theorem alpha_duality_involution {σ : ℝ} (hσ : σ ≠ 0) (hσ1 : σ ≠ 1) :
alphaOfSigma (σ / (σ - 1)) = -alphaOfSigma σ := by
unfold alphaOfSigma
have h1 : σ - 1 ≠ 0 := sub_ne_zero.mpr hσ1
have h2 : σ / (σ - 1) ≠ 0 := div_ne_zero hσ h1
field_simp
ring### The Boltzmann-Escort Identification