Isotropy of the 24-Cell Hinge System and the Einstein–Cartan Sector of the Hurwitz Lattice

The Hurwitz quaternion lattice ΓH (the F4 root lattice) tessellates R4 by its Voronoi cells, theregular 24-cells, in the self-dual honeycomb {3, 4, 3, 3}. In the present programme the discretelattice is taken as fundamental and the continuum as a coarse-grained approximation; earlierwork in this spirit showed that this honeycomb fixes the Yang–Mills kinetic normalisation andthe Regge form of the Einstein–Hilbert action with no free coefficients. Here we treat the torsionsector of Einstein–Cartan gravity, which is forced once spin- 12 matter is present, entirely on thelattice: every quantity is defined from the hinge holonomy of a constant axial contorsion andfrom the K¨ahler–Dirac hopping operator, with no continuum background imported at any stage.We establish four lattice statements, all reducing to the single four-dimensional Levi-Civitacontraction εmabσεmabc = 6 δσc. (A) The torsion-squared curvature scalar built from the con-torsion is Rtors = cSS2 with cS = −32exactly (symbolic), and the 96 hinge-planes of the origin24-cell form an exact 2-design on the Grassmannian Gr(2P , 4): for every antisymmetric tensor F,h⟨F, eˆh ⊗ eˆh⟩ = 16 Fabab. The design property is the statement that the torsion contributionis exactly isotropic on the lattice — the {3, 4, 3, 3} honeycomb introduces no preferred direc-tion into the matter–torsion coupling. (B) The fermion–torsion vertex from the covariantisedhopping operator satisfies Pµ̸=σ Ckinσµ =34L(γ5γσ) exactly, giving the Dirac–torsion coefficientcD =34; the Wilson (“TT”) completion is exactly taste-twisted ({CTT, ξ5} = 0) and infrared-irrelevant. (C) The lattice-to-continuum current identification is fixed, without constructing theintertwiner explicitly, by a basis-independent free-field bubble ratio, c2 = −4 (over-determined16:1), whose value c2 = ε 2d/2is representation-theoretic — the Cl(4) taste-spinor dimension2d/2 = 4 times the pseudo-real reality sign ε = −1 at d mod 8 = 4; the induced four-fermioncoupling is λa = −332 (a−1/M¯P )2, so its sign is determined, s = −1. (D) Brillouin-zone aver-aging of the same vertex suppresses the contact estimate by a geometric factor 16, leaving thePlanck-scale lattice vacuum subcritical in both reference channels. The coefficients cS, cD andthe induced c2D/(2|cS|) = 316 are properties of four-dimensional antisymmetric torsion; what isspecific to ΓH is the exact hinge isotropy, the protected K¨ahler–Dirac taste structure, and theresulting current normalisation and induced-coupling sign — all fixed with no free parameter.Coarse-graining reproduces the standard Einstein–Cartan–Sciama–Kibble coefficient 316 ; thiscontinuum agreement is reported as a consistency check, not as the basis of the lattice results.