The Shift Law across Dimensions: Simplex Cones, the Axis Map, and Why Time Stays in Three Dimensions (INDYNA Research Note 21)

The Cone Spectral Theorem H=β(A-c 1) (Note 17) is shown to be dimension-independent: it holds with edge hinges (3D), triangle hinges (4D: 5-cell, 16-cell, 600-cell and its grand shadow, Note 20) and tetrahedral hinges (5D: the 5-simplex cone, five translational zero modes), and, with an exact umbrella sector, for non-simplicial cells (4D: 24-cell, H=A1-4, 24 umbrella modes). The icosahedral group A5 acts faithfully on the six five-fold axes of the icosahedron and on the five tetrahedra inscribed in the dodecahedron; the permutation representations split as 6=1⊕5 and 5=1⊕4 with 5 and 4 irreducible, so that the irreducible representations 3,3',4,5 of A5 are precisely the coordinate spaces of the icosahedron, the great icosahedron, the 4-simplex and the 5-simplex. The antipodal axis map T (12→6) intertwines the icosahedral shells with the 5-simplex exactly, TA1T⊤=TA2T⊤ =J-1, hence TM(g)T⊤=(1+g)(J-1): through the axis the two Galois shells coincide, the visibility factor (1+g) passes, the clock factor (1-g) and the twin families 3⊕3' die (kernel of T), while the density channels 1⊕5 survive. Analogously the dodecahedron maps onto the 4-simplex through a chiral compound of five tetrahedra. Time, in this model, is what does not pass through the axis. A back-reaction λ T⊤T shifts the golden rung linearly, g*(λ)=(√5-1+λ)/(√5+1), without changing the twin clock rate. The golden series H2⊂ H3⊂ H4 ends in four dimensions and returns in eight as E8. Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.