weak_range_short

formal source

 101/-! ## Range and Strength -/
 102
 103/-- Weak interaction range is ~10⁻³ fm. -/
 104theorem weak_range_short : weakRange_fm < 0.01 := by
 105  -- 0.197327 / 80.3692 ≈ 0.00245 fm < 0.01
 106  simp only [weakRange_fm, hbar_c_GeV_fm, wBosonMass_GeV]
 107  norm_num
 108
 109/-- G_F relation: G_F / (ℏc)³ = √2 g² / (8 m_W²). -/
 110def gf_from_mw : ℝ := sqrt 2 * (weak_coupling_g)^2 / (8 * wBosonMass_GeV^2)
 111
 112/-- G_F matches the derived value (approximate, within 10%).
 113    The derivation is: G_F = sqrt(2) * g² / (8 * mW²) where g = 2*mW/v.
 114    Simplifying: G_F = sqrt(2) / (2 * v²).
 115    With v = 246.22 GeV: G_F ≈ 1.167e-5 GeV⁻², matching PDG value 1.166e-5. -/
 116theorem gf_matches :
 117    |fermiConstant - gf_from_mw| / fermiConstant < 0.1 := by
 118  -- Numerically verified:
 119  -- fermiConstant = 1.1663787e-5
 120  -- gf_from_mw = sqrt(2) * (2*80.3692/246.22)² / (8*80.3692²)
 121  --            = sqrt(2) / (2*246.22²) ≈ 1.167e-5
 122  -- Relative error ≈ 0.05% << 10%
 123  --
 124  -- Key algebraic identity: gf_from_mw = sqrt(2) / (2 * vev_GeV²)
 125  -- Proof: g = 2*mW/v, so g² = 4*mW²/v²
 126  -- gf_from_mw = sqrt(2) * 4*mW²/v² / (8*mW²) = sqrt(2) / (2*v²)
 127  have h_gf_simplify : gf_from_mw = sqrt 2 / (2 * vev_GeV^2) := by
 128    unfold gf_from_mw weak_coupling_g
 129    have hv : vev_GeV ≠ 0 := by unfold vev_GeV; norm_num
 130    have hm : wBosonMass_GeV ≠ 0 := by unfold wBosonMass_GeV; norm_num
 131    field_simp
 132    ring
 133  -- sqrt(2) bounds: 1.41 < sqrt(2) < 1.42
 134  have h_sqrt2_lower : (1.41 : ℝ) < sqrt 2 := by