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theorem proved term proof

alpha_s_two_loop_pos_when_denom_pos

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formal statement (Lean)

  92theorem alpha_s_two_loop_pos_when_denom_pos
  93    (alpha_0 : ℝ) (N_f : ℕ) (mu_GeV mu_0_GeV : ℝ)
  94    (h_alpha_pos : 0 < alpha_0)
  95    (h_denom_pos : 0 < 1 + b0 N_f * alpha_0 * Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2) +
  96                    (b1 N_f / b0 N_f) * alpha_0 *
  97                    Real.log (1 + b0 N_f * alpha_0 * Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2))) :
  98    0 < alpha_s_two_loop alpha_0 N_f mu_GeV mu_0_GeV := by

proof body

Term-mode proof.

  99  unfold alpha_s_two_loop
 100  exact div_pos h_alpha_pos h_denom_pos
 101
 102/-! ## At-Z bracket sanity check
 103
 104The PDG value `alpha_s(M_Z) = 0.1179 +- 0.0009`. The RS prediction is
 105`alpha_s_geom = 2/17 ≈ 0.11765`, sitting inside the PDG bracket. The two-loop
 106formula, evaluated at mu_0 = M_Z = mu, returns alpha_0 exactly (the log
 107vanishes), so the two-loop framework is self-consistent at the anchor scale.
 108-/
 109
 110/-- At `mu = mu_0`, the running coupling equals the input coupling. -/

depends on (16)

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