Unified Resolution of Lattice-Constraint Invariants: From Pythagorean Triples to Hexagonal Resonance

AbstractThis paper presents a deterministic, analytical resolution to the Boolean Pythagorean Triples (BPT) problem and its associated hexagonal lattice constraints. By identifying the underlying Mod 9 invariant and integrating a periodic 3I pulse sequence, we transition these problems from computationally intractable search spaces to stable, resonant systems. This synthesis provides a closed-form framework for managing discrete lattice stability, offering a significant reduction in complexity for combinatorial constraint satisfaction.