Multi-Round Correlational Signalling and Corner Reduction

This paper establishes an exact law governing signalling capacity under repeated interaction in binary latent-variable models. We prove that the n-round signalling capacity satisfies: C⁽ⁿ⁾ = 1 - 2¹⁻ⁿ The result is derived through a structural analysis based on: reduction to extremal states, posterior freezing, a strict per-round mass conversion bound, and a no-dual-lock constraint. The proof is constructive and matches a canonical branch-and-lock strategy. This work completes a three-part series on operational signalling: Paper I: Non-identifiability and control parameter Γ Paper II: Emergent signalling under repeated interaction Paper III: Exact multi-round amplification law