Masses Fermi From Rsinputs Gf Tree Inv Pos

Positivity of the Fermi constant

The Fermi constant, written G_F, is the number that fixes how often particles decay through the weak nuclear force, the interaction responsible for radioactive beta decay. In the standard model of particle physics it is not a free parameter but is derived from three measured inputs: the Z boson mass, the weak mixing angle, and the electromagnetic fine-structure constant. The relation is G_F = 1/(√2 · v²), where v is the vacuum expectation value of the Higgs field, the energy scale at which electroweak symmetry breaks.

Recognition Science (RS) is a framework that derives physical constants from a single forced cost function, and its machine-checked library of formal theorems contains a module called FermiFromRSInputs. That module takes the three standard-model inputs, treats them as RS-derived quantities with zero free parameters, and proves a tree-level closed form for the inverse Fermi constant: G_F⁻¹ = √2 · z² · (8−φ)/36 · α⁻¹/π, where z is the predicted Z mass, φ is the golden ratio, and α is the fine-structure constant. The declaration gf_tree_inv_pos is a theorem in that module, and it proves the simplest structural fact about this quantity: the inverse Fermi constant is strictly greater than zero.

That positivity is not a numerical accident. The proof runs by unfolding the definition of gf_tree_inv and then applying a positivity theorem for the squared vacuum expectation value, which itself follows because z_pred, (8−φ)/36, αInv, and π are all positive. In plain language: the derivation chain never cancels or flips a sign, so the tree-level Fermi constant is guaranteed to be a positive number, never zero and never negative. The library records this as a theorem with no unproved assumptions, and the same module also certifies that the chain uses zero free parameters.

What the declaration does not claim is just as important. It proves positivity of the tree-level value, not a numerical match to the measured Fermi constant. It says nothing about radiative corrections, which shift the physical value by a few percent. It does not assert that the closed form itself is experimentally confirmed; the comparison to PDG data is an empirical check, not a theorem. And the positivity result depends on the RS identification of the inputs, so it inherits that identification's status as a model choice, not a derived coupling.

/-- 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

/-- 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

/-- The VEV squared is positive (since z_pred > 0, (8-φ)/36 > 0, αInv > 0, π > 0). -/
theorem vev_tree_sq_pos : 0 < vev_tree_sq := by
  unfold vev_tree_sq
  have hz : 0 < z_pred := by linarith [z_mass_bounds.1]
  have hs2 : 0 < sin2_theta_W_rs := sin2_theta_positive
  have hc2 : 0 < cos2_theta_W_rs := cos2_theta_positive
  have hα : 0 < alphaInv := by linarith [Numerics.alphaInv_gt]
  have hπ : 0 < Real.pi := Real.pi_pos
  have hz2 : 0 < z_pred ^ 2 := sq_pos_of_ne_zero (ne_of_gt hz)
  have hsc : 0 < sin2_theta_W_rs * cos2_theta_W_rs := mul_pos hs2 hc2
  have h1 : 0 < z_pred ^ 2 * sin2_theta_W_rs := mul_pos hz2 hs2
  have h2 : 0 < z_pred ^ 2 * sin2_theta_W_rs * cos2_theta_W_rs := mul_pos h1 hc2
  have h3 : 0 < z_pred ^ 2 * sin2_theta_W_rs * cos2_theta_W_rs * alphaInv := mul_pos h2 hα
  exact div_pos h3 hπ

/-- 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

This answer does not claim that the tree-level Fermi constant numerically matches the measured PDG value; that comparison is an empirical check, not a theorem. This answer does not claim that the positivity result applies to the radiatively corrected Fermi constant; it applies only to the tree-level value. This answer does not claim that the RS identification of the three inputs is itself derived; that identification is a model choice, not a theorem.

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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