Unbounded Irreversibility at Fixed Low-Order Dynamics and Generator Spectrum

This paper establishes an exact mathematical counterexample showing that extensive agreement between the statistical and dynamical properties of finite-state stochastic systems does not determine their time-reversal irreversibility. A one-parameter family of irreducible, 20-state population-replacement Markov processes is constructed. Throughout the family, the systems possess identical stationary distributions, statewise holding rates, drift, quadratic variation, generator eigenvalue spectra, and specified linear-response functions. Conditional count moments through degree two agree at every time, while stationary temporal count moments of total polynomial degree at most three are also identical. Despite these shared properties, the stationary path-space irreversibility ranges from zero to arbitrarily large values, while the mean transition activity remains fixed. The construction therefore demonstrates that these commonly used statistical summaries cannot determine whether a system is reversible or provide a finite upper bound on its irreversibility. An explicit fourth-degree, two-time correlation statistic is derived that distinguishes the otherwise indistinguishable processes. The paper provides analytical proofs, exact symbolic calculations, numerical verification, and reproducible computational experiments. The result contributes a finite-state identifiability counterexample and a mathematical benchmark for investigating hidden nonequilibrium dynamics, stochastic inference, and the limitations of reduced-order descriptions. The findings concern mathematical path irreversibility. Interpreting this quantity as physical thermodynamic entropy production requires additional assumptions about the physical realization. Research preprint — not peer reviewed.