phi_ladder_div_closed

formal source

 101  rw [zpow_add_one₀ (ne_of_gt phi_pos)]
 102
 103/-- The ladder is closed under division by φ -/
 104theorem phi_ladder_div_closed (x : ℝ) (hx : x ∈ PhiLadder) :
 105    x / φ ∈ PhiLadder := by
 106  obtain ⟨n, rfl⟩ := hx
 107  use n - 1
 108  rw [zpow_sub_one₀ (ne_of_gt phi_pos)]
 109  rw [div_eq_mul_inv]
 110
 111/-! ## J-cost at φ-Ladder Positions -/
 112
 113/-- J-cost formula applied to φ -/
 114theorem J_at_phi : (φ + 1/φ) / 2 - 1 = (Real.sqrt 5 - 2) / 2 :=
 115  Inequalities.J_cost_phi
 116
 117/-- J-cost at φ^n (for n ≥ 1) -/
 118noncomputable def J_at_phi_pow (n : ℕ) : ℝ :=
 119  (φ^n + φ^(-(n : ℤ))) / 2 - 1
 120
 121/-- J-cost at φ is approximately 0.118 -/
 122theorem J_at_phi_approx : (φ + 1/φ) / 2 - 1 < 0.12 := by
 123  rw [J_at_phi]
 124  -- (√5 - 2)/2 ≈ (2.236 - 2)/2 = 0.118
 125  have h_sqrt5_bound : Real.sqrt 5 < 2.24 := by
 126    have h5_sq : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
 127    have h_target : (2.24 : ℝ)^2 > 5 := by norm_num
 128    nlinarith [Real.sqrt_nonneg 5, sq_nonneg (Real.sqrt 5)]
 129  linarith
 130
 131/-! ## Connection to Consciousness Threshold -/
 132
 133/-- HYPOTHESIS: The consciousness threshold C = 1 emerges from φ-quantization.
 134