Gao's dyadic density-one coalescence conjecture

We give a short proof of the density-one assertion posed by Gao for consecutive Collatz starts, independent of the first public proof, due to Said Duran (September 2026). Let C be the ordinary map and let H(n, m) count halving branches among the first m ordinary steps. Write Gk(n) when n and n+1 have, by time k, a common endpoint, equal halving counts, and have stayed above one up to that endpoint. If d̄k = 2−k·#{1 ≤ n < 2k − 1 : Gk(n)}, then d̄k → 1. A uniform absorption estimate for the 2-adic pair process gives the quantitative estimate 1 − d̄k = O(k−1/2), and the exact leading constant is determined in a companion paper. The argument concerns synchronization of two paths; it proves neither termination of every path nor the Collatz conjecture. Companion papers: Exact rates in Collatz synchronization; Quantitative synchronization of Collatz trajectories; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.