holographicBits

formal source

 112/-! ## §3. Holographic Capacity Recovery -/
 113
 114/-- The total holographic capacity (bits) of area A. -/
 115noncomputable def holographicBits (area : ℝ) : ℝ :=
 116  area / (4 * planckArea)
 117
 118/-- Bandwidth equals holographic bits divided by (k_R · 8τ₀). -/
 119theorem bandwidth_eq_bits_over_cost {A : ℝ} (_hA : 0 < A) :
 120    bandwidth A = holographicBits A / (k_R * eightTickCadence) := by
 121  unfold bandwidth holographicBits
 122  ring
 123
 124/-- Each recognition event within the bandwidth budget consumes k_R bits
 125    of holographic capacity per 8-tick cycle. -/
 126theorem bandwidth_times_cost_eq_rate {A : ℝ} (hA : 0 < A) :
 127    bandwidth A * (k_R * eightTickCadence) = holographicBits A := by
 128  rw [bandwidth_eq_bits_over_cost hA]
 129  have h : 0 < k_R * eightTickCadence := mul_pos k_R_pos eightTickCadence_pos
 130  exact div_mul_cancel₀ (holographicBits A) (ne_of_gt h)
 131
 132/-! ## §4. Connection to ILG Parameters -/
 133
 134/-- ILG coherence energy C_lag = φ⁻⁵ equals the coherence exponent E_coh.
 135    This is the energy quantum per recognition event in RS-native units. -/
 136theorem Clag_eq_phi_neg5 : Clag = 1 / phi ^ (5 : ℕ) := by
 137  unfold Clag
 138  ring
 139
 140/-- The ILG modification parameter α = (1−1/φ)/2 is between 0 and 1. -/
 141theorem alpha_locked_in_unit : 0 < alpha_locked ∧ alpha_locked < 1 :=
 142  ⟨alpha_locked_pos, alpha_locked_lt_one⟩
 143
 144/-! ## §5. Demanded Recognition Rate -/
 145