A Massless Information-Entropy Framework for Modeling Collatz Trajectories via Algorithmic Bit-Scrambling and Diophantine Constraints.

The Collatz Conjecture posits that the arithmetic iteration T(n) = n/2 for even n, and T(n) = (3n+1)/2 for odd n, maps any positive integer to the trivial limit cycle {4, 2, 1}. Traditional approaches treating n purely as a quantitative magnitude have failed to produce a global proof. This paper introduces a Massless Information-Entropy Framework, redefining integers as finite binary strings undergoing local data manipulation. We analyze the process as a deterministic finite state machine subject to pseudo-random bit scrambling. By mapping the transition dynamics to 2-adic algebraic structures, we demonstrate that non-trivial finite loops are strictly constrained by Baker’s Theorem on Linear Forms in Logarithms. Furthermore, we apply Shannon entropy bounds to demonstrate that the +1 perturbation acts as an information-randomizing agent, driving long-term geometric decay and precluding stable divergence to infinity.