Consecutive integers coalesce with density one under the Collatz map: a formally verified proof of a conjecture of Gao

Let C be the Collatz map, C(n) = n/2 for even n and C(n) = 3n+1 for odd n. Following Gao (1993), two consecutive integers n and n+1 coalesce if C^k(n) = C^k(n+1) for some k, with the same number of halvings on both trajectories and without either trajectory passing through 1. Gao computed the normalized count d̄_k of the n < 2^k − 1 for which this happens within k steps, proved that d̄_k is nondecreasing, and conjectured that d̄_k → 1. In the form recorded by Lagarias, the conjecture states that the set of n with C^k(n) = C^k(n+1) for some k ≤ log_2 n has natural density one. We prove both statements, and also that the proportion d(x) of n < x for which n and n+1 coalesce tends to 1; Gao did not know whether this limit exists. The proof follows a pair of trajectories through a Markov chain on states {v, 3^k v + c} driven by fair coin flips, which describes the residue classes modulo 2^t exactly, and shows that the chain is absorbed almost surely. The ingredients are a contraction of the square root of |c|/3^k, a harmonic function for the level k, a clock controlling long runs of equal parity, a window in which the offset returns to a bounded range, a universal absorbing word of length O(log^2(2+|c|)), and a counting argument for repeated trials that never conditions on the future. A short lemma shows that a meeting within log_2 n steps is already a coalescence in Gao's sense. Every step is formalized in Lean 4 with Mathlib, and the final theorems depend only on the axioms propext, Classical.choice and Quot.sound. The results concern coalescence only: they do not show that any integer reaches 1, and the convergence is extremely slow.