On reversible observational transformation group generated by base automorphisms, phase gauge and reversible filtering, proves invariance of time scale within blocks constructed from windowed delay integral; gives non-asymptotic error closure and stable principle of ``singularity non-increasing/pole = dominant scale'' under ``finite-order'' Euler--Maclaurin and Poisson discipline. Thus, in EBOC static block universe replaces external elapsed time with intrinsic T_{inv}; obtains unified metric under RCA reversible computation's isomorphic renormalization. Scale identity of this system holds almost everywhere on absolutely continuous spectrum: $ \ \frac{\varphi'(E){\pi} = \rho_{rel}(E) = 1{2\pi}trQ(E)\ }, where S(E)\in U\big(N(E)\big) scattering matrix, Q(E):=-i\,S(E)^\dagger \partial_E S(E) Wigner--Smith delay matrix, \varphi(E):=1{2}\Arg\det S(E), \rho_{rel} relative state density relative to reference channel/free Hamiltonian. Windowed readout defined by Toeplitz/Berezin compression map C_w and its covariant symbol K_w(E): T_w(E):=1{2\pi}tr\big(K_w(E)Q(E)\big),\qquad T_{inv}(I):=\int_I T_w(E)\,dE, $ remaining invariant under reversible observational transformations, becoming universal time scale within EBOC blocks. Above trinity scale connected by Wigner--Smith delay matrix and Birman--Kreĭn spectral shift--determinant formula, providing unified coordinate from phase to density, from scattering to measure, and stable benchmark for variational/optimization.
Using the de Branges--Kreĭn (DBK) canonical system and weighted Mellin space as carriers, we explicitly incorporate the finite bandwidth/time window of real instruments into spectral measures, forming a windowed readout framework; we characterize ``commit (collapse/commit)'' through KL/Bregman information geometry; use scattering phase--spectral density--Wigner--Smith delay as energy scale; close non-asymptotic errors via Nyquist--Poisson--Euler--Maclaurin (three-term decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is $ \,\varphi'(E)=-\pi\,\rho_{\mathrm{rel}(E)=1{2}trQ(E)\,}\quad(a.e.) $ unifying (single/multi-channel) scattering phase derivative, relative local density of states (LDOS) and Wigner--Smith delay; under information geometry we obtain Born probability = minimal-KL projection (I-projection), and ``pointer basis'' is spectral minimum of windowed readout operator. Above criteria consistent with Herglotz--Weyl, Birman--Kreĭn, Wigner--Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable.
Recent experimental developments provide important empirical context for phase-based approaches to physical dynamics. The direct measurement of quantum phase accumulation during free fall using a Quantum Galileo Interferometer (QGI) reported by Dobkowski et al., Science Advances (2026) DOI: 10.1126/sciadv.adj8028, demonstrates that gravitational dynamics can be directly encoded in a measurable phase evolution. Independently, the realization of magnetic-free non-reciprocity through parametric phase modulation reported by Estep, Sounas, Soric, and Alù, Nature Physics (2014) DOI: 10.1038/nphys3134, demonstrates that the relative temporal phase of spatially distributed resonators ( for ) can generate synthetic angular momentum and modify collective dynamics. These results do not validate the Universal Applied Time (UAT) or Unified Principle of Causality (UPC) frameworks. Both experiments were developed and interpreted within established frameworks. Nevertheless, they provide robust experimental precedents for treating discrete phase geometry and temporal modulation as primary experimental variables. Within this context, this work analyzes the eight-coil rotational matrix developed under the UAT/UPC program as an exploratory macroscopic platform. The system implements a synthetic rotational configuration with discrete phase increment , where physical coil opposition is replaced by controlled phase relationships. For eight ideal equal-amplitude components, . Any measured residual must therefore be decomposed into instrumental, phase, frequency, electronic, and potentially structural contributions. During development, a candidate residual was identified under phase/frequency confrontation, alongside a central-coil RMS scale . Both values are treated strictly as candidate invariants requiring independent reproduction. This work explicitly defines the falsification protocol (ideal signals, real channels, random permutation, phase inversion, and frequency variation ) and the hypotheses : residual explained by systematics vs : persists after controls. The detector, calibration code and methodology are documented in the 8+1 coil report. Related identifiers: UAT: 10.5281/zenodo.17729221 UPC: 10.5281/zenodo.18210808 Resonant Hunter v8.4: 10.5281/zenodo.18446712 8+1 Coil Rotational Detector & Calibration: 10.5281/zenodo.19646349 Alù et al. (2014): 10.1038/nphys3134 Dobkowski et al. (2026): 10.1126/sciadv.adj8028
We establish a rigorous theory unifying ``finite throughput--queueing delay--apparent dilation (redshift)'' within windowed scattering and information geometry (WSIG) scale system. In pure theoretical language of operator--measure--function, using ``trinity'' scale identity \varphi'(E)/\pi=\rho_{\rm rel}(E)=-1{2\pi}tr\mathsf Q(E) as mother scale, define observer window's Toeplitz/Berezin compression K_{w,h} and its readout functional, characterize ``finite throughput'' abstract form via passivity and causality-preserving scattering axioms. Prove three main theorems: (T1) Throughput--Delay Dilation Theorem: Under passive load monotonicity, any windowed group delay readout monotonically non-decreasing with load; when baseline readout \Phi_w[\lambda_0]>0, induces Mellin dilation of energy and time scales scale-equivalent to redshift factor 1+z_w=\Delta_w\ge1; (T2) Group Delay--Bandwidth Product Upper Bound: Windowed energy spectrum total readout upper-bounded by product of ``group delay flux'' and ``effective bandwidth constant'', with non-asymptotic error bound from Nyquist--Poisson--Euler--Maclaurin (NPE) three-term closure; (T3) Scale Gauge Equivalence: Cosmological expansion and internal resolution enhancement scale-equivalent to gauge pairing a(t)=R(t)^{-1}, all readouts remaining invariant on same mother scale. Further provide ``front slope--group delay'' correspondence at reversible cellular automaton (RCA) dynamics layer, embedding queueing--throughput--delay discrete dynamics into EBOC static block encoding. Throughout use only finite-order Euler--Maclaurin and Poisson discipline, maintaining ``singularity non-increasing/poles = principal scales'' error control; any numerical and experimental interfacing relegated to appendices.