On the typicality of the coarse-grained arrow on ballistic Poisson particles: a non-vanishing counterexample measure, the limiting density of unmarked particles, and partial monotonicity theorems around an open problem

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 rule; in [ZERMELO, 10.5281/zenodo.22313877] the rule was made pathwise reversible (the model M^†_3), so that only the initial condition carries an arrow, and expectation monotonicity of the number N_0 of unmarked particles was refuted for a fixed finite configuration even with symmetric coins (Proposition Z2-C′ there, a certified overshoot of order 10^{-6}). This paper asks whether such counterexamples become negligible as n→∞ under typical Poisson geometry and coins — whether the coarse-grained arrow becomes a theorem — and answers in two layers, in the static case σ=0. Layer 1 (quantitative). For a single seed the answer is negative in the quenched sense: with symmetric coins, λ=1 and T_s≥ T_0(K) (frozen signs included), the probability P_n that the quenched expectation curve t↦ E^{g,U}[N_0(t)] fails to be non-increasing on [0,t_2(K)] has an explicit rational lower bound that converges, at fixed density, to a positive Palm value (Theorem ST-M; 1-1). The bound is obtained by placing certified counterexample configurations — the four-particle triangle-plus-leaf (Proposition ST-C4; 1-3) and the six-particle configuration of Z2-C′ — into rational boxes; P_n itself is not shown to converge. The frozen-sign duality P(v_j(t)=0)=1/2(1+E^{A^{T}}_{e_j}[θ^{|v(t)|}]) (Lemma ST-L; 1-2) and a catalogue of frozen dips over all connected graphs with k≤6 (ST-CAT; 1-5, a measurement claim relative to its grid and threshold: 71 dip pairs, none for k≤3, none detected in the model at T_s∈{3,30,300}) delimit the mechanism. For broad low-entropy initial conditions (i.i.d. Bernoulli(ε) marks) the unmarked fraction N_0(t)/n converges in probability, uniformly on [0,T], to a deterministic Palm density x(t) with an explicit C^1 derivative, and monotonicity of the empirical curves reduces to the sign of x' (Lemmas ST-P, ST-Δ', ST-C, ST-V, ST-LLN, ST-D; 1-6). Layer 2 (limit). For frozen signs and symmetric coins with ρ c_0≤1 we establish the structure of x — Palm cluster decomposition, root uniformity, a seed-sum representation with its spectral form and the exact pair solution (2-1 to 2-4) — and the Palm duality x(t)=1/2+1/2E_{G}[θ^{N(t)}] with its first-order expansion (2-10; the expansion for ρ c_0<1). Monotonicity x'<0 holds on an initial interval [0,t_0) for any T_s (Theorem L2-T1; 2-5), in the dilute regime ρ<ρ_0(T) on [0,T] (Theorem L2-T2; 2-6, T_s=∞), and for near-uniform initial conditions 1-2ε<θ_1(ρ,T) on [0,T] (Theorem L2-T4; 2-12, ρ c_0<1), while the long-time limit x(∞)=E_{G}[φ_ε(|C(0)|)]<x(0) is explicit (Theorem L2-T3; 2-8). At the cluster level monotonicity fails already at k=4 (recorded, t≤200) and at k=5 (certified, Proposition L2-C5; 2-7), but holds for all t when θ≤θ_0(G) (Lemma L2-P6; 2-11), the higher-order terms being controlled by a flux bound. Whether x is monotone on all of (0,∞) remains open: we record, as a confirmed statement (U; 2-9), what is missing — a cancellation principle across cluster types, or a certified positive window — in falsifiable form, with the conditions under which the verdict would change. Numerical reconnaissance (k≤6, t≤1000, ρ≤0.3) neither proves nor refutes it, and no claim of typical monotonicity is made. All 18 confirmed claims passed independent review (8 proof-and-record review cycles; the manuscript reviews are recorded in the same ledger); the normative text is byte-invariant, and citations, numbers and ledgers are machine-checked. The role of AI systems is disclosed. 要旨. 弾道運動する Poisson 粒子上の接触過程は、前作で規則から非可逆性を抜いた模型 M^†_3 において、矢は初期条件だけが運び、未標識粒子数 N_0 の期待値の単調性は固定した有限配置では対称コインでも破れること(命題 Z2-C′、認証済み)が示された。本稿は「その反例は n→∞、典型的な幾何とコインの下で消えるか — 粗視化した矢は定理になるか」を静的な場合 σ=0 で二層に分けて問う。層1(定量層): 単一シードの quenched 曲線については定理にならない — 認証済みの反例配置(4 粒子の三角形 + 葉〔命題 ST-C4〕と Z2-C′ の 6 粒子)を有理数の箱に置くことで、quenched 期待値曲線が [0,t_2] で非増加でない確率 P_n に明示的な有理数の下界を与え、それは密度固定で正の Palm 値に収束する(定理 ST-M; P_n 自身の収束は主張しない)。符号凍結の双対恒等式(補題 ST-L)と k≤6 の連結グラフの凍結 dip の目録(ST-CAT、測定主張: 71 対、k≤3 には無い、模型 T_s∈{3,30,300} では検出されない)が機構を限定する。広範な低エントロピー初期条件(i.i.d. Bernoulli(ε) の標識)では N_0(t)/n が極限密度 x(t) に [0,T] 上一様に確率収束し、単調性は x' の符号に還元される(補題群 ST-P〜ST-D)。層2(極限層): 符号凍結・対称コイン・ρ c_0≤1 で x の構造(Palm クラスタ分解・根の一様性・シード和表現とスペクトル表示・Palm 双対恒等式と一次展開〔一次展開は ρ c_0<1〕)を確立し、x'<0 を初期区間(任意の T_s)・希薄領域 ρ<ρ_0(T)・近一様初期条件 1-2ε<θ_1(ρ,T)(有限地平)で証明、長時間極限 x(∞)=E_{G}[φ_ε(|C(0)|)]