Masses Electroweak Masses Cos2 Theta Positive

The Weinberg angle in the framework

The Weinberg angle, often written θ_W, is the standard model parameter that describes how the electromagnetic and weak nuclear forces mix. Its sine-squared value is measured to be about 0.231, and it is not predicted by the standard model itself; it is an input. The Recognition Science framework, a machine-checked library of formal theorems, offers a derivation of this number from its own principles. The declaration cos2_theta_positive establishes a simple but necessary fact about that derivation: the cosine-squared of the framework's Weinberg angle is a positive number.

The framework's value for the sine-squared of the Weinberg angle is (3 − φ)/6, where φ is the golden ratio. The declaration cos2_theta_positive proves that the cosine-squared, defined as 1 minus that sine-squared, is greater than zero. This is not a derivation of the angle itself; it is a sanity check on the derived value. The proof is a short argument that uses the fact that the sine-squared value is less than one half, which is itself a proved theorem. The positivity of the cosine-squared ensures that the framework's Weinberg angle is a real, physical angle, not an imaginary one.

In the standard model, the W boson mass is related to the Z boson mass by the cosine of the Weinberg angle: m_W = m_Z cos θ_W. The framework uses this relation to derive the W mass from its Z mass prediction. The positivity of cos²θ_W is a prerequisite for taking the square root to get cos θ_W, which is needed for that mass calculation. The declaration is therefore a small but load-bearing piece of the framework's electroweak sector: it guarantees that the W mass prediction is a real number.

The declaration does not claim that the framework's Weinberg angle matches the measured value. The framework's sine-squared value is approximately 0.236, which is about 2% higher than the measured 0.231. The framework does not claim to derive the standard model's Weinberg angle; it derives its own value from its own principles. The agreement with experiment is not part of the theorem; it is a separate empirical check. The positivity theorem is a purely mathematical statement about the framework's own definitions.

theorem cos2_theta_positive : 0 < cos2_theta_W_rs := by
  unfold cos2_theta_W_rs; linarith [sin2_theta_lt_half]

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

theorem sin2_theta_lt_half : sin2_theta_W_rs < 1/2 := by
  unfold sin2_theta_W_rs
  have hphi : 0 < Constants.phi := phi_pos
  linarith

What this page does not claim

The framework's Weinberg angle matches the measured value. The framework derives the standard model's Weinberg angle. The positivity theorem is a derivation of the angle 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/Masses/ElectroweakMasses.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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