Operational Signalling under Adaptive Interaction: Persistence, Level Cascades, and Control of Multiscale Profiles

This paper studies operational signalling in measurement-dependent models under adaptive multi-round interaction. Unlike fixed-profile analyses, the underlying probability distribution is allowed to evolve dynamically across rounds, leading to fundamentally different behaviour. Previous work established the limitations of single-shot inference, the emergence of signalling under repeated interaction, and the structural theory governing one-step amplification. In particular, a complete one-step theory for the k-ary setting identified optimal signalling coefficients and extremal profile structures. However, these results assume that the probability profile remains fixed across rounds. The present work addresses the adaptive setting in which the profile evolves in response to interaction. Key results Persistence of signallingSignalling remains possible under adaptive interaction even when one-step coefficients are small. Level cascade mechanismA structural process is identified in which probability mass concentrates across rounds, driving amplification. Multiscale dynamicsMulti-round signalling capacity depends on the evolution of the full profile, not just scalar coefficients. Control formulationThe problem is formulated as a controlled process on the probability simplex, revealing a Bellman-type recursive structure. Significance These results show that adaptive evolution fundamentally alters the nature of operational signalling. Static optimality does not determine dynamic capacity; the ability to reshape the underlying distribution across rounds introduces new mechanisms that are invisible in one-step analyses. This paper extends a broader research programme on operational signalling under measurement dependence, providing the first systematic treatment of adaptive profile dynamics. Keywords: measurement dependence, operational signalling, adaptive interaction, Bellman recursion, probability simplex, multiscale dynamics