Constants Fermi Constant Score Card Sqrt2 Pos

The square root of two

The Fermi constant, G_F, measures the strength of the weak nuclear force, the interaction responsible for radioactive beta decay. In natural units, it is defined by the equation G_F = 1 / (sqrt 2 * v^2), where v is the vacuum expectation value of the Higgs field, about 246 GeV. The square root of two appears because the weak interaction's mathematical structure involves it in the relationship between the coupling constant and the physical decay rate.

The declaration sqrt2_pos is a theorem in the framework's machine-checked library of formal theorems. It proves a basic fact: the square root of two is greater than zero. This is not a discovery about physics. It is a necessary technical step, a guardrail, ensuring that the denominator in the Fermi constant formula is positive. Without this, the division would be undefined in the formal system, and the later proof that the predicted value lies in a specific interval could not proceed.

In Recognition Science, this step supports a broader claim. The framework's library proves that its predicted Fermi constant, computed from the formula, falls within the interval from 1.16 x 10^-5 to 1.17 x 10^-5 GeV^-2. The measured CODATA value, 1.1663787 x 10^-5 GeV^-2, sits inside that bracket. The positivity of the square root of two is one of the small, necessary ingredients in establishing that bracket. The framework's library shows the prediction is consistent with the measured value to within the stated range.

The declaration does not claim that the framework has derived the value of the Higgs vacuum expectation value from first principles. The value of 246 GeV is taken as a canonical input, a definitional choice, not a derived result. The framework's library proves the interval for the Fermi constant given that input. The fully derived bridge from the framework's fundamental constants to the SI/GeV scale of 246 GeV remains an open target, not a proved theorem.

private theorem sqrt2_pos : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)

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 declaration does not claim the framework has derived the 246 GeV vacuum expectation value from first principles. The declaration does not claim the Fermi constant prediction is exact; it claims a bracketing interval. The declaration does not claim the square root of two is a framework-specific constant; it is a classical mathematical object.

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)

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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