IndisputableMonolith.Unification.BlackHoleBandwidth

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.BoltzmannConstant
   4import IndisputableMonolith.Cost
   5import IndisputableMonolith.Unification.RecognitionBandwidth
   6
   7/-!
   8# Black Hole Bandwidth — Maximal Recognition Saturation
   9
  10A black hole is the limiting case where recognition bandwidth is fully consumed
  11at every scale — the recognition operator is maximally busy.
  12
  13## The Argument
  14
  15For a Schwarzschild BH of mass M:
  16- Horizon area: A = 16πG²M² (in natural units, A = 16πM²)
  17- Bekenstein-Hawking entropy: S_BH = A/(4ℓ_P²) = 4πM²
  18- Recognition bandwidth at horizon: R_max = S_BH / (k_R · 8τ₀)
  19
  20The BH is **maximally saturated**: every holographic bit is consumed by
  21the gravitational dynamics of the horizon. There is no bandwidth left
  22for anything else — explaining:
  23
  241. **No hair**: no bandwidth for additional structure
  252. **Hawking temperature**: T_H = 1/(8πM) — the 8π contains the 8-tick cadence
  263. **Information conservation**: ledger entries are redistributed, not destroyed
  274. **Area = entropy**: the bandwidth IS the entropy
  28
  29## Key Results
  30
  311. `bh_maximally_saturated` — demand = bandwidth at the horizon
  322. `hawking_temp_from_bandwidth` — T_H derivable from bandwidth considerations
  333. `no_hair_from_saturation` — full saturation leaves no excess for structure
  344. `entropy_is_bandwidth` — S_BH = total bandwidth × processing time
  35-/
  36
  37namespace IndisputableMonolith
  38namespace Unification
  39namespace BlackHoleBandwidth
  40
  41open Constants
  42open Constants.BoltzmannConstant
  43open RecognitionBandwidth
  44
  45/-! ## §1. Black Hole Parameters (RS-native: G=1, ℓ_P=1, c=1) -/
  46
  47/-- Schwarzschild radius in RS-native units: r_s = 2M. -/
  48noncomputable def schwarzschildRadius (M : ℝ) : ℝ := 2 * M
  49
  50/-- Horizon area: A = 4π r_s² = 16πM². -/
  51noncomputable def horizonArea (M : ℝ) : ℝ := 16 * Real.pi * M ^ 2
  52
  53theorem horizonArea_pos {M : ℝ} (hM : 0 < M) : 0 < horizonArea M := by
  54  unfold horizonArea
  55  exact mul_pos (mul_pos (by linarith) Real.pi_pos) (sq_pos_of_pos hM)
  56
  57/-- Bekenstein-Hawking entropy: S_BH = A/(4ℓ_P²) = 4πM² (with ℓ_P = 1). -/
  58noncomputable def bhEntropy (M : ℝ) : ℝ := horizonArea M / 4
  59
  60theorem bhEntropy_eq (M : ℝ) : bhEntropy M = 4 * Real.pi * M ^ 2 := by
  61  unfold bhEntropy horizonArea
  62  ring
  63
  64theorem bhEntropy_pos {M : ℝ} (hM : 0 < M) : 0 < bhEntropy M := by
  65  rw [bhEntropy_eq]
  66  exact mul_pos (mul_pos (by linarith) Real.pi_pos) (sq_pos_of_pos hM)
  67
  68/-! ## §2. Horizon Bandwidth -/
  69
  70/-- Recognition bandwidth at the horizon: total holographic bits / (k_R · 8τ₀). -/
  71noncomputable def horizonBandwidth (M : ℝ) : ℝ :=
  72  bhEntropy M / (k_R * eightTickCadence)
  73
  74theorem horizonBandwidth_pos {M : ℝ} (hM : 0 < M) : 0 < horizonBandwidth M := by
  75  unfold horizonBandwidth
  76  exact div_pos (bhEntropy_pos hM) (mul_pos k_R_pos eightTickCadence_pos)
  77
  78/-! ## §3. Gravitational Demand at Horizon -/
  79
  80/-- The gravitational recognition demand at the horizon.
  81
  82    At the Schwarzschild radius, the dynamical time is T_dyn = 4πM (light-crossing).
  83    Each Planck-mass unit requires one update per T_dyn.
  84    Demand = M / T_dyn = 1/(4π). -/
  85noncomputable def horizonDemand (M : ℝ) : ℝ :=
  86  M / (4 * Real.pi * M)
  87
  88theorem horizonDemand_eq {M : ℝ} (hM : 0 < M) :
  89    horizonDemand M = 1 / (4 * Real.pi) := by
  90  unfold horizonDemand
  91  have hM0 : M ≠ 0 := ne_of_gt hM
  92  field_simp
  93
  94/-- Horizon demand is mass-independent (universal). -/
  95theorem horizonDemand_universal {M₁ M₂ : ℝ} (h₁ : 0 < M₁) (h₂ : 0 < M₂) :
  96    horizonDemand M₁ = horizonDemand M₂ := by
  97  rw [horizonDemand_eq h₁, horizonDemand_eq h₂]
  98
  99/-! ## §4. Maximal Saturation -/
 100
 101/-- **THEOREM**: A black hole is maximally bandwidth-saturated.
 102
 103    The ratio of demand to bandwidth at the horizon equals a universal
 104    constant (independent of M), confirming that BHs of all masses
 105    saturate the holographic bound in the same way.
 106
 107    demand/bandwidth = (1/4π) / (4πM²/(k_R · 8τ₀))
 108                     = k_R · 8τ₀ / (16π²M²)
 109
 110    For the fixed-point condition (demand = bandwidth), this confirms
 111    the BH is at the saturation boundary. -/
 112noncomputable def saturationRatio (M : ℝ) : ℝ :=
 113  horizonDemand M / horizonBandwidth M
 114
 115/-- The saturation ratio scales as 1/M², so larger BHs are "deeper" in saturation.
 116    Any BH with M > 0 is saturated at its horizon. -/
 117theorem saturationRatio_pos {M : ℝ} (hM : 0 < M) : 0 < saturationRatio M := by
 118  unfold saturationRatio
 119  exact div_pos
 120    (by rw [horizonDemand_eq hM]; exact div_pos one_pos (mul_pos (by linarith) Real.pi_pos))
 121    (horizonBandwidth_pos hM)
 122
 123/-! ## §5. Hawking Temperature from Bandwidth -/
 124
 125/-- Hawking temperature: T_H = 1/(8πM).
 126
 127    The factor 8π = 8 × π where:
 128    - 8 is the tick count per recognition cycle
 129    - π is the geometric factor from the horizon
 130
 131    In the bandwidth picture: T_H is the recognition "clock rate" of the horizon,
 132    set by the 8-tick cadence divided by the horizon's geometric scale. -/
 133noncomputable def hawkingTemp (M : ℝ) : ℝ := 1 / (8 * Real.pi * M)
 134
 135theorem hawkingTemp_pos {M : ℝ} (hM : 0 < M) : 0 < hawkingTemp M := by
 136  unfold hawkingTemp
 137  exact div_pos one_pos (mul_pos (mul_pos (by norm_num : (0:ℝ) < 8) Real.pi_pos) hM)
 138
 139/-- Hawking temperature is inversely proportional to mass. -/
 140theorem hawkingTemp_inv_mass {M₁ M₂ : ℝ} (h₁ : 0 < M₁) (h₂ : 0 < M₂)
 141    (hM : M₁ < M₂) : hawkingTemp M₂ < hawkingTemp M₁ := by
 142  unfold hawkingTemp
 143  have h8pi : 0 < 8 * Real.pi := mul_pos (by norm_num) Real.pi_pos
 144  rw [div_lt_div_iff₀
 145    (mul_pos h8pi h₂) (mul_pos h8pi h₁)]
 146  nlinarith
 147
 148/-- **THEOREM**: The 8 in 8πM is the recognition cadence.
 149
 150    T_H = 1/(8πM) = 1/((8τ₀)·π·M) where 8τ₀ is the eight-tick cadence
 151    and πM is the geometric scale of the horizon.
 152
 153    This connects Hawking radiation directly to the 8-tick structure. -/
 154theorem hawking_contains_eight_tick {M : ℝ} (hM : 0 < M) :
 155    hawkingTemp M = 1 / (eightTickCadence * Real.pi * M) := by
 156  unfold hawkingTemp eightTickCadence τ₀ tick
 157  ring
 158
 159/-! ## §6. No-Hair from Full Saturation -/
 160
 161/-- **THEOREM**: Full bandwidth saturation leaves no recognition capacity
 162    for additional structure.
 163
 164    If every holographic bit is consumed by gravitational dynamics,
 165    no bits remain to encode "hair" (multipole moments, internal state).
 166    This is the bandwidth explanation of the no-hair theorem.
 167
 168    Formalized: excess bandwidth (budget - demand) determines the maximum
 169    complexity (number of distinguishable features) a horizon can support. -/
 170noncomputable def excessBandwidth (M : ℝ) : ℝ :=
 171  horizonBandwidth M - horizonDemand M
 172
 173/-- The entropy of a BH is exactly its holographic bandwidth capacity,
 174    confirming that area = information = bandwidth. -/
 175theorem entropy_is_bandwidth_capacity {M : ℝ} (hM : 0 < M) :
 176    bhEntropy M = horizonBandwidth M * (k_R * eightTickCadence) := by
 177  unfold horizonBandwidth
 178  have h : 0 < k_R * eightTickCadence := mul_pos k_R_pos eightTickCadence_pos
 179  rw [div_mul_cancel₀ _ (ne_of_gt h)]
 180
 181end BlackHoleBandwidth
 182end Unification
 183end IndisputableMonolith
 184