Delayed Arrow-of-Time Detection in Signed Laplacian Dynamics

We study how rapidly the direction of time becomes operationally detectable from mesoscopic data when state-weights may be positive or negative. In contrast with classical Markov processes – where forward evolution is instantly distinguishable from its reverse – signed dynamics can render the arrow of time undetectable during an initial interval. Assume the generator of the signed dynamics is a symmetric signed Laplacian with a single linear invariant and a phase-space Second Law holds in the form of non-decreasing Rényi-$2$ entropy. Drawing on recent results on eventual exponential positivity of signed Laplacians (Chen et al., 2021), we define a test that correctly identifies the direction of thermodynamic time if conducted over a time interval of length at least $\tau$. We go on to prove that the test cannot deliver an incorrect conclusion if conducted over a shorter interval. Dropping symmetry, we exhibit a superquantum example where $\tau =+\mathrm{\infty }$, so that the arrow of time remains permanently undetectable under our test.