Placement Theorems for the Gauge Sector: Necessity of SU(3)×SU(2)×U(1)

We establish by exhaustion the placement of the Standard Model gauge sectors on the eigenspaces of the truncated octahedron's face Laplacian. Theorem 58.1 (Electroweak Placement): the Eg doublet at eigenvalue 4 is the unique candidate for the electroweak sector: it is the only even-parity doublet, it is annihilated by the torsion operators (no FCNC channel), and it carries 100% square-face content. Theorem 58.2 (Colour Placement): the T₂g triplet at eigenvalue 7 is the unique candidate for the colour sector: it is the only even triplet, it is torsion-active, and its face content is exactly 100% hexagonal (the square subspace decomposes as A₁g ⊕ Eg ⊕ T₁u only, so T₂g cannot occur on squares), the geometric reading of confinement. Together with the fermion and Higgs placements of Paper #57, all six eigenspaces are uniquely assigned, forcing the group content SU(3)×SU(2)×U(1). Result 58.3 (Weinberg Angle, Tier 2): the on-shell effective mixing angle formula sin²θ_W = (17 − 3√17)/20 = 0.23153 matches the LEP effective value at 0.00σ and satisfies the exact structural identity cos²θ_W = 2C_A√Δ·s₁/(Δ + C_A), where s₁ is the T₁u(r₁) square-face content; the derivation of the mixing weight itself remains open, and Result 58.3 is stated as a derived formula with an open derivation, not a theorem.Changes in version 2.0: Theorem 58.3 is demoted to Result 58.3 (Tier 2): the v1.0 proof contained an abandoned intermediate derivation and a bracketed placeholder, both removed; the exact structural identity is stated and the missing normalisation step is named as an open problem, with the formula and its 0.00σ LEP match unchanged. The "3/13" section title is corrected (stale draft value). Torsion columns now name their operators (T_hex vs inter-type T, per Paper #57 v2.0) and the ±2i band entries are corrected to equal-weight superpositions. The MS-bar comparison is corrected to 7.75σ. A selection-criteria honesty note is added to Section 2. T₂g face content is corrected to exactly 100% hexagonal, strengthening both placement theorems. All corrections are machine-verified in verify_FtF_audit_2026-07.py and verify_Weinberg_obstruction_2026-07.py in the repository.