Complementary Methods for Equilibrium Diagnostics: Geometric Ratio Analysis and Variance-Based Deviation Measures

This extends the Thales-based variance diagnostic to connected triloop configurations, including linear chains, branching structures, and closed cycles. The analysis shows how deviation quantified by the coupling channel propagates, dampens, or amplifies when diagnostic contexts are linked. A simple geometric condition governs cyclic closure, distinguishing stable, neutral, and runaway variance regimes. These results do not introduce dynamics or physical laws; they classify how variance behaves under connected orthogonal decompositions. The extension clarifies why supercritical operating points require heterogeneity or damping to avoid systemic amplification, strengthening the diagnostic scope of the method while preserving its domain-agnostic and non-causal character.