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

Abstract

The Cone Spectral Theorem $H=\beta(A-c\,\mathbb1)$ (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, $H=A_1-4$, $24$ umbrella modes). The icosahedral group $A_5$ 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 $6=1\oplus5$ and $5=1\oplus4$ with $5$ and $4$ irreducible, so that the irreducible representations $3,3',4,5$ of $A_5$ are precisely the coordinate spaces of the icosahedron, the great icosahedron, the $4$-simplex and the $5$-simplex. The antipodal axis map $T$ ($12\to6$) intertwines the icosahedral shells with the $5$-simplex exactly, $TA_1T^{\!\top}=TA_2T^{\!\top} =J-\mathbb1$, hence $TM(g)T^{\!\top}=(1+g)(J-\mathbb1)$: through the axis the two Galois shells coincide, the visibility factor $(1+g)$ passes, the clock factor $(1-g)$ and the twin families $3\oplus3'$ die (kernel of $T$), while the density channels $1\oplus5$ 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 $\lambda\,T^{\!\top}T$ shifts the golden rung linearly, $g^\ast(\lambda)=(\sqrt5-1+\lambda)/(\sqrt5+1)$, without changing the twin clock rate. The golden series $H_2\subset H_3\subset H_4$ ends in four dimensions and returns in eight as $E_8$. 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.