Recursive Interaction Systems: Non-Identifiability, Emergence, and Metastability

We define Recursive Interaction Systems (RIS), a class of dynamical systems in which a high-dimensional latent state is projected through a non-invertible emission map and the emitted observation is recursively re-injected into the next-step dynamics. Under explicit assumption packages, we derive three structural consequences. First, non-identifiability: if the emission map has nontrivial kernel and the initial distribution assigns positive mass to two distinct states in the same emission fiber, observable trajectories do not uniquely determine latent trajectories. Second, typical-set separation: if the component predictors satisfy positive divergence rate and cross-entropy separation, the mixture concentrates on commitment chains to which each component assigns exponentially small probability. Third, spectral metastability: under exact or approximate lumpability with slow/fast spectral separation, the induced kernel exhibits near-unit eigenvalues and slow mixing. Independence: the three textbook sufficient conditions for typical-set separation (Δi > 0, hard token partition, geometric basin separation) are each individually non-necessary; loop-driven divergence accumulation holds in concrete systems where each fails (Theorem 6.16, formally verified in Lean). For the separation and metastability results, the δ-separated regime provides an explicit solvable instance (spectral gap Θ(δ), positive divergence rate). We construct an explicit finite-state RIS exhibiting all three phenomena in one inspectable model. The contribution is identifying one operator class (recursive lossy feedback) in which all three phenomena arise from a single recursive mechanism. Empirical realization in trained systems is outside the scope of the present work.