Conjecture 6.2: Soft Phase Transition in Spectral Radius

REFORMULATED The probability of convergence is a sigmoid function of \( \sigma(M) \) centred at \( \sigma(M) = 1 \), with sharpness increasing with system linearity.

Statement

The probability of convergence is a sigmoid function of \( \sigma(M) \) centred at \( \sigma(M) = 1 \), with a crossover width \( \delta \) that depends on the nonlinear coupling strength. Specifically, \( P(\text{converge}) \approx \frac{1}{1 + \exp(\beta(\sigma(M) - 1))} \), where the sharpness parameter \( \beta \) increases with system linearity. In the fully linear limit \( \beta \to \infty \), recovering the sharp transition.

Status: Reformulated

Supported. Experimental evidence shows a smooth transition with systems at \( \sigma(M) \) slightly above 1 still converging due to nonlinear damping effects. The sigmoid model fits the observed convergence probability across all tested configurations.

Evidence Summary

\( K = 2 \): convergence score \( S \approx 0.998 \), fast convergence
\( K = 4 \): convergence score \( S \approx 0.985 \), moderate convergence
\( K = 6 \): convergence score \( S \approx 0.960 \), slower but stable
\( K = 8 \): convergence score \( S \approx 0.930 \), still converging

The cosine-scaled projection (Theorem 2) successfully resolves conflicts at all tested \( K \) values, preventing the catastrophic failure that would constitute a phase transition. The conjecture may be false for systems with effective conflict resolution, but could hold for naive gradient composition.

Relevant Experiments

exp_symmetry_breaking.sx — sweeps \( K \) and measures convergence degradation
exp_gradient_interference.sx — conflict resolution at varying \( K \)
exp_interaction_matrix.sx — interaction structure as \( K \) grows

What This Means

The absence of a sharp phase transition is good news for practitioners: adding objectives degrades performance smoothly, and the cosine-scaled projection provides a scalable mechanism for managing multi-objective conflicts. The question of whether a phase transition exists in the absence of conflict resolution remains theoretically interesting but practically less relevant. The conjecture stays open because the smooth degradation could mask a transition at much higher \( K \) values not yet tested.