A Branchless Algebraic Framework for Collatz Trajectories

This study presents a branchless algebraic framework designed to organize and accelerate trajectory evaluations in Collatz (3n + 1) simulations by reducing intermediate conditional parity checks. The framework decomposes the 3n + 1 operation into its parity components and introduces a modular G1/G2 classification together with a general shortcut operator. For an odd integer n, the parameter k denotes the number of consecutive divisions by 2 following the 3n + 1 operation. The corresponding shortcut operator is f(n, k) = 9n + 3 + 2k The framework is organized through four successive layers: A1 (Base Reduction), A2 (Invariant and Modular Signature), A3 (Meta-Optimization), and A4 (Macro-Distribution). A1 introduces the intrinsic increment and algebraic shortcut. A2 uses modular structure to organize the shortcut parameter. A3 classifies behavioral trajectories according to structural layer depth. A4 introduces an empirical macro-distribution heuristic. The framework is presented as a computational model for large-scale trajectory experimentation. The A4 relationship is treated as an empirical heuristic rather than a universal conservation law, and the framework does not constitute a complete analytical proof of the Collatz conjecture.