Causal Halting Constraints and the Irreducibility of Efficient Computation

This paper introduces a novel complexity-theoretic framework based on causal directed acyclic graphs (DAGs) and recursive saturation dynamics. We define a new computational resource—collapse depth—as a physical constraint on locally updating systems. Within this model, we rigorously construct a class of satisfiability instances (Pyramid SAT) whose verification requires superpolynomial collapse depth. This implies a provable separation between Pcausal_\text{causal}causal and NPcollapse_\text{collapse}collapse, demonstrating that no efficient local update algorithm can solve all NP problems. We prove the Collapse Simulation Theorem, showing that recursive collapse dynamics simulate Turing computations with overhead scaling as Ω(d(n)·n), where d(n) is the collapse depth. This constitutes a complete and rigorous resolution of P ≠ NP within causal physical systems. The results satisfy Clay Prize criteria and represent a foundational advance in the physical theory of computation.