Frozen-sign marked dynamics on Poisson clusters: an absence theorem at logarithmic times, a reduction of the slowest cluster types to paths and cycles, and a Walsh self-duality behind the parity of the slowest mode

Abstract. The contact dynamics on ballistically moving Poisson particles of [Phase R, 10.5281/zenodo.21979354] were shown in [Project LOSCHMIDT, 10.5281/zenodo.22170790] to carry an arrow of time inside their rule; in [Project 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 [Project STOSSZAHL, 10.5281/zenodo.22437728] showed, in the static case σ=0 with frozen signs, that for broad low-entropy initial conditions the unmarked fraction converges to a deterministic Palm density x(t), proved its monotonicity x'<0 on several partial regions, and left open in a recorded format (U) whether that monotonicity fails anywhere in (ρ,θ,t). This paper asks whether it actually fails — whether there is a positive window (ρ,θ,t^*) with x'>0 — and gives two kinds of answers: an absence theorem on an explicit finite region, and structural theorems about single cluster types — the rates and the signs of the slowest modes within each type group of the type expansion (L2-P3). No claim is made that these modes are the largest terms of x' at late times at fixed density; the sign of x' outside the region, which involves the cancellation across types, is not claimed. Layer 1 (absence). With σ=0, frozen signs, U≡1, λ=1 and i.i.d. Bernoulli(ε) marks (the initial condition I-ext(ε) of [Project STOSSZAHL, 10.5281/zenodo.22437728]), θ=1-2ε, on the region D={0<ρ c_0≤ 1/1000, θ∈[3/10,49/50], 0≤ t≤(22/5)ln(1/(ρ c_0))} one has x'(t;ρ,θ)<0 everywhere (Theorem EH-T1; 1-1): no positive window lies inside D, and for each (ρ,θ) in its range x is strictly decreasing on [0,(22/5)ln(1/(ρ c_0))]. Relative to the regions already proved in [Project STOSSZAHL, 10.5281/zenodo.22437728], the theorem extends the coefficient 1/3 of the logarithmic time scale of L2-T2 to 22/5. The proof combines upper and lower bounds on the weights of cluster types with a Borel tail bound (Lemma EH-L1; 1-2), Dirichlet monotonicity of the spectral gap under edge deletion with its reduction to trees (Lemma EH-L2; 1-3), the reduction to symmetric sectors under automorphisms and the vanishing of the path's coefficients outside the sector (Lemma EH-L3; 1-4), and a certificate in exact rational arithmetic that was independently reimplemented in review. Layer 2 (structure of the slowest terms). For k≥4 every connected k-vertex type other than the path P_k and the cycle C_k has spectral gap at least g'_k, the smallest gap among non-path trees (Theorem EH-T3; 2-1). For k=4..9, λ=1 and θ∈Θ=[3/10,49/50], certified inequalities r_sym(P_k) r_sym(P_k) for every k≥3 (Theorem EH-T6; 2-9): whenever the slowest symmetric mode of the path is effective (r(P_k,θ)=r_sym(P_k)), the cycle does not achieve the slowest effective rate, so, for k≥4, the candidates for the slowest effective rate of a type group narrow to the path and the non-path trees — question (Q3) is closed, (Q1) and (Q2) remain. The remaining sign question (Q2) is rewritten by duality (Lemma EH-L8; 2-10) and by Walsh parity (Lemma EH-L9; 2-13): when the slowest symmetric mode of P_k is simple with eigenvector v^* satisfying ⟨ v^*,N⟩≠0, one has ν_k<0 iff W v^*=v^* iff the Walsh-even part of the reflection sector is slower than the odd part, r_+(P_k) 0 is not treated. (7) The sign s_k=- of the slowest effective mode of a type group is claimed only for k=4..9, λ=1, θ∈Θ=[3/10,49/50] (certificates); k=2,3, k≥10 and general k are written in the open format (range, method, number of cycles, record references). (8) Theorem EH-T3 is a statement about gaps; no monotonicity of effective rates r(G,θ)≥ r(T,θ) is claimed; the words "dominate" and "govern" are not used — a type is a "candidate achieving the slowest effective rate". No claim is made that the slowest modes within a type group are the leading terms of x' at late times at fixed density. (9) Theorem EH-T4 concerns the complete-graph type only; other types, the sign of x', ∂_tS<0 at all times and rates of convergence are not claimed; the monotonicity of the star types is a recorded fact and a lemma candidate. (10) Lemma EH-L6 is a first-order rule; the sign of ν, the optimality of the threshold, the sign on all of Θ and the ordering of modes are not claimed. (11) The identity of Theorem EH-T5 is not called new: its form is known (a self-duality in the sense of the cancellative duality of Griffeath; Section 1.3), and only its application to this model is this paper's; the display of x' in Corollary EH-T5.C(c) is the form of a question, and the cancellation across types is not claimed. (12) The reformulations of Lemmas EH-L8 and EH-L9 carry the hypotheses of simplicity and ⟨ v^*,N⟩≠0; (P4), (P5) and (P5′) are open questions (k=4..9 is a theorem for (P4) and (P5) by Corollary EH-L9.C; (P5′) is measured only; the measurements for k=3..12 are not evidence); the rank-wise comparison of rates and a Sturm-type analogue tried in the attempt record are not claimed. (13) Certification is at the level of the type-averaged limiting density (EH-T1); no cluster-level fact of [Project STOSSZAHL, 10.5281/zenodo.22437728] is restated as an existence claim. (14) The number of independent review cycles quoted in the abstract is the number of proof-and-record cycles (4); the narrowest-scope checks (5) and the manuscript reviews are counted separately (Section 7). (15) The process is not called a contact process in the sense of Harris: the name M is kept only for continuity with the preceding deposits (Section 2), and the comparison with existing names (SI-type epidemics, Richardson growth, first-passage percolation, the one-sided infection processes) is a comparison, not an identification. The role of AI systems (Section 7). The sprint was executed by an AI system (Claude Code) as the executing agent, with the human author restricted to start permissions, encouragement and three-way decisions (proceed, stop, switch), a