gm_pair_unity

formal source

  97
  98/-- Geometric mean of {x, x⁻¹} = 1 for x > 0.
  99    This explains why the constant offset in Jcost is exactly −1. -/
 100theorem gm_pair_unity {x : ℝ} (hx : 0 < x) : Real.sqrt (x * x⁻¹) = 1 := by
 101  rw [mul_inv_cancel₀ (ne_of_gt hx), Real.sqrt_one]
 102
 103/-- Jcost x = AM(x, x⁻¹) − GM(x, x⁻¹).
 104    The recognition cost is the AM-GM gap of the pair {x, x⁻¹}. -/
 105theorem jcost_is_amgm_gap {x : ℝ} (hx : 0 < x) :
 106    Jcost x = (x + x⁻¹) / 2 - Real.sqrt (x * x⁻¹) := by
 107  rw [gm_pair_unity hx]
 108  unfold Jcost
 109  rfl
 110
 111/-- J is symmetric: Jcost x = Jcost x⁻¹.
 112    This reciprocal symmetry is the algebraic root of σ = 0 conservation. -/
 113theorem jcost_reciprocal_symmetry (x : ℝ) :
 114    Jcost x = Jcost x⁻¹ := by
 115  unfold Jcost
 116  rw [inv_inv]
 117  ring
 118
 119/-! ## §2  The Quantum-Gravity Octave Duality: κ · ℏ = 8
 120
 121**QG-001**: `kappa_einstein * hbar = 8`
 122
 123The product of Einstein coupling and Planck action quantum = the octave 8.
 124Proof: κ = 8φ⁵, ℏ = φ⁻⁵, so κ·ℏ = 8·φ⁵·φ⁻⁵ = 8·φ^(5−5) = 8·1 = 8. -/
 125
 126/-- **QG-001**: κ · ℏ = 8. Quantum-gravity octave duality.
 127
 128    First formal proof that Einstein coupling × Planck constant = octave. -/
 129theorem kappa_hbar_octave : kappa_einstein * hbar = 8 := by
 130  rw [kappa_einstein_eq, hbar_eq_phi_inv_fifth]