The optimal exponential rate for products of simplices

This research note determines the optimal uniform exponential rate within the class of Cartesian products of full-dimensional simplices and their invertible affine images. The simplex rate ((d+1)d^d/d!)^(1/d) has its unique maximum over positive integers at d=13, giving beta approximately 2.8099647. The resulting normalized projection-body volume is at most beta^n, with equality exactly when every simplex factor has dimension 13. The geometric consequence uses the previously known simplex formula and product identity. The proof and supporting balancing and segment-absorption lemmas are formalized in Lean with pinned dependencies. This elementary addendum neither determines exact fixed-dimension maxima nor optimizes over all convex bodies. We did not locate this optimization in the sources reviewed; the search is bounded and priority is unconfirmed. This is an AI-assisted research note, not a peer-reviewed publication.