Masses Electroweak Masses Sin2 Theta Lt Half

The Weinberg angle bound

The Weinberg angle, written θ_W, is a parameter in the standard model of particle physics that describes how the weak force and the electromagnetic force mix. Its sine-squared value, sin²θ_W, is not predicted by the standard model itself; it must be measured. The measured value is approximately 0.231, which is less than one half.

In Recognition Science, the framework models this angle from its own structure. Its definition sets sin²θ_W = (3 − φ)/6, where φ is the golden ratio. Since φ is about 1.618, this gives roughly (3 − 1.618)/6 ≈ 0.2303, close to the measured value.

The declaration sin2_theta_lt_half is a formal theorem in the framework's machine-checked library of formal theorems. It proves that this defined value is strictly less than one half. The proof is short: it uses the fact that φ is positive and less than 2, which forces (3 − φ)/6 to be less than 1/2. This is a mathematical fact about the definition, not a new physical measurement.

What this theorem does not claim is important. It does not prove that the physical Weinberg angle is less than one half; that is an empirical fact already known from experiment. It also does not derive the value of the Weinberg angle from first principles, nor does it show that the framework's definition matches experiment to any stated precision. The theorem only establishes an inequality for a number defined within the framework.

The consequence is modest but real: within the framework, the Weinberg angle definition is internally consistent with the known physical bound. This consistency is a small check, not a derivation. The framework's larger claim, that its constants and masses reproduce measured values, remains a separate empirical comparison.

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

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

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

What this page does not claim

The theorem does not prove the physical Weinberg angle is less than one half; that is an empirical fact. The theorem does not derive the Weinberg angle value from first principles. The theorem does not establish any numerical agreement between the framework's value and experiment.

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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