Conjecture 6.1: Convergence Ratio Class Universality

REFORMULATED For systems sharing the same spectral radius \( \sigma(M) \) and number of subsystems \( K \), the ratio \( R \) converges to a class-dependent constant \( R^*(\sigma, K) \).

Statement

For systems sharing the same spectral radius \( \sigma(M) \) and number of subsystems \( K \), the ratio \( R = \frac{V(\theta_T)}{V(\theta_0)} \) converges to a class-dependent constant \( R^*(\sigma, K) \) as \( T \to \infty \). That is, universality holds within equivalence classes defined by \( (\sigma(M), K) \), not globally.

Status: Reformulated

Supported. Experimental evidence shows R converges for each system with the limit depending on \( \sigma(M) \) and \( K \), consistent with a classification rather than a single universal constant.

Evidence Summary

Experiments across multiple problem domains show consistent convergence of \( R \) within each problem instance, but different asymptotic values across problem classes:

Belief networks: \( R \approx 0.9999 \)
Game-theoretic systems: \( R \approx 0.9998 \)
Compiler pass composition: \( R \approx 0.9997 \)

The differences are small but systematic and reproducible, ruling out universality in the strict sense. Whether a weaker form of universality holds (e.g., universality within topological equivalence classes) remains open.

Relevant Experiments

exp_convergence_order.sx — measures \( R \) convergence dynamics
exp_contraction.sx — subsystem contraction rates that determine \( R \)
exp_iratio_proof.sx — interaction ratio at equilibrium

What This Means

The convergence ratio is a powerful diagnostic — its rapid stabilisation within any given system confirms that the system has entered the contraction regime. However, it cannot serve as a universal constant for cross-system comparison. The conjecture as originally stated is likely false, but a refined version restricting universality to topologically equivalent systems may hold. This remains an active research question.