Shadow-flow rigidity of D_n polytopes and quartic descent at the 24-cell

For every n >= 4, we prove that the D_n root polytope and its polar have only globally linear odd admissible vertex speeds in every nonzero direction. Admissibility means affine compatibility on every facet transverse to that direction. For the regular 24-cell, every fixed nontrivial odd one-direction linear motion of its original 24 vertices strictly increases the volume product for all sufficiently small nonzero times, even when facets may split. The time neighborhood depends on the chosen direction and speed. Nevertheless, the relative rotation Q_theta = conv(B_1^4, (1/2) R_theta B_infinity^4), with R_theta rotating the first two coordinates, strictly decreases the volume product for 0 < theta <= pi/4. We compute both volumes exactly and obtain 16 - (10/9) theta^4 + O(theta^6) at zero. The note distinguishes this fixed original-vertex, one-direction obstruction from local minimality among all origin-symmetric convex bodies. It does not settle or refute Mahler's conjecture, improve any general lower bound, or rule out shadow systems with additional generating points. Version 1 contains the ten-page English preprint and its complete editable LaTeX source. AI systems, including OpenAI Codex, were used extensively in mathematical derivation, manuscript preparation, and internal analytical checking, as disclosed in the manuscript. Internal checks do not constitute independent human peer review, external peer review, or formal proof-assistant verification. Targeted literature checks do not certify global priority.