Same Shadows, Different Forests: Local Predecessor Structure and Global Dynamics 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 this question for the maps \( x\mapsto(qx+c)/2 \) and give three complementary answers. First, every local statistic of the predecessor trees of \( 3x+1 \) (branching classes, Leftward and Upward children, cotree densities, densities of finite parity patterns) coincides with that of \( 3x-1 \), which is \( 3x+1 \) on the negative integers and has three cycles. Local branching statistics therefore determine neither the cycle structure nor basin membership. Second, cycles are characterised by an exact global condition: a parity word \( B \) of length \( l \) with \( k \) odd steps yields an integer cycle if and only if \( D=2^l-q^k \) divides \( c\,z_q(B) \). Reversing the perspective, we ask for which maps a given word yields a cycle. One pass over all \( 2.2\cdot10^{12} \) words of length at most 40 then gives the complete list of 1097 primitive cycles of all maps \( 3x+c \) with \( |c|\le1000 \). For 0 < \( c \) < 1000 this list agrees exactly with the conjecturally complete tables of Belaga, Mignotte and Morley (OEIS A226607) for all cycles with at most 40 elements. An equidistribution heuristic predicts 1149 cycles, and the observed counts scatter less than a Poisson model allows. Third, the sizes of the basins of attraction differ strongly and remain largely unexplained. A divisibility filtration determines all rational basin densities exactly. The empirical shares of the remaining cycles, measured up to 1010 for 18 maps, are stable to three digits, but their limits are not known to exist. In $3x+5$, two cycles of the same type attract about 50% and 9% of all integers.