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

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.