A General Structural Principle for Coherence and the Emergence of an Interior

This paper introduces a general structural rule for when a system forms and maintains an interior. The rule applies across physical, biological, cognitive, and symbolic scales. The central idea is that a system persists when its internal restoring activity exceeds the disruptive influence of its environment. This condition is expressed by the coherence ratio R = λ_self / λ_env. A system reaches a coherence boundary when R ≥ R★, where R★ is a geometry dependent threshold. When this boundary is crossed, the system maintains its own pattern across time and gains a self preserving interior that provides a reference frame for information. The same condition can be written in timescale form. Let τ_self be the internal restoration timescale, let Γ_disruption be the environmental disruption rate, and let κ be a boundary coefficient. The coherence condition becomes τ_self⁻¹ ≥ κ Γ_disruption. Both forms describe the same structural threshold. This paper extends the relation across multiple scales by allowing each level to inherit stability from the level beneath it. The internal restoring rate at level n is λ_self,n. A stabilizing contribution c strengthens the self term at the next level, producing the update λ_self,(n+1) = √( λ_self,n² + c² ). This induces a recursive update of the coherence ratio. If the disruptive rate varies slowly across neighboring levels, the resulting relation is R_(n+1) = R_n √( 1 + k / R_n² ), where k is a positive constant determined by c and λ_env,n. If the sequence {R_n} reaches or exceeds the threshold R★ across levels, then the system maintains a coherent interior through multiple layers of organization. This provides a structural basis for persistence, boundary formation, and the unified reference frame required for conscious experience. This paper presents the first formulation of this coherence threshold, its multi level recursion, and the specific mathematical rule R_(n+1) = R_n √( 1 + k / R_n² ) as a general condition for the emergence of an interior.