Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

We prove that, in ordinary natural density, almost every positive integer has shortcut and unaccelerated Collatz iterates reaching the same value below (log n)^A_FP U(n) for every prescribed positive function U(n) tending to infinity. The shortcut and raw witnessing times are centered respectively at c_* log n and 3c_* log n / 2, with error O(sqrt(log n log log n)). For every prescribed positive R(n) tending to infinity, both trajectories remain below n R(n) through the common witness.

For every fixed A>A_FP, the complete target-clock-height conclusion has exceptional count O(X/(log X)^(kappa_* A-1/2)). With the balanced relative height ceiling, the same order is forced from below by the height condition; this is a joint lower bound, not a target-only optimality result. A stretched-logarithmic companion gives stretched-exponential exceptional decay with a common relative-height allowance of the same stretched-logarithmic order.

The proof combines exact parity-word counting, deterministic first-passage transport along decreasing thresholds, an O(sqrt(M log M)) cumulative-time corridor, and a finite ballot estimate giving timeout mass O(m^(-3/2) 2^(-kappa_* m)). The pure target C(log n)^A_FP with bounded multiplier remains unproved. All conclusions are almost-all statements and do not prove pointwise Collatz convergence, exclude exceptional cycles or divergent trajectories, or control the orbit after the selected witness.