Time reversal for contact processes on ballistic Poisson particles: chain-level duality, degenerate path-level entropy production, and the velocity dependence of the quenched arrow

Abstract

We study time reversal for the contact process on ballistically moving Poisson particles introduced in (entry-driven, fire-and-forget transmission at range R, success probability p, delay tau, each ordered pair attempts at most once, lifetimes mathrm{Exp}(T_s), re-marking of dead particles allowed). Three layers of questions are addressed. Structure of double reversal: under the velocity-symmetry assumption A-nu the law of the driving data (geometry, lifetime crosses, coins) is invariant under the double reversal Θ_T (Lemma 2) --- informally, the arrow lives in the exploration rule, not in the data --- and the reversed dynamics requires an unbounded look-ahead into the reversed future (Lemma 4). At the level of interval-annotated genealogies, forward and reversed feasibility and open-walk probabilities correspond exactly (Lemma D-c2, telescoping identity p^n e^{-(T-ntau)/T_s}), yet the hitting event admits no representation by open paths (Theorem D-c3, a five-particle witness certified in exact rational arithmetic). Expected numbers of alive-at-T tau-interior open walks are reversal symmetric, quenched and annealed (Theorem D-c4): the alive-at-T hitting asymmetry lives only in boundary walks and interference terms. In the static limit the quenched hitting pair identity holds pathwise (Theorem P7) while its alive-at-T version fails through re-marking (Proposition P7'). Path-level entropy production: the joint laws of the forward and reversed records are mutually singular, even after discarding all boundary events (Theorem P9, Corollary P9.2); the path-measure entropy production is +infty a.s. and the process is absolutely irreversible (lambda_S=1). Velocity dependence of the arrow: the estimator-free quenched hitting asymmetry D^ever_T vanishes identically below a geometry-dependent velocity scale sigma_0(G)>0 (Theorem H3.1: for sigma Preprint v0.3.1 (2026-08-30). All 42 mathematical claims were accepted under a claim-driven verification protocol (fifteen proof-review cycles and two manuscript-review cycles by fresh AI instances; three mutually agreeing simulator implementations); the complete method record, error ledger and the empty steering-intervention log are disclosed in the appendices. The bundled repository archive contains the normative sprint documents, all review cycles, the simulation code and the raw Monte-Carlo output.