Joint laws and local synchronization for Collatz minima and arrival times

Let m(n) be the least value on the ordinary Collatz orbit of n, and let τ(n) be its first-attainment time. We prove a spatial transfer theorem for the entire discrete pair (m(n), τ(n)). From uniform pair coalescence and a limiting minimum law, its empirical distribution becomes independent of the starting residue modulo every prescribed qX with log qX = o(log X). Simultaneously, almost every block of LX+1 consecutive starts has identical minima and exact first-attainment times when LX = o(√(log X)). The proof combines merger before a truncated minimum with an elementary random-translation coupling, avoiding a loss proportional to the modulus. Composing this transfer with the Gaussian law of the time to the minimum gives joint asymptotic independence of the minimum, the normalized clock and the residue, uniformly in the clock threshold and in aggregate over all minima and residues. The Gaussian block limit is diagonal. We also prove weighted versions and an explicit spatial rate from a uniform minimum-tail bound. The pair-coalescence estimate, the minimum law, the Gaussian law and the minimum tail are theorems of companion papers, imported with their precise statements, so the main theorems are unconditional. The same transfer gives growing-modulus laws for every label preserved by early merger, including the first hitting time of every fixed target. Companion papers: Brownian motion in Collatz orbits; Quantitative synchronization of Collatz trajectories; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.