Exact rates in Collatz synchronization

Gao conjectured that his dyadic proportions of coalescing adjacent integers tend to one. We determine the exact deficiency: 1 − d̄k ~ β/√k, with β = √6·E[M]/√π and 1 ≤ E[M] < ∞, where M counts the excursions of the canonical adjacent-pair coupling. The balanced shortcut absorption time has coefficient 2E[M]/√π; the first ordinary meeting time has coefficient β. The same β governs the fraction of integers n < N with no ordinary meeting by ⌊log2 n⌋. The proof passes from stationary rigidity to actual killed occupation and then determines the physical clock of a long excursion. A weighted chronological assembly lemma handles dependent excursions using only their expected total weight. This weight also yields the harmonic survival capacity from every integer canonical state and extends the coefficient to the exact admissible offset domain ℤ[1/3]. A uniform finite-horizon estimate gives synchronized growing-block versions of Korec's and Tao's descent theorems. No termination assumption is used. Companion papers: Late synchronization and the geometry of Collatz families; Quantitative synchronization of Collatz trajectories; Gao's dyadic density-one coalescence conjecture; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.