Masses Electroweak Masses Cos2 Theta W Rs Eq

The Weinberg angle in RS

The Weinberg angle θ_W is the quantity in the standard model of particle physics that measures how much the Z boson and the photon mix when electroweak symmetry breaks. Its sine squared, sin²θ_W, is measured to be about 0.223. The Recognition Science (RS) framework defines this quantity as (3 − φ)/6, where φ is the golden ratio, and then defines cos²θ_W as 1 minus that value.

The declaration cos2_theta_W_rs_eq proves, in the machine-checked library of formal theorems, that this definition simplifies to (3 + φ)/6. This is a pure algebraic identity: it follows from the definitions by ring arithmetic, with no physical input. The same file proves that sin²θ_W lies between 0 and 1/2, and that cos²θ_W is positive, which are the bounds one expects for a valid mixing angle.

In Recognition Science, the Z boson mass is set at a specific rung of the mass ladder, giving m_Z = 2 × φ^51 / 10^6 MeV, and the W mass is defined as m_Z × cos(θ_W). The theorem wz_ratio_eq_cos states that the ratio w_pred / z_pred equals cos_theta_W_rs by construction. This is not a prediction of the ratio; it is a definitional choice that builds the Weinberg relation into the framework.

The framework's Z mass prediction lands in the interval (91075.09, 91075.10) MeV, and the file proves this lies within 0.13% of the PDG 2024 value of 91187.6 MeV. The W mass, derived from Z times the cosine, is therefore tied to the same golden-ratio structure. But the identification of sin²θ_W with (3 − φ)/6 is a definition, not a derived theorem; the file labels it as coming from the gauge embedding geometry without proving that geometry.

What the declaration does not claim is that the standard model's measured Weinberg angle has been derived from first principles. The identity cos²θ_W = (3 + φ)/6 is exact within the framework's definitions, but those definitions are chosen, not forced. The connection between the golden ratio and electroweak mixing remains a model choice, and the measured value is an empirical check, not a theorem.

theorem cos2_theta_W_rs_eq : cos2_theta_W_rs = (3 + Constants.phi) / 6 := by
  unfold cos2_theta_W_rs sin2_theta_W_rs; ring

noncomputable def sin2_theta_W_rs : ℝ := (3 - Constants.phi) / 6

/-- The Z boson mass prediction lies in (91075.09, 91075.10) MeV. -/
theorem z_mass_bounds :
    (91075.09 : ℝ) < z_pred ∧ z_pred < (91075.10 : ℝ) := by
  rw [z_pred_eq]
  constructor
  · rw [lt_div_iff₀ (by norm_num : (0 : ℝ) < 1000000)]
    calc (91075.09 : ℝ) * 1000000 = (91075090000 : ℝ) := by norm_num
      _ < 2 * (45537548334 : ℝ) := by norm_num
      _ < 2 * Constants.phi ^ 51 := by nlinarith [phi51_gt]
  · rw [div_lt_iff₀ (by norm_num : (0 : ℝ) < 1000000)]
    calc 2 * Constants.phi ^ 51 < 2 * (45537549354 : ℝ) := by nlinarith [phi51_lt]
      _ = (91075098708 : ℝ) := by norm_num
      _ < (91075100000 : ℝ) := by norm_num
      _ = (91075.10 : ℝ) * 1000000 := by norm_num

/-- The Z boson prediction is within 0.13% of the PDG value. -/
theorem z_relative_error :
    |z_pred - m_Z_exp| / m_Z_exp < 0.0013 := by
  have hb := z_mass_bounds
  have hexp_pos : (0 : ℝ) < m_Z_exp := by unfold m_Z_exp; norm_num
  rw [div_lt_iff₀ hexp_pos, abs_lt]
  unfold m_Z_exp
  constructor <;> nlinarith [hb.1, hb.2]

/-- The W/Z mass ratio equals cos(θ_W) by construction. -/
theorem wz_ratio_eq_cos : w_pred / z_pred = cos_theta_W_rs := by
  unfold w_pred
  have hzne : z_pred ≠ 0 := ne_of_gt (by linarith [z_mass_bounds.1])
  exact mul_div_cancel_left₀ _ hzne

What this page does not claim

The measured Weinberg angle has been derived from first principles. The identification of sin²θ_W with (3 − φ)/6 is a theorem rather than a definition. The W/Z mass ratio is a prediction; it is fixed by construction.

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/Masses/ElectroweakMasses.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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