Reversible contact dynamics on ballistic Poisson particles: pathwise time reversal, the Gibbs arrow, and certified counterexamples to Boltzmann-type monotonicity

Abstract. The contact process on ballistically moving Poisson particles of [Phase R, 10.5281/zenodo.21979354] was shown in [LOSCHMIDT, 10.5281/zenodo.22233373] to carry an arrow of time inside its exploration rule (mutual singularity of forward and reversed path laws). Here we remove the irreversibility from the rule and ask where an arrow can then come from. Layer 1 (reversibility). Keeping the driving data of the original model (Poisson geometry, lifetime crosses, ordered-pair coins) we prove that each of the three rule ingredients — transmission latency τ>0, the state-driven entry trigger, and absorbing death — alone separates the forward and reversed record laws (in the class of record functionals with a standard reversal, under the respective positive-probability conditions; Theorem Z-L, Propositions Z-E and Z-D, Lemma A-ac; claims 1-6), so that the original requirements R1–R3 are strictly incompatible, while a minimal modification M^†_3 (states {0,+,-}, crosses acting as the involution σ=(+ -), Poisson arrows on contact whose effect is the involution ι=(0 +) controlled by the sender's state, no latency, no pair memory) is pathwise reversible: the record of the reversed data started from the final state is exactly the reversed record (Theorem Z-1; 1-4), its transition kernel is symmetric with the uniform equilibrium (Theorem Z-2; 1-5), and the only remaining asymmetry is the initial-condition edge (Lemma Z-B; 1-7). Layer 2 (the coarse-grained arrow). For the Boltzmann entropy S_M of the number of marked particles (Definition Z2-S; 2-1) we prove: the relative entropy of the law to the uniform measure is non-increasing for every initial law, quenched, and the averaged relative entropy is non-increasing annealed (Theorem Z2-G; 2-2); the reversed initial condition retraces the same quantity backwards, pathwise (Theorem Z2-R; 2-3); in a finite static system no non-constant state function is pathwise monotone, with mean recurrence time |R|/q, equal to (3^n-1)/q under strong connectivity (T_s<∞) (Proposition Z2-P; 2-4). Expectation monotonicity of N_0 is not a theorem: for a fixed asymmetric coin field it fails, together with that of S_M (Proposition Z2-C; 2-6), and for N_0 it fails with an overshoot of order 10^-6 even for symmetric coins in the frozen-sign limit and hence in the model for T_s≥ T_0<7.918×10^8 (Proposition Z2-C' with Lemma Z2-Δ; 2-8); the S_M side for symmetric coins remains open (2-5b). Both counterexamples are certified in exact rational arithmetic. Partial positive results (two particles, complete graph, mean-field ODE, initial derivative; 2-9 to 2-12) and the open conjectures (S_M side; coin-averaged) delimit what survives: the H-theorem-type arrow is a property of what is averaged and of two time scales, not of the reversible rule. Layer 3 (attribution) measures this typicality in periodic boxes (five families of 100 fields in a field-pair design; independently re-implemented and re-measured in review): from a seed the arrow is monotone within resolution while the uniform law is stationary (3-2); no decrease of the sign-resolved entropy is detected at n=40 (3-3); the slow-phase drop decreases with T_s ≥ 1 (3-4); motion speeds the arrow, the paired differences at the horizon being negative in all families (3-5); and the recurrence identity is reproduced (3-6). These are measurement claims, not theorems (Section 5). 要旨. 弾道運動する Poisson 粒子上の接触過程は、前作で探索規則の内部に時間の矢を持つことが示された(前向きと反転の経路法則の相互特異性)。本稿では規則から非可逆性を抜き、矢がどこから来るかを問う。層1: 遅延・進入トリガ・吸収死亡のそれぞれが単独で法則を分離すること(設定 (S) の類、各正確率条件の下; 定理 Z-L、命題 Z-E・Z-D)、最小改変模型 M^†_3 が経路ごとに可逆であること(定理 Z-1)、核が対称で一様平衡を持つこと(定理 Z-2)、初期条件の縁(補題 Z-B)。層2: 標識数の Boltzmann エントロピー S_M について、Gibbs 側の単調性(定理 Z2-G)、反転による逆行(定理 Z2-R)、経路単調性の不成立と再帰(命題 Z2-P)を証明し、期待値の単調性は、非対称コインでは N_0・S_M とも(命題 Z2-C)、対称コインの符号凍結極限では N_0 について(命題 Z2-C'; S_M 側は未反証、2-5b)一般には成り立たないことを厳密有理数の反例で示す。層3: 矢の帰属(初期条件・粗視化・符号緩和・運動・再帰)を周期箱の測定で典型性として述べる — シードからの単調整合と一様初期状態の定常性(3-2)、S_B の減少の非検出(3-3)、T_s ≥ 1 での遅い相の降下の減少(3-4)、運動が矢を速めること(地平での場対差が全族で負、3-5)、再帰恒等式の再現(3-6)。これらは定理ではなく測定主張である。 Version note. v0.5.1 (2026-09-05): deposit version after ten context-free review cycles (eight proof and measurement reviews, two manuscript reviews; 25 confirmed claims; no cycle remanded a document). The bundle zermelo-repo-v3.tar.gz contains the normative Japanese sprint documents, the claim and error ledgers, every review cycle (briefs, findings, records, reviewer scripts), the machine-generated records, the certificates in exact rational arithmetic, and the scripts. Third paper of the series after Phase R (10.5281/zenodo.21979354) and Project LOSCHMIDT (10.5281/zenodo.22233373).