Minimum Dissipation from Scalar Gaussian Spectra and Finite-Lag Bounds for Incomplete Observations

Can a completely time-reversible observed signal impose a strictly positive minimum dissipation on its hidden source? This paper derives an exact minimum entropy-production rate for a family of scalar Gaussian spectra, optimized over all finite-dimensional stable Ornstein–Uhlenbeck sources with positive-definite diffusion and an instantaneous linear readout. The proof establishes that every admissible source can be reduced to a minimal realization preserving the complete observed Gaussian law without increasing entropy production. An explicit example exhibits identical observed signals compatible with every entropy-production rate from 1/24 upward, while excluding equilibrium within the specified class. The observation therefore fixes a positive dissipation floor without identifying the source or its actual dissipation. For bounded observations of more general stationary Markov sources, the paper also derives finite-lag covariance bounds, an information-theoretic strengthening, and an optimized prediction-residual witness. Brownian-rotor examples demonstrate detection beyond specified existing scalar estimators. A further analysis identifies a missing positivity condition in particular published equilibrium reconstructions. The physical interpretation depends on the appropriate time-reversal convention and thermodynamic identification. Hidden odd-parity variables, detector memory, and finite measurement bandwidth are explicitly examined. Motivated by Shadow Theory’s source/readout distinction, the results are formulated independently within stochastic thermodynamics. This research preprint includes mathematical proofs, worked examples, and a discussion of scope and limitations.