The Greene-Kleitman shadow and the two-spine lift of covering growth

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Every finite poset carries a Greene-Kleitman shape. Deleting one maximal element at a time produces a standard Young tableau, because each deletion removes exactly one Ferrers box. The resulting map is a shadow, not a retraction onto the Young sector, and not a selector of that sector by the covering kernel. The Plancherel growth law on Young-sector covering histories equals the law of the Robinson-Schensted-Knuth recording tableau of N i.i.d. uniform points in a null rectangle, inserted in u-order. A labeled 1+1 sprinkling corresponds bijectively to a pair of standard Young tableaux of one common shape: given the shape, the two tableaux are independent and uniform, and one does not determine the other. In any causal sub-diamond, the longest chain equals the first row of the recording shape and, over square root of the number of points, tends to 2 independently of aspect ratio. Companion 40, Theorem 10, remains the correct statement that the graded box order fails that test. A history-independent law on Young's graph whose first row and first column are both o(n) is Plancherel. Dilworth width of Minkowski sprinkling in a causal diamond of dimension d at least 2 is of order N to the (d-1)/d in probability, so the Greene shadow of a d at least 3 sprinkling is not history-independent. That corollary is unconditional. It is not a no-go for 3+1. In a labeled 1+1 sprinkling the causal past of a point determines both Robinson-Schensted-Knuth spines of that past; the complement is not the missing global insertion tableau. The shadow is seed-level covering-to-covering. Its composite with a same-shape lift factors through the Young sector and does not take the covering-law family as a set. Companion 41, Theorem 3 and Proposition 5, are unchanged. 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 and does not assert that covering structures cannot produce geometry. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-41. It is not a remint of Companion 36.