Masses Vevconsistency

The VEV consistency module

The vacuum expectation value (VEV) of the Higgs field is the energy scale at which the electroweak symmetry breaks, roughly 246 GeV. In the standard model it is an input, measured from other quantities. In the Recognition Science framework, a module called VEVConsistency proves it is not an independent parameter at all. The module derives the VEV from three framework-derived inputs: the Z boson mass, the weak mixing angle, and the inverse fine-structure constant. The core result is a closed-form formula for the tree-level VEV squared, which depends only on the Z mass prediction, the golden ratio φ, the inverse coupling, and π, with zero free parameters.

The derivation starts with the standard electroweak tree-level relation, v² = m_Z² · sin²θ_W · cos²θ_W · α⁻¹ / π. The module proves a clean algebraic identity: the product sin²θ_W · cos²θ_W equals (8 − φ)/36, where φ is the golden ratio. This is a theorem in the machine-checked library. The module also proves that this product lies between 0.176 and 0.178, which pins down the mixing angle contribution. Combining these with the Z mass prediction, which is in the range (91075.09, 91075.10) MeV, and the inverse coupling band (137.030, 137.039), gives a tree-level VEV of about 253 GeV.

That tree-level value overshoots the measured PDG value of 246.22 GeV by about 2.8%. The module explains this discrepancy precisely: it is the QED vacuum polarization running from α(0) to α(M_Z). At the Z pole, the inverse coupling is about 128.9 instead of 137.036. The correction factor √(128.9/137.036) ≈ 0.970 brings the VEV to about 245.5 GeV, within 0.3% of the measured value. The module proves the running ratio lies between 0.940 and 0.942, and this standard loop calculation accounts for the difference.

The module bundles these results into a consistency certificate, a structure that holds the closed form, the interval bounds, the zero-free-parameter formula, and the running ratio bounds. The certificate proves that the VEV is determined by the framework's inputs, not fitted to the measurement. The machine-checked library shows this with zero axioms and zero sorry, meaning every step is formally verified. The framework's claim is structural: the VEV is a consequence of the framework's derived quantities, and the only missing ingredient is a standard, well-established quantum correction.

In plain language, the module says this: if you accept the framework's derived values for the Z mass, the weak mixing angle, and the coupling, then the Higgs VEV is forced to be a specific number. The raw calculation gives 253 GeV, and the known running of the coupling from low to high energy scales corrects it to 245.5 GeV, matching the measured value within a third of a percent. The framework does not claim to derive the fine-structure constant itself; that remains an open problem. But given that constant as a boundary datum, the VEV is no longer a free parameter in this account.

/-- 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 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]

/-- The running ratio is in (0.940, 0.942). -/
theorem running_ratio_bounds :
    (0.940 : ℝ) < running_ratio ∧ running_ratio < (0.942 : ℝ) := by
  unfold running_ratio alphaInv_MZ
  have halpha_gt := Numerics.alphaInv_gt       -- 137.030 < αInv
  have halpha_lt := Numerics.alphaInv_lt        -- αInv < 137.039
  have halpha_pos : (0 : ℝ) < alphaInv := by linarith
  constructor
  · -- 128.9 / αInv > 0.940 ⟺ 128.9 > 0.940 × αInv
    rw [lt_div_iff₀ halpha_pos]
    -- 0.940 × αInv < 0.940 × 137.039 = 128.816... < 128.9
    nlinarith
  · -- 128.9 / αInv < 0.942 ⟺ 128.9 < 0.942 × αInv
    rw [div_lt_iff₀ halpha_pos]
    -- 128.9 < 0.942 × αInv. αInv > 137.030, so 0.942 × 137.030 = 129.084... > 128.9
    nlinarith

noncomputable def vevConsistencyCert_holds : VEVConsistencyCert where
  sin2_cos2_closed := sin2_cos2_product
  sin2_cos2_interval := ⟨sin2_cos2_gt, sin2_cos2_lt⟩
  closed_form := vev_tree_sq_closed_form
  running_bounded := running_ratio_bounds

What this page does not claim

The module does not derive the fine-structure constant itself; it treats the inverse coupling band as a certified construction window. The module does not claim the tree-level VEV of 253 GeV is the physical value; it explicitly requires the QED running correction to match the measurement. The module does not prove the QED vacuum polarization running calculation; it cites it as a standard well-established result.

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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