The Triviality Pattern of Compensation-Avoiding Towers in Collatz Dynamics: A Profinite-Symbolic Obstruction Framework

Any Collatz counterexample must sustain a compensation-avoiding orbit where v2 (3Tʲ (n) +1) ∈ 1, 2 indefinitely. We investigate such towers through two exhaustive tests. Finite Residue Exhaustion: For every M ≤ 21, exhaustive enumeration of all 2^ (M-1) odd residues modulo 2M shows UM = 1. Every odd residue except r ≡ 1 receives compensation within 53 steps. Periodic Tower Obstruction: For all primitive words w ∈ 1, 2ᵖ with 2 ≤ p ≤ 21, the divisibility condition 2A (w) - 3ᵖ | Δ (w) fails. The only solution is w = (2), x = 1. Any remaining counterexample must be a nonperiodic critical balanced 1, 2-tower satisfying global 2-adic compatibility — a highly rigid structure for which no candidate has been observed. Paper D of a four-part series. Paper A: Moon, K. -U. (2026). Zenodo. https: //doi. org/10. 5281/zenodo. 20068553Paper B: Moon, K. -U. (2026). Zenodo. https: //doi. org/10. 5281/zenodo. 20068640Paper C: Moon, K. -U. (2026). Zenodo. https: //doi. org/10. 5281/zenodo. 20068757