Adaptive Internal Propagation (AIP): A Structural Scaling Framework with a Prime-Graph Testbed

We introduce Adaptive Internal Propagation (AIP) as a structural scaling framework for discrete systems with interacting components. AIP assigns to a system three quantities: internal capacity κ, effective dependency α, and a scaling exponent β describing how realized dependency changes as interaction capacity is enlarged. The aim of this note is not to propose a completed universal theory of computational hardness or collective structure, but to provide a compact language for describing internal geometry in a way that is mathematically explicit and empirically testable. To obtain a concrete and nontrivial testbed, we instantiate AIP in prime threshold graphs, where interactions are controlled by a distance parameter Δ and clique structures can be enumerated exactly. In this setting we observe stable power-law behaviour of the form α_k(Δ) ≈ C_k Δ^β across multiple clique sizes and scales, with a negative exponent that remains strikingly stable. The present version therefore emphasizes the framework and its controlled prime-graph evidence, while treating broader links to SAT, CSPs, and complexity theory as future directions rather than as established consequences.