Collatz Analysis: Two-Stage Tree and Multiset Calculus Farhad Aliabdali Additionally, we develop a signed-multiset calculus on generators {g_j} that encodes binary arithmetic via local rewrite rules. We prove this system is terminating and confluent, yielding unique canonical binary normal forms. Within this calculus, we derive an explicit bit-complement formula for 2^D-3^k and reformulate the classical cycle equation in multiset language, enabling digit-by-digit analysis of cycle constraints. By applying a Multiset Calculus, we derive a polynomial obstruction showing that any cycle's algebraic structure is incompatible with positive-coefficient polynomial division. While this does not strictly rule out integer solutions due to carry propagation, computational verification suggests, we establish rigorous residue-class locking conditions (Theorem 7.2) that constrain the trajectory growth. Central to our findings are new proofs establishing structural obstructions to cycle formation: we prove the impossibility of cycles with monotone odd-growth phases (Theorem 19.10) and demonstrate that pure-even return paths are algebraically inconsistent with the required cycle denominators (Theorem 19.5). These results collectively define a new class of non-divisibility barriers (Theorem 19.2) that rule out broad categories of potential non-trivial cycles, providing a refined algebraic map of the conjecture’s remaining complexity. This work establishes a framework for Collatz analysis; it does not resolve the conjecture. The computational synthesis in Section 19 presents empirical observations and heuristic patterns that require further investigation.