Weinberg Angle from Dζ Channel Structure

Dζ decomposes into SY (symmetric) and AF (antisymmetric) channels: SY_coeff = 2+ζ₆-ζ₆²-2ζ₆³-ζ₆⁴+ζ₆⁵ = 6 (real) AF_coeff = ζ₆+ζ₆²-ζ₆⁴-ζ₆⁵ = 2i√3 (pure imaginary)

normSq: |6a + AF·b|² = 12(3a² + b²). Channel weights = |coeff|²/12: SY = 36/12 = 3, AF = 12/12 = 1. sin²θ_W = AF/(SY+AF) = 1/4.

normSq decomposition and weight extraction

|6a + AF·b|² = 12(3a² + b²). The normSq of each channel coefficient determines its weight after normalization by GCD = 12.

theorem FUST.WeinbergAngle.normSq_decomposition (a b : ℝ) : Complex.normSq (6 * ↑a + DζOperator.AF_coeff * ↑b) = 12 * (3 * a ^ 2 + b ^ 2)

theorem FUST.WeinbergAngle.SY_normSq : Complex.normSq 6 = 36

theorem FUST.WeinbergAngle.SY_weight_from_normSq : Complex.normSq 6 / Complex.normSq DζOperator.AF_coeff = 3

SY weight = |SY_coeff|² / |AF_coeff|² = 36/12 = 3

theorem FUST.WeinbergAngle.AF_weight_from_normSq : Complex.normSq DζOperator.AF_coeff / Complex.normSq DζOperator.AF_coeff = 1

AF weight = |AF_coeff|² / |AF_coeff|² = 1

Complete derivation chain

theorem FUST.WeinbergAngle.weinberg_from_Dζ : 2 + ζ₆ - ζ₆ ^ 2 - 2 * ζ₆ ^ 3 - ζ₆ ^ 4 + ζ₆ ^ 5 = 6 ∧ DζOperator.AF_coeff.re = 0 ∧ DζOperator.AF_coeff ≠ 0 ∧ (∀ (a b : ℝ), Complex.normSq (6 * ↑a + DζOperator.AF_coeff * ↑b) = 12 * (3 * a ^ 2 + b ^ 2)) ∧ Complex.normSq 6 / Complex.normSq DζOperator.AF_coeff = 3