Masses Vevconsistency Vev Tree Sq Closed Form

The VEV formula

The Higgs vacuum expectation value, usually written v, is the constant background strength the Higgs field takes everywhere in empty space. It sets the mass scale for the weak force carriers and for all fundamental particles that interact with the Higgs field. In the standard model, v is an input parameter, measured to be about 246 GeV, not something the theory derives.

In the Recognition Science framework, the declaration vev_tree_sq_closed_form proves that the tree-level value of v squared can be written as a single closed expression. The expression is v² = m_Z² · sin²θ_W · cos²θ_W · α⁻¹ / π, where m_Z is the Z boson mass, θ_W is the weak mixing angle, and α⁻¹ is the inverse fine-structure constant. The proof substitutes a known closed form for the mixing angle product, sin²θ_W · cos²θ_W = (8 − φ)/36, where φ is the golden ratio. The result is a formula that depends only on the Z mass, the golden ratio, the inverse fine-structure constant, and π, with zero free parameters.

The theorem does not claim that this tree-level expression matches the measured value. At zero momentum it gives v_tree ≈ 253 GeV, which overshoots the Particle Data Group value of 246.22 GeV by about 2.8 percent. The discrepancy is exactly the QED vacuum polarization running from α(0) to α(M_Z): the inverse coupling at the Z pole is about 128.9 rather than 137.036. Applying the correction factor √(128.9/137.036) ≈ 0.970 gives v_physical ≈ 245.5 GeV, within 0.3 percent of the measured value.

The framework's library also proves bounds on the mixing product: it lies between 0.176 and 0.178, and the running correction ratio lies between 0.940 and 0.942. These facts bundle into a consistency certificate showing the VEV is determined by Recognition Science inputs, not fitted. The certificate does not claim that the fine-structure constant itself is derived: the framework explicitly proves that no normalization-blind condition can pin the coupling, and the band used here is a construction window, not a measurement claim.

/-- The VEV formula uses the RS-derived closed form for the mixing product.
    This shows v_tree² is algebraically equivalent to a formula involving
    only (z_pred, φ, αInv, π), with zero free parameters. -/
theorem vev_tree_sq_closed_form :
    vev_tree_sq = ElectroweakMasses.z_pred ^ 2 * ((8 - phi) / 36) *
    alphaInv / Real.pi := by
  unfold vev_tree_sq
  have h := sin2_cos2_product
  -- Left-associativity: z² * sin² * cos² = (z² * sin²) * cos²
  -- Regroup to expose sin² * cos² for substitution
  have hassoc : ElectroweakMasses.z_pred ^ 2 * ElectroweakMasses.sin2_theta_W_rs *
      ElectroweakMasses.cos2_theta_W_rs =
      ElectroweakMasses.z_pred ^ 2 * (ElectroweakMasses.sin2_theta_W_rs *
      ElectroweakMasses.cos2_theta_W_rs) := by ring
  rw [hassoc, h]

/-- sin²θ_W · cos²θ_W = (8-φ)/36.
    Proof: (3-φ)/6 × (3+φ)/6 = (9-φ²)/36 = (9-(φ+1))/36 = (8-φ)/36. -/
theorem sin2_cos2_product :
    ElectroweakMasses.sin2_theta_W_rs * ElectroweakMasses.cos2_theta_W_rs =
    (8 - phi) / 36 := by
  unfold ElectroweakMasses.cos2_theta_W_rs
  unfold ElectroweakMasses.sin2_theta_W_rs
  have hsq : phi ^ 2 = phi + 1 := phi_sq_eq
  field_simp
  nlinarith [hsq]

/-- The tree-level VEV squared from RS inputs (in MeV²). -/
noncomputable def vev_tree_sq : ℝ :=
  ElectroweakMasses.z_pred ^ 2 *
  ElectroweakMasses.sin2_theta_W_rs *
  ElectroweakMasses.cos2_theta_W_rs *
  alphaInv / Real.pi

/-- The running correction ratio: α⁻¹(M_Z) / α⁻¹(0).
    This corrects the tree-level VEV to the physical scale. -/
noncomputable def running_ratio : ℝ := alphaInv_MZ / alphaInv

What this page does not claim

The tree-level formula does not match the measured VEV without the QED running correction. The framework does not derive the fine-structure constant; the band used here is a construction window, not a measurement claim. The theorem does not prove that the physical VEV equals the measured value; it proves a structural formula with zero free parameters.

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)

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Derived articles

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