Algebraic Obstructions to Residue-Only Lyapunov Certificates for the Shortcut Collatz Map

Abstract

This paper studies residue-only Lyapunov potentials of the form V(n) = log_2 n + c(n mod 2^m) for the shortcut Collatz map. The main result is a fixed-depth obstruction theorem: for every fixed residue modulus 2^m, every finite block length B, and every correction function c, there exist arbitrarily large integers n for which the potential strictly increases along all of the first B shortcut iterates. The obstruction arises from integers congruent to -1 modulo 2^(m+B), whose first B shortcut steps are all odd, leaving the residue correction unchanged while the logarithmic component increases.

The paper also presents complementary algebraic and computational diagnostics, including a positive-drift representative cycle modulo 128, an exact lifted all-ones self-loop, representative and lifted linear-programming scans, partial residue-tree data, and remarks on QUBO and GPU-based search infrastructure. These diagnostics illustrate why low-bit residue models can falsely suggest feasible Lyapunov corrections when higher 2-adic dependence is ignored.

The results do not prove or disprove the Collatz conjecture. Rather, they identify a precise structural limitation of a natural finite-dimensional Lyapunov ansatz and provide a reproducible framework for testing richer certificate templates.