Steiner-Centroid Pedal Area Ratios in Elliptic Billiards

For an odd-periodic billiard in an ellipse with a fixed confocal elliptic caustic, let C be the Steiner curvature centroid: the weighted average of the vertices with weights sin(2 theta_i), where theta_i is the internal reflection angle. Let A be the signed area of the original chord polygon and B_C the signed area of its pedal polygon with respect to C. We prove that C is always defined, that B_C is nonzero, and that A/B_C is invariant throughout the family. The result covers every coprime odd winding and odd repeated traversal, with an explicit signed-star convention; circular billiards are treated separately. This resolves invariant k501 in Table 6 of Reznik, Garcia and Koiller's Fifty New Invariants of N-Periodics in the Elliptic Billiard. The proof combines the classical pedal-area quadratic with reflection trace identities, complementary elliptic-function divisors, a character argument on an odd quotient torus, and a strict odd-grid theta inequality. The quotient-torus argument also identifies the moving centroid's axis-aligned elliptic locus. Exact standard-library checks accompany the proof but are not used as a substitute for the general analytic argument. This is an unrefereed, AI-assisted preprint; no absolute priority, independent human review, or formal proof-assistant verification is asserted.