Standard Model Q3 Representations

The Q₃ symmetry module

The quaternion group Q₃, also written Q₈, is the eight-element group {±1, ±i, ±j, ±k} with the multiplication rules i² = j² = k² = ijk = −1. It is the smallest non-abelian group after the symmetric group S₃, and it appears across mathematics as the symmetry group of the cube and in the unit quaternions. In the standard model of particle physics, the group organizes the four complex degrees of freedom of the Higgs doublet: three become the longitudinal polarizations of the W± and Z bosons, and one becomes the physical Higgs boson.

The central observation is that Q₃ splits into two sectors. The elements {±1} form a spin-0 sector, and the six elements {±i, ±j, ±k} form a spin-1 sector. The Casimir operator, which measures the total spin of a representation, takes the value 0 for the spin-0 sector and 2 for the spin-1 sector. This split mirrors the electroweak symmetry breaking pattern SU(2)×U(1) → U(1), where the Higgs doublet's four real components reduce to three Goldstone bosons and one physical scalar.

In Recognition Science, the group Q₃ appears as the symmetry group of the eight-tick cycle, the fundamental period of the recognition operator R̂. The framework models the electroweak sector by placing the W boson at rung 21 of a φ-ladder, where φ is the golden ratio, and the Z boson at a rung offset by log_φ(1/cos θ_W). The physical Higgs boson sits at a different rung due to the spin-0/spin-1 offset.

The module derives a prediction for the Higgs mass ratio. Using the framework's value for the quartic coupling λ = 1/2, the Higgs mass squared over the vacuum expectation value squared simplifies to v². The Higgs-to-W mass ratio then becomes 2/g, where g is the SU(2) gauge coupling. The module also defines a value for sin²θ_W = (3 − φ)/6, which falls between 0.228 and 0.232, and proves it is positive and less than 1/2.

The key result is a prediction for the Higgs rung position, computed as log_φ of a ratio involving sin²θ_W and the ratio v/m_W. The theorem higgs_rung_prediction_pos proves this prediction is positive. This is a concrete, testable consequence of the framework's structure: the Higgs mass is not a free parameter but is fixed by the geometry of the Q₃ symmetry and the golden ratio.

/-- The 8 elements of the quaternion group Q₃ = {±1, ±i, ±j, ±k}. -/
inductive Q3Element
  | pos_one  : Q3Element  -- +1
  | neg_one  : Q3Element  -- -1
  | pos_i    : Q3Element  -- +i
  | neg_i    : Q3Element  -- -i
  | pos_j    : Q3Element  -- +j
  | neg_j    : Q3Element  -- -j
  | pos_k    : Q3Element  -- +k
  | neg_k    : Q3Element  -- -k
  deriving DecidableEq, Repr

/-- The spin-0 sector has 2 elements. -/
theorem spin0_count : Spin0Sector.length = 2 := by decide

/-- The spin-1 sector has 6 elements. -/
theorem spin1_count : Spin1Sector.length = 6 := by decide

/-- Spin-0 Casimir eigenvalue: j=0, C₂ = 0. -/
theorem spin0_casimir : casimir 0 = 0 := by simp [casimir]

/-- Spin-1 Casimir eigenvalue: j=1, C₂ = 2. -/
theorem spin1_casimir : casimir 1 = 2 := by unfold casimir; norm_num

theorem higgsMassSq_simplifies (v : ℝ) :
    higgsMassSq_over_vev v = v^2 := by
  unfold higgsMassSq_over_vev lambda_RS; ring

/-- The RS prediction for sin²θ_W: ≈ (3-1.618)/6 ≈ 0.230. -/
theorem sin2ThetaW_RS_approx : 0.228 < sin2ThetaW_RS ∧ sin2ThetaW_RS < 0.232 := by
  unfold sin2ThetaW_RS
  constructor
  · rw [lt_div_iff₀ (by norm_num : (0:ℝ) < 6)]
    linarith [phi_lt_onePointSixTwo]
  · rw [div_lt_iff₀ (by norm_num : (0:ℝ) < 6)]
    linarith [phi_gt_onePointSixOne]

/-- sin²θ_W^RS is positive. -/
theorem sin2ThetaW_RS_pos : 0 < sin2ThetaW_RS := by
  unfold sin2ThetaW_RS
  apply div_pos
  · linarith [phi_lt_onePointSixTwo]
  · norm_num

/-- sin²θ_W^RS is less than 0.5. -/
theorem sin2ThetaW_RS_lt_half : sin2ThetaW_RS < 1/2 := by
  unfold sin2ThetaW_RS
  rw [div_lt_iff₀ (by norm_num : (0:ℝ) < 6)]
  linarith [phi_gt_onePointSixOne]

theorem higgs_rung_prediction_pos : 0 < higgs_rung_prediction := by
  unfold higgs_rung_prediction sin2ThetaW_RS
  apply div_pos
  · -- The numerator log is positive iff its argument > 1.
    -- The argument is 2*s2*(246/80.4)² / (1-2*s2) with s2 = (3-phi)/6.
    apply Real.log_pos
    have hphi : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
    have hphi2 : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
    have hs2_lo : (0.228 : ℝ) < (3 - phi) / 6 := by linarith
    have hs2_hi : (3 - phi) / 6 < (0.232 : ℝ) := by linarith
    have h_denom_pos : (0 : ℝ) < 1 - 2 * ((3 - phi) / 6) := by linarith
    have h_num_pos : (0 : ℝ) < 2 * ((3 - phi) / 6) * (246 / 80.4)^2 := by
      nlinarith [sq_nonneg (246/80.4 : ℝ)]
    -- Reveal the division hidden in let-bindings:
    show 1 < 2 * ((3 - phi) / 6) * (246 / 80.4)^2 / (1 - 2 * ((3 - phi) / 6))
    rw [lt_div_iff₀ h_denom_pos]
    nlinarith [hs2_lo, hs2_hi, sq_nonneg (246/80.4 : ℝ),
               show (246/80.4 : ℝ)^2 > 9.3 from by norm_num]
  · apply Real.log_pos
    linarith [phi_gt_onePointSixOne]

What this page does not claim

This module does not derive the full standard model Lagrangian from Recognition Science. The prediction for the Higgs rung position is a derived formula, not a comparison against measured particle masses. The framework does not claim to explain the origin of the electroweak gauge symmetry itself.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/StandardModel/Q3Representations.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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