Fixed time-asymmetric correlations with arbitrarily weak dissipation

Unrefereed research candidate. Experiments often estimate how much energy a driven system dissipates from the time asymmetry of measured correlations. A popular shortcut assigns the observed covariance the entropy-production rate of a Gaussian process with the same covariance. Seara, Machta and Murrell (Nature Communications 12, 392, 2021) stated that for nonlinear systems this Gaussian value lower-bounds the true rate. This package shows that the unrestricted statement is false. A family of fully observed four-state Markov chains shares one fixed, nonzero, time-asymmetric covariance at every lag, with bounded rates and uniform mixing, while their entropy-production rates take every positive value. Planar Markov diffusions with the same linear drift as the Gaussian comparator and state-dependent noise reproduce the same covariance exactly, with rates filling (0, 1/3]; a Student-type member has rate α/(3(α+1)). The Gaussian value of the covariance is 1/3. Rare, large excursions carry the covariance, while dissipation scales with how often the system is active. A reversible-gating construction gives the general mechanism, the paper locates the missing step in the published justification, and an established fourth-order bound (Raux, Maier and Seifert) recovers up to 93.5% of the true rate on the example. Scope: fixed covariance with vanishing dissipation needs growing readout amplitudes or heavy-tailed noise, and energies or affinities diverge in the limits; diffusions with constant noise remain open; the Gaussian formula and the published numerical estimates are not disputed. The archive contains the 16-page manuscript (PDF and LaTeX), Python verification scripts (exact rational, symbolic and numerical), their result records, a replay driver with claim checks, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; exact and symbolic checks; a producer-side replay from a clean copy; internal AI-assisted review (a five-perspective review, two adversarial correctness audits by a different model, a proof audit and a re-review). This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.