Counting backwards from infinity: contraction geometry of Collatz-type maps

We study iterated non-injective maps on residue classes modulo 2M. Two map-intrinsic quantities capture the relevant structure: the contraction defect D =|domain|/|image| and the drift γ (average growth per step, defined via the v2-sum).We prove γ = q/4 for all single-step maps n 7→odd(qn+ c) with odd q ≥3, indepen-dent of c, and DM ≥4/3 uniformly for all M ≥3, for both the single-step and thecycle Collatz map. Combined with an explicit local-global conjecture (the Contrac-tion Principle), (D,γ) predicts convergence or divergence; applied to twelve mapsin the qn+ c family and four piecewise maps near the critical threshold γ = 1, everyprediction is consistent with the operationally observed outcome. We also provea deterministic countdown theorem for the embedding depth ed(n) = v2(n + 1),decomposing every orbit into predictable countdown segments and reset events.