The Necessity of the Standard Model: Placement Theorems for the Higgs and Fermion Chirality

We prove that the Standard Model particle content is not merely consistent with the geometry of the truncated octahedron (Kelvin cell): it is the only possible outcome. Two placement theorems are established. The framework uses two distinct torsion operators, named explicitly in this version: the scalar torsion charge is the eigenvalue under T_hex = (1/3)·A_hh, the degree-normalised adjacency of the hexagonal subgraph (the cube graph); the chiral structure lives in the inter-type operator T = P_sq·L·P_hex − P_hex·L·P_sq. Theorem 57.1 (Higgs Necessity): of the two scalar (spin-0) irreps of O_h, namely A₁g and A₂u, only A₂u can drive spontaneous symmetry breaking. Under T_hex, A₁g carries charge +1 (positive effective mass², stable vacuum) and A₂u carries charge −1 (the bipartite minimum of the cube graph: negative effective mass², origin unstable). SSB is forced for A₂u and forbidden for A₁g. The Higgs is A₂u by exhaustion of alternatives. Theorem 57.2 (Chiral Structure): from T²|T₁u = −4·I (Theorem 56.1), the inter-type operator T has purely imaginary eigenvalues ±2i on the T₁u subspace, and T is purely cross-block between the two T₁u bands: chirality eigenstates are equal-weight superpositions of the r₁ and r₂ bands, exactly as γ⁵ eigenstates superpose mass eigenstates in Dirac theory. The labelling T₁u(r₁) = left-handed, T₁u(r₂) = right-handed is fixed by the B+V=D orientation of T together with the Eg-coupling argument (T₁u(r₁) carries square-face content (1+1/√17)/2 ≈ 62%, hence the stronger coupling to the pure-square electroweak Eg sector). The chiral structure is a theorem; the band labelling is a derived identification conditional on that coupling argument.Changes in version 2.0: the two torsion operators conflated in v1.0 are now named and separated (T_hex scalar charges vs the inter-type operator T); the premises of Theorem 57.1 are corrected (conclusion unchanged); the v1.0 claim that the +2i eigenvectors localise in the T₁u(r₁) band is withdrawn (they are exactly 50/50 across the two bands, and the band labelling is now a stated conditional identification); the Section 2 dimension table and Section 6 Weinberg angle line are corrected. All corrections are machine-verified in verify_FtF_audit_2026-07.py in the repository.