Internal repetitions, drift and factor complexity in the 3x+1 conjugacy map

Let T be the 3x+1 map on the 2-adic integers ℤ₂, and let v(x) denote the parity vector of x. Lagarias' Periodicity Conjecture asserts that v(x) is eventually periodic for every rational x. Let x = p/q, with q odd, be rational with a parity vector that is not eventually periodic. Write Γ*(N) for the maximal expansion exponent of T over a segment of the first N steps, and p(n) for the factor complexity of v(x). Main result (drift–complexity trade-off). Γ*(p(n)) + log₂ p(n) ≥ n − O_x(1). It holds regardless of the sign of the orbit, the size of q and any density: a rational aperiodic orbit must pay either in symbolic complexity or in dynamical expansion. In particular h(1+γ) ≥ 1 for the entropy h and the logarithmic drift rate γ. Forced growth of the drift. Combined with a finite-window entropy estimate based on the identity E[2^X] = 1 for the drift increments, this yields Γ*(W) ≥ log₂log₂W − 2 log₂log₂log₂W − O_x(1), hence liminf Γ*(W) / log₂log₂W ≥ 1. The tool is an elementary inequality comparing two iterates of the same rational point, which share the denominator q. As a benchmark, an elementary harmonic estimate gives Γ*(W) ≥ (1 − q′/6) log₂W − O_x(1) for positive orbits, where q′ is the 3-free part of q; it is stronger for q′ ∈ {1, 5}, in particular for integers, but becomes vacuous for q′ ≥ 7, where the repetition argument still applies. Typical parity words have Γ*(W) ~ log₂W. These results are related to J. Allikvere's periodic-shadow obstruction (Zenodo, 25 July 2026), whose priority for that criterion is acknowledged in the paper. 7 pages. Ancillary exact-arithmetic verification scripts (Python 3, standard library) are included; nothing in the proofs depends on them. AI tools (Anthropic Claude, OpenAI ChatGPT) were used as declared in the paper; they are not authors.