The companion paper 1 compresses the Collatz dynamics into a self-map F on states (ω,d), with forward per-step arithmetic governed by a 2-adic anchor. Here we run the map backward, in three movements. First, the inverse structure: the predecessors of a state are parameterized completely by its representatives (doors) and one free exponent s, and the backward depth obeys the exact law d = 1 + v3(s −M3(y)), where the backward anchor M3(y) is an affine discrete logarithm base 2 in the 3-adic exponent group — the corresponding 3-adic mirror of the forward law s = 2 + v2(d −M(ω)), with the primes 2 and 3 and the coordinates s and d exchanged. The forward per-step theory admits a systematic 3-adic mirror in this sense, with four structural asymmetries whose causes are identified.Second, applications of the inverse parameterization: backward generation dies only through a single mortal door per state, dead on exactly two of the four admissible residue–parity classes; the states with no predecessors at all are precisely the depth-one states whose unique door is divisible by 3; and steering laws show that backward branching controls the 3-adic data completely, freezes the 2-adic residues, and through the freeze places the predecessor’s forward anchor on an exact lattice to any prescribed precision, so that the anchor walk, unsolved in the forward direction, is controllable in reverse. Third, a density application: a deliberately impoverished sub-tree yields a self-contained proof that at least 2−3.6X0.3 odd integers below X reach 1. The bound is far weaker than the linear-programming results of Krasikov–Lagarias; its contribution is a derivation in which the local branching law is exact rather than bounded by inequalities.
Benjamin James Macindoe (Fri,) studied this question.