Protocols, Limits, and Silence: A Metamathematical Model of Bounded Reflection Using Dynamic Hierarchies

We develop a rigorous formal model for a reflective cognitive process, modeled as a dynamic hierarchy of formal theories with operations of expansion (push) and contraction (pop). Building on the author's previous work (Osipenkov 2026a, 2026b), we define a formal protocol as any theory appearing in such a hierarchy. Using the semantic isolation theorem (2026b), we prove that after a pop operation a protocol cannot define truth for the language it has just abandoned — a dynamic analogue of Tarski's undefinability theorem. We then analyze the limit of a protocol undergoing infinite oscillation (alternating pushes and pops) and prove that this limit collapses to the base theory Q. This limit state, which we call formal silence, is characterized by the absence of any non‑trivial semantic predicate: by Tarski's theorem, Q cannot express its own truth. The construction provides a precise mathematical analog for the limits of introspection and the “ineffable ground” discussed in transcendental philosophy. We show that the framework unifies the impossibility of internal semantic critique (2026a) with the dynamics of hierarchical reflection (2026b), offering a unified perspective on bounded cognition. Philosophical correlates are drawn with Kant's notion of the Ding an sich (interpreted as the process of abstraction rather than a transcendent object) and Wittgenstein's dictum that “whereof one cannot speak, thereof one must be silent” — now realized as a provable theorem about the limit of a formal protocol.