Coincidence of critical thresholds for Collatz-type maps: why the divisor prime must be two

We study the Collatz-type map Tₐ,ᵦ,q(n) = (an+b)/qᵛᑫ⁽ᵃⁿ⁺ᵇ⁾ through two numerical invariants of fundamentally different character. The first is the Archimedean mean logarithmic drift δ = log a − E[v_q(an+b)] log q, which governs whether typical orbits grow or shrink. The second is the non-Archimedean similarity dimension dimₛ = H(ν_q)/log a of the invariant measure of an associated iterated function system on the a-adic integers. We prove that the critical multipliers determined by these two invariants coincide if and only if q = 2, while in general they satisfy the exact relation a_drift = (q − 1)a_dim. The proof rests on the pointwise identity −log ν_q(m) = m log q − log(q − 1), valid for every m ≥ 1, which expresses an exact correspondence between the information content of a q-adic valuation and its Archimedean cost. We further show that the condition q = 2 is equivalent to three additional properties of the family: the triviality of (ℤ/qℤ)*, the vanishing of an associated free energy, and the map being everywhere defined on the units. Thus, the arithmetic feature distinguishing the classical 3x + 1 case appears in four equivalent forms. We derive a closed form for the full Rényi spectrum of the invariant measure, with the similarity dimension occurring as its value at t = 1. For q ≥ 3, we identify a nonempty band of multipliers for which the map contracts on average while its invariant measure is singular. We also show that the associated transfer operator, although quasi-compact on Hölder spaces, has no eigenvalue other than 1; its remaining spectrum is entirely essential and occurs at the contraction rate. In addition, we prove a purity theorem: the invariant measure μₐ,ᵦ,q is either purely absolutely continuous or purely singular with respect to Haar measure, never a nontrivial mixture. Finally, among all integer pairs a ≥ 2, with q prime and gcd(a,q) = 1, we obtain a complete classification of the supercritical regime a < a_dim: exactly two pairs, (a,q) = (3,2) and (a,q) = (2,3), satisfy this condition; every other admissible pair has an unconditionally singular invariant measure. The first exceptional pair is the classical Collatz map. None of these results resolves the Collatz conjecture; we explain precisely why the invariants and dynamical structures studied here cannot establish the required global statement about every individual orbit.