Masses Fermi From Rsinputs Fermi From Rsinputs Cert

The Fermi certificate

The Fermi constant G_F is the coupling strength of the weak nuclear force, the interaction responsible for radioactive beta decay. It is one of the most precisely measured quantities in physics, with a value of about 1.166 × 10⁻⁵ GeV⁻². In the standard model, G_F is not a fundamental input but is derived from three other measured quantities: the Z boson mass m_Z, the weak mixing angle sin²θ_W, and the fine-structure constant α_EM. The tree-level relation is G_F = π·α / (√2 · m_Z² · sin²θ_W · cos²θ_W), a standard textbook formula.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration called FermiFromRSInputsCert, a certificate that bundles three proved statements about this relation. The first is a closed form: G_F⁻¹ = √2 · z² · (8−φ)/36 · α⁻¹/π, where φ is the golden ratio and z is a predicted mass scale. The second is positivity: the inverse Fermi constant is strictly greater than zero. The third is the count of free parameters in the derivation chain, which is exactly zero. The certificate is a structure that holds these three facts together, and the framework provides a proof that the certificate is satisfied.

In Recognition Science, the framework models the weak mixing angle through the closed form sin²θ_W · cos²θ_W = (8−φ)/36, a relation proved in its library. This means G_F inherits its structural origin from the framework's derived inputs, rather than being fitted to experiment. The zero-parameter count is a formal statement about the derivation chain: every input in the chain is itself derived within the framework, none is a free empirical adjustment. The certificate is the framework's way of bundling these three facts into a single object that can be referenced and audited.

What the certificate does not claim is as important as what it proves. It does not claim that the framework's prediction of G_F matches the measured value to any particular precision; the certificate is about the structural derivation, not about numerical agreement with experiment. It does not claim that the tree-level relation is the full story, since radiative corrections modify the relation at higher orders. And it does not claim that the framework's derivation of α, m_Z, or sin²θ_W is itself complete; those are separate results in the library, and the certificate simply assumes them as inputs. The certificate is a precise, narrow object: it says that given the framework's derived inputs, G_F follows with zero free parameters, and it proves the positivity and closed form of that result.

structure FermiFromRSInputsCert where
  closed_form :
    gf_tree_inv = Real.sqrt 2 *
      (z_pred ^ 2 * ((8 - phi) / 36) * alphaInv / Real.pi)
  positivity : 0 < gf_tree_inv
  zero_params : free_parameters_in_gf_chain = 0

/-- G_F expressed through the sin²·cos² closed form.
    G_F⁻¹ = √2 · z² · (8-φ)/36 · α⁻¹/π. -/
theorem gf_tree_inv_closed_form :
    gf_tree_inv = Real.sqrt 2 *
      (ElectroweakMasses.z_pred ^ 2 * ((8 - phi) / 36) * alphaInv / Real.pi) := by
  unfold gf_tree_inv
  rw [vev_tree_sq_closed_form]

/-- G_F⁻¹ is positive (since it is √2 times a positive quantity). -/
theorem gf_tree_inv_pos : 0 < gf_tree_inv := by
  unfold gf_tree_inv
  exact mul_pos (Real.sqrt_pos.mpr (by norm_num)) vev_tree_sq_pos

/-- All RS inputs in the G_F chain are structural. -/
theorem gf_zero_free_params :
    free_parameters_in_gf_chain = 0 := rfl

What this page does not claim

The certificate does not claim numerical agreement between the framework's G_F and the measured value. The certificate does not claim that the tree-level relation is the complete physical story, since radiative corrections are not included. The certificate does not claim that the framework's derivation of its input quantities is itself complete or proved.

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/FermiFromRSInputs.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.

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