Error Control and Spectral Windowing Readout\\ in Computational Universe:\\ Time--Frequency--Complexity Role of PSWF/DPSS Window Functions\\ Under Unified Time Scale

Error Control and Spectral Windowing Readout in Computational Universe: Time–Frequency–Complexity Role of PSWF/DPSS Window Functions Under Unified Time Scale Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous series works on computational universe Ucomp = (X, T,C,I), we have established discrete complexity geometry (complexity distance, volume growth, and discrete Ricci curvature), discrete information geometry (task information manifold (SQ, gQ) and embedding ΦQ), 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 κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), unifying physical time density, spectral shift function derivative, and Wigner– Smith group delay trace. However, above structures are still “ideal limits”: radius Tof complexity ball BT(x0) 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) = ZΩ W(ω)f(ω) dω, where W(ω) is window function, f(ω) is frequency domain quantity related to universe evolution (e.g., κ(ω) or its weighting), can naturally introduce class of time–band-limited–complexity-limited joint extremal problems. 1 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 Nand finite bandwidth W. Under language of unified time scale–complexity geometry, we obtain following conclusions: 1. For computational universe readouts with band-limited unified time scale frequency (e.g., scattering delay spectrum), if complexity budget only allows 2TW/π level degrees of freedom, then PSWF window functions give optimal error–complexity tradeoff under this budget; 2. On discrete complexity graph Gcomp, DPSS provides optimal readout sequence under finite step length Nand finite bandwidth W, whose error decay and spectral concentration constants controlled by DPSS eigenvalues; 3. 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 EQ, thereby giving variational characterization of “optimal observation strategy under finite resources”. 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/δ-ring. Keywords: Computational universe; Error control; Spectral windowing; PSWF; DPSS; Time-frequency concentration; Unified time scale; Complexity geometry 1 Introduction In any actual physical or computational system, readouts and decisions cannot proceed over infinite time, infinite complexity budget, and infinite frequency band. Unified time scale–complexity geometry–information geometry give geometric structures of “infinite ideal universe”:  Complexity ball BT(x0) can expand as T→ ∞;  Geodesic worldlines on control manifold (M, G) can extend to infinity;  Unified time scale density κ(ω) in frequency domain can be observed over infinite frequency band; 2  Fisher structure of information manifold (SQ, gQ) can be completely identified under infinite data. But in reality, we must work under following restrictions: 1. Time–complexity limitation: Total complexity budget Tfinite, regions outside complexity ball unreachable in finite time; 2. Frequency band limitation: Effective frequency band of unified time scale density κ(ω) finite, or actual readout device can only respond within finite frequency band; 3. Readout order limitation: Realizable readout operator dimension finite, e.g., can only sample finite time moments or finite frequency points, or can only execute finite-order moment–filtering operations; 4. Error tolerance: System must ensure error under these limitations does not exceed some admissible threshold. In signal processing and time–frequency analysis, PSWF/DPSS become classical tools with their property of “optimal energy concentration under finite time and finite band”. However, these results mostly discussed in pure signal spaces (e.g., in L2([−T, T]) and L2(R)), not yet systematically embedded into unified time scale–complexity geometry framework. Purpose of this paper threefold: 1. Formalize readout process in computational universe as window function problem in unified time scale frequency domain; 2. Under unified time scale–complexity budget constraints, give optimality theorems of PSWF/DPSS window functions in error control sense; 3. Embed these results into time–information–complexity joint variational principle, constructing theoretical framework of “optimal observation under finite resources”. Paper organization: Section 2 defines readout operators and error models in computational universe. Section 3 introduces spectral windowing readout in unified time scale frequency domain. Section 4 reviews and restates energy concentration and finite time– band optimality properties of PSWF/DPSS, translating them to error upper bounds for “finite complexity readout”. Section 5 discusses “complexity–time–bandwidth” triple unified constraints in computational universe. Section 6 incorporates window function choice into joint variational principle, giving variational form of “optimal observation strategy”. Appendices give detailed proofs of PSWF/DPSS definitions, key eigenvalue properties, and main error upper bound theorems. 2 Readout Operators and Error Models in Computational Universe This section formalizes readout operators on computational universe Ucomp = (X, T,C,I), defining error and error budget. 3 2.1 Path-Level Readout Operators Consider evolution path starting from initial state x0 Γ = (x0, x1, x2, . . . ),(xk, xk+1)∈T. Under unified time scale, each step has physical time increment ∆tk=C(xk, xk+1)/λ, where λis unit conversion constant (can be absorbed into Cbelow). Definition 2.1 (Path Readout Operator).A path readout operator is map R:{path Γ} → Cm, writable as composition of finite-time linear functionals, e.g., R(Γ) = ⟨r1,Γ⟩,...,⟨rm,Γ⟩, where each rjis finitely-supported “kernel”, e.g., ⟨rj,Γ⟩=X k rj(k, xk). In continuous limit, can view Γ as curve (θ(t), ϕ(t)) on control manifold and information manifold, readout becomes time integral ⟨rj,Γ⟩=ZT 0 Rj(θ(t), ϕ(t)) wj(t) dt, where wj(t) is weight function (window function). 2.2 Ideal Readout and Truncated Readout Ideally, we want readout over infinite-length path or infinite time window: Rideal(Γ) = Z∞ 0 R(θ(t), ϕ(t)) dt. However under finite complexity budget Tmax, can only readout in finite time: Rtrunc(Γ) = ZTmax 0 R(θ(t), ϕ(t)) W(t) dt, where W(t) is window function supported on [0, Tmax], used to smoothly truncate at time boundary. Difference between them defines readout error: Err(Γ; W) = Rideal(Γ) − Rtrunc(Γ). In frequency domain description, choice of Wdirectly determines error decay and energy leakage properties, thus choosing optimal window function Wis core of error control. 4 2.3 Error Norm and Worst-Case Error For unified discussion, we introduce semi-norm on path space (e.g., L2norm induced by unified time scale frequency spectrum). Suppose for each path Γ there exists corresponding frequency domain object fΓ(ω) (e.g., combined from scattering data and unified time scale density on control manifold), satisfying |Γ|2=ZΩ |fΓ(ω)|2dµ(ω). Readout error can be written as frequency domain window form (see Section 3), whose worst-case norm defined as E(W) = sup Γ=0 |Err(Γ; W)| |Γ|. We care about finding window function family {W}making E(W) as small as possible under given time–bandwidth and complexity budget constraints. 3 Spectral Windowing Readout in Unified Time Scale Frequency Domain This section introduces frequency domain description under unified time scale master scale, writing readout operator as inner product of window function and frequency spectrum. 3.1 Unified Time Scale Frequency Domain Representation In previous work, we have connected unified time scale with scattering data on physical universe side. Pulling this structure back to computational universe, can consider each path Γ corresponds to frequency domain object fΓ(ω) = κ(ω) Φ(Γ; ω), where κ(ω) is unified time scale density, Φ(Γ; ω) encodes path response to this frequency mode through control–scattering structure. For given readout kernel, can define window in frequency domain W(ω) such that R(Γ) = ZΩ W(ω)fΓ(ω) dω. Ideal readout corresponds to Wideal(ω)≡1 (or some fixed weight), while finite complexity readout corresponds to restricting W(ω) to finite bandwidth [−W, W] or finite degree-of-freedom space. 5 3.2 Time–Frequency Dual Restriction and Window Function Design Problem Suppose we can only readout within time interval [−T, T], corresponding to time window wT(t) (e.g., wT(t) = 1 on |t| ≤ T, otherwise 0), and only interested in frequency band [−W, W]. In frequency domain, readout sensitivity to energy outside frequency band determines error: if path spectral components leave [−W, W], then window function W(ω) needs to suppress them as much as possible; but simultaneously should maintain as good pass characteristics as possible within [−W, W]. Thus we obtain typical dual-restriction window function design problem:  Time restriction: Readout window wT(t) supported on [−T, T];  Frequency restriction: Readout window spectrum bwT(ω) concentrated on [−W, W];  Objective: Under given constraints, minimize out-of-band energy or worst-case error. In signal analysis, this precisely classical problem of PSWF/DPSS; this paper interprets it as “best finite complexity readout” under unified time scale–complexity geometry. 4 PSWF/DPSS and Energy Concentration: From Time–Frequency to Time–Complexity This section reviews definitions and energy concentration of continuous PSWF and discrete DPSS, translating them to error control results in computational universe. 4.1 Definition and Time–Band Concentration of Continuous PSWF Definition 4.1 (Continuous PSWF).Fix time window T > 0 and bandwidth W > 0. Define integral operator (Kf)(t) = ZT −T sin W(t−s) π(t−s)f(s) ds, |t| ≤ T. This operator equivalent to composition of “first time-limit to [−T, T], then band-limit to [−W, W]”. Its eigenfunctions ψn(t) and eigenvalues λnsatisfy ZT −T sin W(t−s) π(t−s)ψn(s) ds=λnψn(t),|t| ≤ T. ψncalled Prolate Spheroidal Wave Functions under time window [−T, T] and bandwidth [−W, W]. Proposition 4.2 (Energy Concentration of PSWF).PSWF satisfy: 1. They constitute orthogonal basis on [−T, T]; 6 2. For each ψn, define frequency domain energy concentration αn=RW −W|b ψn(ω)|2dω R∞ −∞ |b ψn(ω)|2dω, then αn=λn, and λ0≥λ1≥. . . ; 3. For given time window and bandwidth, {ψn}are optimal basis in sense of “energy concentration under simultaneous time and frequency restrictions”: total energy concentration of any other same-dimensional subspace does not exceed sum of PSWF subspace. Proof of Proposition ?? given in Appendix A. 4.2 Definition and Finite Step–Finite Bandwidth of Discrete DPSS In discrete case, consider sequence of length N:x[0], . . . , x[N−1], whose discrete time– frequency restriction problem can be characterized through DPSS (Discrete Prolate Spheroidal Sequences). Definition 4.3 (DPSS).Fix sequence length Nand normalized bandwidth W∈(0,1/2). Define Toeplitz matrix Kmn =sin 2πW(m−n) π(m−n),0≤m, n ≤N−1, on diagonal define Kmm = 2W. Solve eigenvalue problem N−1 X n=0 Kmnv(k) n=λkv(k) m. Normalized eigenvectors v(k)are DPSS under (N, W), eigenvalues λkare energy concentrations: λk=P|ω|≤W|bv(k)(ω)|2 P|ω|≤1/2|bv(k)(ω)|2. Proposition 4.4 (Discrete Energy Concentration of DPSS).DPSS {v(k)}maximize energy concentration under discrete time–frequency dual restriction: for any subspace V⊂CNof dimension K, its energy concentration sum within frequency band [−W, W] does not exceed concentration sum of space spanned by first KDPSS. Proof see Appendix A. 7 4.3 Interpretation in Computational Universe as “Finite Complexity Readout” In computational universe, path segment of length Nviewable as complexity step number limitation N; corresponds to time window T≈N∆tunder unified time scale. Frequency band Wcorresponds to effective support of unified time scale density κ(ω). Under this perspective:  PSWF corresponds to optimal continuous window function under given complexity window [−T, T] and frequency band [−W, W];  DPSS corresponds to optimal discrete readout sequence under discrete complexity step length Nand frequency band W. Therefore, under finite complexity budget any readout operator hoping to faithfully capture information in unified time scale frequency spectrum, its time window or sequence should approximate PSWF/DPSS as much as possible. 5 Complexity–Time–Bandwidth Triple Unified Constraints This section discusses relationship among complexity budget T, time window T, and frequency band W, giving “computational universe version of Landau–Pollak–Slepian restriction”. 5.1 Time–Frequency–Complexity Degree-of-Freedom Counting In classical time–frequency analysis, effective degree-of-freedom number of signal subspace with bandwidth Wand time restriction Tis Neff ≈2WT π. This result can be obtained through asymptotic behavior of PSWF eigenvalues: eigenvalues λnclose to 1 when n < 2WT/π, rapidly decline to 0 when n > 2WT/π. In unified time scale–complexity geometry, we can interpret this degree-of-freedom count as:  For given complexity budget Tand unified time scale frequency band W, number of independent modes that can be reliably encoded or read out is approximately Neff;  On task information manifold, this corresponds to number of Fisher modes identifiable under finite complexity budget. 5.2 Complexity–Time–Bandwidth Constraint Inequality We formalize computational universe version: 8 Theorem 5.1 (Complexity–Time–Bandwidth Degree-of-Freedom Upper Bound).Suppose computational universe has unified time scale frequency domain representation fΓ(ω), whose support or effective energy concentrated on [−W, W]. For complexity budget T, restricting path within complexity ball BT(x0), consider all readout operators Rj(Γ) = ZW −W Wj(ω)fΓ(ω) dω, j = 1, . . . , K, where {Wj}are orthogonal window function family in L2([−W, W]). Then under error tolerance ε, there can exist optimal window function family {W⋆ j} (given by first KPSWF spectra), such that for all paths Γsatisfying fΓ(ω)− K X j=1 ⟨fΓ, W⋆ j⟩W⋆ j(ω)L2([−W,W ]) ≤ε|fΓ|L2([−W,W ]) only when K≳2WT π+Olog(1/ε). That is, under complexity–time–bandwidth triple constraint, reliably distinguishable degree-of-freedom number does not exceed ≈2WT/π. Proof see Appendix B. This result understandable as “computational universe’s Nyquist–Slepian restriction”: unified time scale frequency band Wtogether with complexity budget Tdetermine effective mode number readable in finite time. 6 Variational Principle of Window Function Choice: Optimal Observation Strategy This section introduces window function choice into time–information–complexity joint variational principle, constructing variational form of “optimal observation strategy”. 6.1 Extended Joint Manifold and Window Function Degrees of Freedom Previous joint manifold EQ=M×SQ where curve z(t)=(θ(t), ϕ(t)) describes control–information state evolution. Now add window function degrees of freedom: let Wbe some window function space (e.g., subspace in L2([−T, T]) satisfying band-limit constraints), for each readout channel jchoose window function Wj∈ W. Extended joint configuration as bz(t)=(θ(t), ϕ(t),{Wj}K j=1). Window function itself can have no explicit time evolution (viewed as static part of strategy), or can be updated on slow variable scale. 9