J_at_phi

formal source

 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
 135    The exact relationship needs derivation. Some possibilities:
 136    1. C = 1 is the point where J-cost integration over one octave equals unity
 137    2. C = 1 relates to φ^n summing to specific values
 138    3. C = 1 is forced by the 8-tick structure interacting with φ
 139
 140    This is marked as a hypothesis until the derivation is complete. -/
 141def H_ThresholdFromPhi : Prop :=
 142  ∃ (mechanism : ℕ → ℝ → ℝ),
 143    mechanism 8 φ = 1  -- Some function of 8 ticks and φ gives threshold 1
 144