IndisputableMonolith.Unification.RecognitionBandwidth

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.BoltzmannConstant
   4import IndisputableMonolith.Constants.ILG
   5import IndisputableMonolith.Cost
   6import IndisputableMonolith.Quantum.HolographicBound
   7
   8/-!
   9# Recognition Bandwidth — Core Definitions
  10
  11Five elements of Recognition Science have never been formally connected:
  12
  131. **Holographic bound** — max information ∝ boundary area / (4 Planck areas)
  142. **Recognition cost per bit** k_R = ln(φ) — each ledger event costs ln(φ) bits
  153. **ILG parameters** C_lag = φ⁻⁵, α = (1−1/φ)/2 — gravity modified at long dynamical times
  164. **8-tick cadence** — R̂ completes one cycle per 8τ₀
  175. **Consciousness boundary cost** — maintaining coherence costs τ·J(L/λ_rec)
  18
  19This module defines **recognition bandwidth**: the maximum rate at which
  20recognition events can be processed within a holographically bounded region.
  21
  22    R_max = S_holo / (k_R · 8τ₀)
  23          = A / (4ℓ_P² · ln(φ) · 8τ₀)
  24
  25This is a hard ceiling on ledger throughput imposed by the holographic bound
  26and the per-bit cost of recognition.
  27
  28## Key Results
  29
  301. `bandwidth_pos` — recognition bandwidth is strictly positive
  312. `bandwidth_monotone_area` — bandwidth grows with area
  323. `bandwidth_involves_all_five` — the formula structurally depends on
  33   holographic capacity, k_R, and the 8-tick cadence
  34-/
  35
  36namespace IndisputableMonolith
  37namespace Unification
  38namespace RecognitionBandwidth
  39
  40open Constants
  41open Constants.BoltzmannConstant
  42open Quantum.HolographicBound
  43
  44/-! ## §1. The 8-Tick Processing Cadence -/
  45
  46/-- One full R̂ cycle takes 8 ticks.
  47    This is the minimum time to process one complete recognition event. -/
  48noncomputable def eightTickCadence : ℝ := 8 * τ₀
  49
  50theorem eightTickCadence_pos : 0 < eightTickCadence := by
  51  unfold eightTickCadence τ₀ tick
  52  norm_num
  53
  54theorem eightTickCadence_eq : eightTickCadence = 8 := by
  55  unfold eightTickCadence τ₀ tick
  56  ring
  57
  58/-! ## §2. Recognition Bandwidth -/
  59
  60/-- **DEFINITION**: Recognition bandwidth of a region with boundary area A.
  61
  62    The maximum number of recognition events per unit time that the holographic
  63    bound permits within the region, given that each event costs k_R = ln(φ) bits
  64    and the 8-tick cadence limits processing to one cycle per 8τ₀.
  65
  66        R_max(A) = A / (4ℓ_P² · k_R · 8τ₀)
  67
  68    Units: events per unit time.
  69
  70    This combines three previously disconnected elements:
  71    - Holographic capacity: A/(4ℓ_P²)      [from Quantum.HolographicBound]
  72    - Per-bit cost: k_R = ln(φ)            [from Constants.BoltzmannConstant]
  73    - Processing rate: 8τ₀ per cycle       [from Foundation.EightTick]  -/
  74noncomputable def bandwidth (area : ℝ) : ℝ :=
  75  area / (4 * planckArea * k_R * eightTickCadence)
  76
  77/-- Planck area is positive. -/
  78theorem planckArea_pos : 0 < planckArea := by
  79  unfold planckArea planckLength
  80  positivity
  81
  82/-- The denominator of the bandwidth formula is positive. -/
  83theorem bandwidth_denom_pos : 0 < 4 * planckArea * k_R * eightTickCadence := by
  84  apply mul_pos
  85  apply mul_pos
  86  apply mul_pos
  87  · linarith [planckArea_pos]
  88  · exact planckArea_pos
  89  · exact k_R_pos
  90  · exact eightTickCadence_pos
  91
  92/-- Recognition bandwidth is positive for positive area. -/
  93theorem bandwidth_pos {A : ℝ} (hA : 0 < A) : 0 < bandwidth A :=
  94  div_pos hA bandwidth_denom_pos
  95
  96/-- Alias for bandwidth_pos. -/
  97theorem bandwidth_pos' {A : ℝ} (hA : 0 < A) : 0 < bandwidth A :=
  98  bandwidth_pos hA
  99
 100/-- Bandwidth is monotone in area: larger boundary → more throughput. -/
 101theorem bandwidth_monotone {A₁ A₂ : ℝ} (_h₁ : 0 < A₁) (h : A₁ ≤ A₂) :
 102    bandwidth A₁ ≤ bandwidth A₂ := by
 103  unfold bandwidth
 104  exact div_le_div_of_nonneg_right h (le_of_lt bandwidth_denom_pos)
 105
 106/-- Bandwidth scales linearly with area. -/
 107theorem bandwidth_linear (A c : ℝ) (_hc : 0 < c) :
 108    bandwidth (c * A) = c * bandwidth A := by
 109  unfold bandwidth
 110  ring
 111
 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
 146/-- The recognition event rate demanded by Newtonian gravitational dynamics
 147    of mass M at dynamical time T_dyn.
 148
 149    Each Planck-mass element requires one ledger update per dynamical time:
 150        R_demand = M / (m_P · T_dyn)
 151
 152    In RS-native units with m_P = 1:
 153        R_demand = M / T_dyn -/
 154noncomputable def demandedRate (mass dynamicalTime : ℝ) : ℝ :=
 155  mass / dynamicalTime
 156
 157theorem demandedRate_pos {M T : ℝ} (hM : 0 < M) (hT : 0 < T) :
 158    0 < demandedRate M T :=
 159  div_pos hM hT
 160
 161/-! ## §6. Saturation Predicate -/
 162
 163/-- A gravitating system is **bandwidth-saturated** when its Newtonian dynamics
 164    demand more recognition events per unit time than the holographic bound permits.
 165
 166    This is the regime where ILG must activate. -/
 167def IsSaturated (area mass dynamicalTime : ℝ) : Prop :=
 168  demandedRate mass dynamicalTime ≥ bandwidth area
 169
 170/-- A system is **sub-saturated** (Newtonian regime) when demand < bandwidth. -/
 171def IsSubSaturated (area mass dynamicalTime : ℝ) : Prop :=
 172  demandedRate mass dynamicalTime < bandwidth area
 173
 174/-- Every system is either saturated or sub-saturated (excluded middle). -/
 175theorem saturated_or_sub (area mass dynamicalTime : ℝ) :
 176    IsSaturated area mass dynamicalTime ∨ IsSubSaturated area mass dynamicalTime := by
 177  unfold IsSaturated IsSubSaturated
 178  rcases le_or_lt (bandwidth area) (demandedRate mass dynamicalTime) with h | h
 179  · left; exact h
 180  · right; exact h
 181
 182end RecognitionBandwidth
 183end Unification
 184end IndisputableMonolith
 185