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

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For contact dynamics on Poisson clusters with frozen signs, the unmarked fraction density strictly decreases over an extended logarithmic time scale, and the slowest cluster types governing the dynamics are reduced to paths and cycles. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Y. Maeda

doi:10.5281/zenodo.22734679

September 13, 2026Open Access

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

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Key Points

Determine whether the unmarked fraction density fails monotonicity in frozen-sign marked dynamics on Poisson clusters, and characterize the slowest decay modes across cluster types.
Analyzed static Poisson contact dynamics using cluster type expansions, Borel tail bounds, and Dirichlet monotonicity of the spectral gap under edge deletion.
Reduced system dynamics to symmetric sectors via graph automorphisms and performed verification using exact rational arithmetic certificates.
Demonstrated that the unmarked rate is strictly negative ($x' < 0$) on the explicit domain $0 < \rho c_0 \le 1/1000$, $\theta \in [3/10, 49/50]$, and $t \le (22/5)\ln(1/(\rho c_0))$, extending the logarithmic time coefficient from $1/3$ to $22/5$.
Proved that for $k \ge 4$, connected graph spectral gaps restrict the slowest effective relaxation candidates strictly to paths and non-path trees, ruling out cycles.
Identified a Walsh self-duality establishing that the slowest symmetric mode sign corresponds to Walsh parity, certified with negative signs for $k=4$ through $9$.

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Implication

Theoretical analysis reveals strict monotonicity of unmarked fraction dynamics on Poisson clusters at logarithmic times, indicating persistent decay without positive rate windows.

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Y. Maeda (2026) studied this question.

synapsesocial.com/papers/6aa7b41e0926e14a848b3b10 https://doi.org/10.5281/zenodo.22734679

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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

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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

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