The KFX Patriarch: A Structural Classification of Residual Unreachability

This work presents a systematic analysis of a class of finite, fully deterministic discrete dynamical systems that exhibit structural residual unreachability under coarse-grained observation. Focusing on a minimal family of models, we derive necessary and sufficient conditions for residual reachability by explicitly analyzing the arithmetic coupling between modular phase evolution, finite observation windows, and parity constraints. Unlike explanations that rely on randomness, noise, or asymptotic limits, the results demonstrate that residual unreachability can arise as a purely structural phenomenon in noise-free, finite systems. The proposed “patriarch-level” formulation compresses an entire family of counterexamples into a small set of congruence relations and interval feasibility conditions, providing a clear and transferable framework for understanding coarse-grained unreachability in deterministic systems.