When Coherence Measures Become Meaningful: Covariance as a Precondition for Structural Signal Interpretability

Existing coherence metrics are frequently interpreted as reflecting intrinsic structural properties of a system or output. Yet this interpretation remains unjustified if the signal varies under transformations that preserve the underlying content. We address this problem by introducing covariance as a necessary condition for the interpretability of structural coherence signals. Within the CMCI framework (St-Louis, 2026), covariance is defined as invariance — or bounded stability — of the signal trajectory under a pre-declared family of admissible context transformations. We formalize this concept mathematically, distinguishing strict covariance from tolerance-bounded covariance, and show how it can be falsified through explicit counterexamples. We then present a reproducible empirical protocol based on controlled prompt transformations, identity and cross-topic controls, and an equivalence-based statistical framework. In a pilot study (K = 30 session pairs, T = 10 turns), the CMCI μ signal remains stably bounded under syntactic paraphrase (95% upper confidence limit on the 90th percentile of maximum pointwise deviation: 0.065, below the pre-registered tolerance ε_μ = 0.10) while remaining strongly discriminative across genuine content changes (cross-topic median deviation: 0.253; 100% of sessions exceeding the tolerance). These results establish covariance not as a claim of correctness, but as a necessary precondition for interpretability: a structural signal that is not stable under admissible reformulations cannot be meaningfully treated as reflecting content-level structure. This reframes coherence evaluation from a problem of score design alone to one of measurement validity under transformation. This preprint is part of the CMCI / Coherix research program and complements the framework paper published in Frontiers in Artificial Intelligence (DOI: 10.3389/frai.2026.1836120).