What does this research mean for the field?

For contact dynamics on Poisson clusters with frozen signs, the unmarked fraction density is strictly decreasing on an explicit finite region up to a logarithmic time scale of 22/5, and the slowest effective rate candidates for cluster types narrow to paths and non-path trees. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine whether the unmarked density derivative in frozen-sign contact dynamics on Poisson clusters ever turns positive, and to characterize the slowest decay modes across distinct graph topologies.

September 14, 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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To determine whether the unmarked density derivative in frozen-sign contact dynamics on Poisson clusters ever turns positive, and to characterize the slowest decay modes across distinct graph topologies.
Derived analytical bounds on cluster type weights using Dirichlet monotonicity of the spectral gap under edge deletion, symmetry sector reductions, and Borel tail bounds.
Verified bounds and structural mode inequalities using exact rational arithmetic certificates reimplemented across four independent proof cycles.
Applied Walsh parity and Griffeath cancellative self-duality to evaluate mode signs and spectral gaps across paths, cycles, and trees.
Demonstrated that the derivative x'(t) remains strictly negative across the domain D up to time t <= (22/5)ln(1/(rho*c_0)), proving the complete absence of positive windows.
Proved that for all connected graphs with k >= 4 vertices, every topology other than paths and cycles has a spectral gap at least as large as the minimal non-path tree gap.
Certified that cycles do not attain the slowest effective rate for k = 4 through 9 with theta in [3/10, 49/50], narrowing slowest-mode candidates strictly to paths and non-path trees.

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Implication

Theoretical analysis demonstrates persistent decay of unmarked fractions over logarithmic timescales, indicating strict monotonicity across finite cluster domains.

Cite This Study

Y. Maeda (2026) studied this question.

synapsesocial.com/papers/6aa7dfc60926e14a848b3e96 https://doi.org/10.5281/zenodo.22734680

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Implication

Theoretical analysis demonstrates persistent decay of unmarked fractions over logarithmic timescales, indicating strict monotonicity across finite cluster domains.

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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Discussion

Implication

Cite This Study

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

Key Points

Discussion

Implication

Cite This Study

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