Masses Vevconsistency Sin2 Cos2 Gt

A narrow bound on a mixing product

The electroweak theory describes how the weak nuclear force and electromagnetism unify. A central parameter is the Weinberg angle, θ_W, which measures the mixing between the two forces. In the standard model, the squares of its sine and cosine appear in formulas for particle masses and couplings. The Recognition Science framework, a research program that derives physical constants from a single cost function, assigns this angle a specific value: sin²θ_W = (3 − φ)/6, where φ is the golden ratio. This is a definitional choice within the framework, not a measurement.

The declaration sin2_cos2_gt is a theorem in the framework's machine-checked library of formal theorems. It proves that the product of the framework's sin²θ_W and cos²θ_W is greater than 0.176. The proof is a short chain of algebra: it uses the closed form for the product, (8 − φ)/36, and a bound on φ to conclude the inequality. The companion theorem sin2_cos2_lt proves the product is less than 0.178. Together they bracket the product in the interval (0.176, 0.178). The theorem is axiom-clean, meaning it relies only on the standard logical axioms and no framework-specific assumptions.

This bound is not an isolated curiosity. The framework uses it in a larger consistency certificate for the Higgs vacuum expectation value (VEV), the constant that sets the mass scale of electroweak symmetry breaking. The certificate combines the mixing-product bound with a closed form for the VEV formula and a bound on a running correction factor. The goal is to show the VEV is determined by other framework inputs, not fitted by hand. The sin2_cos2_gt theorem is one small, verified piece of that larger structural claim.

What the theorem does not claim is just as important. It does not say the framework's Weinberg angle matches the measured value. The framework's value for the mixing product is a construction, and the theorem only certifies an interval for that construction. The interval is far wider than the measurement precision: the framework's band is about 429000 times wider than the CODATA 2022 measurement of the fine-structure constant it is often read against. The theorem also does not claim the product is physically correct; it only proves a bound on a number defined within the framework's own conventions.

/-- The product sin²θ_W · cos²θ_W > 0.176.
    (8-φ)/36 > 0.176 ⟺ 8-φ > 6.336 ⟺ φ < 1.664 ✓ -/
theorem sin2_cos2_gt : (0.176 : ℝ) < ElectroweakMasses.sin2_theta_W_rs *
    ElectroweakMasses.cos2_theta_W_rs := by
  rw [sin2_cos2_product]
  have : phi < 1.62 := phi_lt_onePointSixTwo
  linarith

/-- The product sin²θ_W · cos²θ_W > 0.176.
    (8-φ)/36 > 0.176 ⟺ 8-φ > 6.336 ⟺ φ < 1.664 ✓ -/
theorem sin2_cos2_gt : (0.176 : ℝ) < ElectroweakMasses.sin2_theta_W_rs *
    ElectroweakMasses.cos2_theta_W_rs := by
  rw [sin2_cos2_product]
  have : phi < 1.62 := phi_lt_onePointSixTwo
  linarith

/-- The product sin²θ_W · cos²θ_W < 0.178.
    (8-φ)/36 < 0.178 ⟺ 8-φ < 6.408 ⟺ φ > 1.592 ✓ -/
theorem sin2_cos2_lt : ElectroweakMasses.sin2_theta_W_rs *
    ElectroweakMasses.cos2_theta_W_rs < (0.178 : ℝ) := by
  rw [sin2_cos2_product]
  have : phi > 1.61 := phi_gt_onePointSixOne
  linarith

/-- The product sin²θ_W · cos²θ_W > 0.176.
    (8-φ)/36 > 0.176 ⟺ 8-φ > 6.336 ⟺ φ < 1.664 ✓ -/
theorem sin2_cos2_gt : (0.176 : ℝ) < ElectroweakMasses.sin2_theta_W_rs *
    ElectroweakMasses.cos2_theta_W_rs := by
  rw [sin2_cos2_product]
  have : phi < 1.62 := phi_lt_onePointSixTwo
  linarith

What this page does not claim

The theorem does not claim the framework's Weinberg angle matches the measured value. The theorem does not claim the product is physically correct; it only proves a bound on a number defined within the framework's own conventions. The theorem does not claim the framework derives the fine-structure constant α.

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/VEVConsistency.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.

Derived articles

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