Monotone Loss of Symbolic Freedom in the Collatz Dynamics via HKD Piano Lanesv | IJCT Volume 13 – Issue 1 | IJCT-V13I1P5

We have introduced a structural invariant for Collatz dynamics—symbolic freedom defined via parity-block rank on HKD piano lanes—and shown that this invariant undergoes a monotone, irreversible loss under arithmetic refinement. The proof is elementary, relying only on set inclusion and linear algebra, yet it yields a rigidity mechanism absent from previous approaches. The central result is that for every refinement step m → 2m, the symbolic freedom of each refined lane is bounded above by that of its parent. Since symbolic freedom is finite, it must be exhausted after finitely many refinements. Once exhausted, parity evolution becomes rigid and enforces uniform contraction over fixed-length blocks, ruling out infinite Collatz trajectories. Computational verification on the refinements Z6 → Z12 → Z24 confirms the theoretical mono- tonicity with zero observed violations. A separate comparison demonstrates that HKD-based con- traction strictly dominates logarithmic drift methods in both average and worst-case behavior, without parameter tuning. The resulting picture is that Collatz dynamics are constrained not by typical behavior or prob- abilistic drift, but by a finite symbolic resource that is deterministically depleted. This perspective explains both the limitations of prior methods and the effectiveness of the HKD framework. Taken together, these results provide a deterministic obstruction to non-terminating Collatz trajectories.