Constants Fermi Constant Score Card Row Fermi Pred Lower

The lower bound

The Fermi constant, usually written G_F, sets the strength of the weak nuclear force, the interaction responsible for radioactive beta decay. In natural units, where energy is measured in gigaelectronvolts (GeV), its measured value is 1.1663787 x 10^-5 GeV^-2, a figure from the CODATA and Particle Data Group compilations. The constant is not a free parameter in the Recognition Science framework; the framework derives it from a single electroweak scale, the vacuum expectation value of the Higgs field, which it takes to be 246 GeV.

What the declaration row_fermi_pred_lower proves is the lower half of a bracket. It shows, with a machine-checked proof, that the framework's prediction for G_F is greater than 1.16 x 10^-5 GeV^-2. A companion declaration proves the prediction is less than 1.17 x 10^-5 GeV^-2. Together they place the predicted value between those two bounds, and the measured CODATA value sits inside the same interval. The proof is a formal theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions and no axioms beyond the standard logical ones.

The bracket is real but narrow, and the framework says plainly what it does not claim. The prediction uses the canonical 246 GeV value for the electroweak scale, a number taken from experiment, not derived from first principles within the framework. The bridge that would derive that 246 GeV scale from deeper structure remains open. So the theorem proves the arithmetic of the bracket, not the origin of the scale that feeds into it. The row is labeled PARTIAL_THEOREM in the framework's own planning documents.

The honest test is a falsifier. If a future measurement placed G_F outside the bracket, the prediction would be wrong. If a future derivation of the electroweak scale did not recover the 246 GeV value, the bracket would lose its anchor. Neither has happened. The measured value sits inside the bracket, and the proof that it does is as solid as the proof of the bracket itself. What the reader can take away is a concrete, checkable claim: the framework's Fermi constant prediction lands within one percent of the measured value, with the exact arithmetic verified by machine.

theorem row_fermi_pred_lower :
    (1.16e-5 : ℝ) < row_fermi_pred := by
  unfold row_fermi_pred
  rw [lt_div_iff₀ fermi_den_pos]
  have hs : Real.sqrt 2 < (1.4143 : ℝ) := sqrt2_lt_14143
  have hden :
      Real.sqrt 2 * vev_canonical ^ 2 < (1.4143 : ℝ) * (246 : ℝ) ^ 2 := by
    have hv : vev_canonical = (246 : ℝ) := rfl
    have hvpos : 0 < vev_canonical ^ 2 := sq_pos_of_ne_zero (ne_of_gt vev_canonical_pos)
    rw [hv]
    nlinarith
  have hnum : (1.16e-5 : ℝ) * ((1.4143 : ℝ) * (246 : ℝ) ^ 2) < 1 := by
    norm_num
  nlinarith

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

/-- P1-C01 Fermi constant prediction in GeV^-2 natural units. -/
noncomputable def row_fermi_pred : ℝ :=
  1 / (Real.sqrt 2 * vev_canonical ^ 2)

What this page does not claim

The theorem does not prove that the 246 GeV scale itself is derived within the framework. The bracket does not imply a precise match to the measured value; it only places the prediction within one percent. The proof does not establish the Fermi constant's value in SI units, only in natural units of GeV^-2.

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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