The Four-Dimensional Home and the Three Tesseracts (INDYNA Research Note 41)

We carry the INDYNA vertex action from three to four dimensions without changing it. On the flag doubles of the tesseract, the 24-cell and the 600-cell the area-weighted, spatially shadowed Regge action has a home (a stationary radius field) in all three bodies, and, as in 3D, the golden house lies deepest. Its static Hessian has exactly one positive direction, the breathing. Five exact statements organise the four-dimensional picture: (i) a parity theorem—the four-dimensional exchange pattern xyzw is even under the antipodal map, so the static drift–form block vanishes identically; (ii) the three observer tesseracts of the 24-cell are the three roles of one isoclinically rotating 24-cell, and three isoclinic steps are exactly the inversion u↦−u; (iii) the flat interior rotates as a pure change of coordinates, while the angular shell carries rotational inertia whose shadow/visible ratio is exactly φ⁴; (iv) the theorem of the counter-rotating source—only left and right rotation together reach the form sector (spin 2), one sense of rotation reaches only the breathing, and the drift is never reached; (v) the breathing of the observer bodies is a complex conjugate pair (tesseract 1.32877±0.11230 i, 24-cell 1.22910±0.09651 i), while the golden 600-cell breathes with a real frequency. We state the full non-abelian Routh reduction for the rotating world and report its present status: no holding rate has been found yet. The three-dimensional counterparts (exchange modes of all Platonic houses, the five cubes of the dodecahedron) are included.