Focal Antipedal Centroids of Rectangles Circumscribed about an Ellipse

Let a rectangle be circumscribed about a noncircular nondegenerate ellipse with center O and a focus F. We show that its antipedal quadrilateral with respect to F has vertex centroid O and area centroid O+(F-O)/3, independently of the rectangle's orientation. The quadrilateral is always finite and strictly convex. Applied to the known outer tangent rectangles of simple four-periodic elliptic billiards with a confocal elliptic caustic, this calculation proves both constancy assertions in the experimental invariant k406,b of Reznik, Garcia and Koiller and identifies their values. The proof uses an orthonormal support frame and exact polygon first moments; it does not assert the corresponding general-even-period invariant k407. Source record: AMR-050-0025 (raw ID 5100025, k406,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or formalization is claimed. The source and novelty review credits the known four-periodic outer-rectangle lemma and makes no absolute priority claim. The source ZIP includes the English LaTeX manuscript and a portable standard-library exact rational checker.