A computational verification of the Markov Uniqueness Conjecture for z ≤ 10^500000

We describe a finite computational verification of Markov uniqueness for normalized positive integer solutions of x^2 + y^2 + z^2 = 3xyz with z ≤ 10^{500000}. The archived run reports 239 536 376 556 triples with distinct maximal coordinates. The method combines a disjoint tree decomposition, certified fixed-point range bounds, a deterministic 48-bit candidate screen,and replay modulo 2^{128}. We give the mathematical justification and a reproducibility procedure covering package checks,retained-data auditing, and a full rerun. The conclusion depends on correct execution of the exhaustive screen and replay; the retained audit does not regenerate the deleted screening buckets.