Geometry and complexity scaling

Jincheng Zhang

This paper introduces the Multi-Scale Quantum Geometry (MSQG) framework, a novel computational paradigm that integrates quantum mechanics with geometric principles. We propose a new approach to simulating complex physical systems by leveraging the inherent properties of geometric systems, specifically focusing on multi-scale quantum geometry. The MSQG framework provides a distinct approach to analyzing and predicting the behavior of systems across different scales, offering potential advancements in quantum computation and quantum information processing. This research explores the application of MSQG in simulating systems exhibiting unique geometric properties, ultimately aiming to provide a foundational understanding and practical tools for these fields.

Dietrich Stauffer

Kim Christensen, Nicholas R. Moloney

Haobo Ma, Wenlin Zhang

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.

Haobo Ma, Wenlin Zhang

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.

Jack D. Hidary