Magnetism, Gravity and Inversion Breaking in Four Dimensions (INDYNA Research Note 42)

In the INDYNA vertex action the drift sector (spin 1, magnetism) and the form sector (spin 2, gravitation) of a radius field couple statically only through a lock-in deformation. In three dimensions the cube lock-in xyz provides this coupling. We prove that in four dimensions every radius field that is even under the antipodal map u↦−u gives exactly zero coupling, and identify u↦−u with the inversion of the observer (three isoclinic Hurwitz steps). Among the regular 4D polytopes only the 5-cell is not centrally symmetric. We show that the tesseract and the 600-cell contain no regular 5-cell, while the 120-cell contains 840 of them in two symmetry orbits (120 and 720): observer and inversion-breaking pair sit on the two sides of the golden dual pair. The canonical odd invariant of a simplex, P(u)=Σₖ (u·vₖ)³, reproduces the three-dimensional lock-in exactly (24 xyz for the tetrahedron) and yields the first non-vanishing four-dimensional magnetism–gravity coupling: initial slope 0.03394920 in both orbits, κ = 0.07318514 and 0.07533183 at the crossing points. We also correct the three-dimensional coupling at the true crossing point (κ=0.2138083; the earlier register value 0.1748 mixed two amplitude conventions). None of the rules for α fixed so far reproduces the fine-structure constant; α remains open, and all attempts are listed.