The Δₖ Automaton: A Cycle Exclusion Proof via Explicit Q₀

This paper establishes the complete exclusion of non-trivial cycles in the Collatz map.

The proof combines two key ingredients:

1. An exponential upper bound (“Skeleton inequality”), derived from the telescoping structure of the Collatz orbit.
2. A polynomial lower bound, obtained via Baker–Matveev theory on linear forms in logarithms.

The contradiction between these bounds produces a finite cutoff value Q_0, beyond which no cycles can exist.

Since the finite range k < Q_0 has already been exhaustively verified computationally, the result proves that no non-trivial cycles exist.

Equivalently, in the Δₖ Automaton interpretation, the cumulative jump exponents S(k) and the logarithmic deviation \Lambda(k) demonstrate intrinsic irreversibility of the dynamics, forbidding periodic states.

This extended version provides full derivations, constant comparisons, and appendices summarizing computational verifications and robustness of the proof