RepresentationInvariance

Measurement systems applied to representation-dependent outputs — such as coherence signals derived from large language model (LLM) text — face a fundamental epistemological problem: the measured quantity may vary not because the underlying structure changed, but because its textual realization changed. We formalize this problem within a general theoretical framework. We define a structural state space as the 𝒮quotient of a realization space ᵣ under semantic equivalence, and a measurement Ωfunction: ᵣ → ᵈ. ΦΩℝ A measurement is structurally meaningful if and only if it factors through the quotient map: ᵣ →. We operationalize this ideal criterion as πΩ𝒮-εinvariance under a formally defined class of admissible transformations ₐdm, establish 𝔗connections to Invariant Risk Minimization, equivariant representation learning, and psychometric measurement invariance, and show that -invariance is a necessary but not εsufficient condition for structural interpretability. The Cognitive Multi-scale Coherence Index (CMCI) framework is presented as a concrete instantiation of the theory, with empirical bootstrap-confidence covariance tests providing a falsifiable operationalization of the -invariance criterion. We identify four structural failure modes of the theory, boundε its assumptions explicitly, and outline a research programme for strengthening the framework toward a full equivariance theory. 1.