A full-support central measure on the covering growth graph exists, and seed-rigid-ray mixtures do not supply it

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Write Gamma for the covering growth graph whose vertices are unlabelled covering classes. A central measure is the Kolmogorov measure on infinite rooted paths associated with a coherent function on Gamma. Full support means every finite class carries strictly positive mass. Not every finite class lies on a rigid ray from the seed: the unique predecessor-count-two class at size 4 is already off every such ray, so countable mixtures of seed-rigid-ray extremes vanish there. Every finite class has a unique-maximal descendant by iterated two-parent covering of pairs of maximals. Iterated one-parent covering on that unique maximal is an infinite rigid tail. The associated stem-and-tail functions mix to a strictly positive coherent function. The set of full-support central measures is therefore nonempty, is not a singleton, and is not classified. The mixture is not extreme. Residual mass along a rigid ray peels a non-minimal full-support function. Existence of an extreme full-support central measure remains not found. The construction is not the class-Markov locally-positively-supported pair of Companion 27, Theorem 16, and is not the finite-n path-entropy measure of Companion 37, Theorem 11. Companion 41 cites the mixture as a deposited fact and does not restate the theorem. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. This note does not claim Lorentz invariance. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-46.