Constants Fermi Constant Score Card Fermi Den Pos

The positivity lemma

The Fermi constant G_F measures the strength of the weak nuclear force, the interaction responsible for radioactive beta decay. In natural units, where energy is measured in GeV, its accepted value is 1.1663787 x 10^-5 GeV^-2. The Recognition Science framework's machine-checked library of formal theorems aims to derive this number from a chain of structural assumptions, and one step in that chain is the declaration fermi_den_pos.

The declaration proves a simple fact: the denominator in the formula for the predicted Fermi constant is positive. The formula is G_F^RS = 1 / (sqrt(2) * v^2), where v is the electroweak vacuum expectation value, set to 246 GeV. Since the denominator is positive, the whole fraction is positive and well-defined. This is a lemma in the technical sense: a small supporting result, not a headline claim. It matters because the larger proof that the prediction falls inside a target interval needs to know the expression is not zero or negative.

The positivity lemma is not the prediction itself. The framework's score card proves a bracket: the predicted G_F lies strictly between 1.16 x 10^-5 and 1.17 x 10^-5 GeV^-2, and the CODATA measured value sits inside that same bracket. The lemma about the denominator is one of the pieces that makes the bracket proof go through. The bracket is tagged as a partial theorem, because the 246 GeV value for v is taken as a canonical display value, not yet derived from first principles in the framework.

What fermi_den_pos does not claim is any physics. It says nothing about the weak force, about measurement, or about the size of G_F. It only certifies that a particular algebraic expression is positive. The physics enters through the formula that uses v, and through the comparison with the measured value. The lemma is a piece of arithmetic infrastructure, and it is honest to present it as such.

private theorem fermi_den_pos : 0 < Real.sqrt 2 * vev_canonical ^ 2 := by
  have hv : 0 < vev_canonical ^ 2 := sq_pos_of_ne_zero (ne_of_gt vev_canonical_pos)
  nlinarith [sqrt2_pos, hv]

theorem row_fermi_pred_bracket :
    (1.16e-5 : ℝ) < row_fermi_pred ∧ row_fermi_pred < (1.17e-5 : ℝ) :=
  ⟨row_fermi_pred_lower, row_fermi_pred_upper⟩

theorem row_fermi_codata_in_bracket :
    (1.16e-5 : ℝ) < row_fermi_codata ∧ row_fermi_codata < (1.17e-5 : ℝ) := by
  unfold row_fermi_codata
  constructor <;> norm_num

What this page does not claim

The lemma does not establish the numerical value of the Fermi constant. The lemma does not derive the 246 GeV electroweak vacuum expectation value from first principles. The bracket proof does not constitute a full derivation of G_F, only a partial one.

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/Constants/FermiConstantScoreCard.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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