Same Shadows, Different Forests: Cycles and Basin Densities of Collatz-Type Maps

What determines the global dynamics of a Collatz-type map when its local predecessor structure is indistinguishable from that of another map? We study the maps \( x\mapsto(qx+c)/2 \) and give three answers. First, every local statistic of the predecessor trees of 3\( x \) + 1, including cotree densities, coincides with that of 3\( x \) - 1, which has three cycles. Second, a parity word \( B \) of length \( l \) with \( k \) odd steps yields an integer cycle if and only if \( 2^l-q^k \) divides \( c\,z_q(B) \). One pass over all \( 2.2\cdot10^{12} \) words of length at most 40 lists all 1097 primitive cycles of the maps 3\( x \) + \( c \) with \( |c|\le1000 \) and confirms the published tables (OEIS A226607) up to this length. Third, we separate the existence of basin densities from their values. Every basin of every map 3\( x \) + \( c \) with odd \( c\ge-1 \) has a logarithmic density; in particular, the positive integers whose 3\( x \) + 1 orbit reaches 1 form a set with a logarithmic density. By recent results of Mazur, it is even a positive natural density. For the values, the coset of \( n \) modulo \( \langle2,3\rangle\le(\mathbb Z/c\mathbb Z)^\times \) is an invariant that fixes many densities exactly, and the harmonic mass of the backward tree of a cycle predicts 410 measured densities to about 5% without fitted parameters.