Finite-Window Survival Counts and Residue Refinement for the Odd Collatz Map

This expository and computational technical note develops an exact framework for counting positive odd integers in dyadic windows whose accelerated Collatz trajectories remain above a fixed threshold through a prescribed time. Each valuation word determines an arithmetic progression and a strict survival cutoff, yielding an exact decomposition of the survival proportion into a geometric reference term and a signed rounding error. Conditioning on a prefix expresses suffix survival through residue incidence. Coarse dyadic refinement then provides upper bounds whose excess is identified exactly within each bin. For a specified subfamily of bins, this excess is bounded by a heavy-residue charge. A decomposition into baseline and excess contributions identifies the additional estimates needed for a bound uniform in the window scale. Exact finite computations illustrate the framework, including seven certified horizons and two detailed charge decompositions. The note also explains why a suitable uniform fixed-target survival estimate would imply a natural-density conclusion. That estimate remains unproved. No uniform survival bound, natural-density improvement, or exclusion of divergent orbits is established.