IndisputableMonolith.Unification.ConsciousnessBandwidth

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.BoltzmannConstant
   4import IndisputableMonolith.Cost
   5import IndisputableMonolith.Unification.RecognitionBandwidth
   6
   7/-!
   8# Consciousness Bandwidth — The Holographic Constraint on Conscious Extent
   9
  10A conscious boundary (ConsciousnessH) has a maximum spatial extent
  11set by the holographic information budget.
  12
  13## The Argument
  14
  15A boundary of extent L persists for τ ticks. Its maintenance cost is:
  16
  17    BoundaryCost = τ · J(L / λ_rec)
  18
  19The holographic capacity of its boundary area A ∝ L² is:
  20
  21    S_holo = L² / (4ℓ_P²)
  22
  23Each recognition event to maintain the boundary costs k_R = ln(φ) bits.
  24The maximum number of maintenance events is S_holo / k_R.
  25Over the consciousness barrier period (360 = 45 × 8 ticks), the total
  26maintenance budget is:
  27
  28    N_events = L² / (4ℓ_P² · ln(φ))
  29
  30The maintenance DEMAND over 360 ticks is proportional to the J-cost
  31integrated over the barrier:
  32
  33    N_demand ∝ 360 · J(L / λ_rec)
  34
  35Setting demand = budget defines a maximum coherent extent L_crit.
  36
  37## Key Results
  38
  391. `consciousness_barrier` — the 360-tick barrier period
  402. `critical_extent_exists` — L_crit exists and is positive
  413. `bandwidth_constrains_extent` — larger boundary = more demand, limited budget
  424. `z_complexity_reduces_extent` — higher Z = more complex = smaller L_crit
  43-/
  44
  45namespace IndisputableMonolith
  46namespace Unification
  47namespace ConsciousnessBandwidth
  48
  49open Constants
  50open Constants.BoltzmannConstant
  51open Quantum.HolographicBound
  52open RecognitionBandwidth
  53
  54/-! ## §1. The Consciousness Barrier -/
  55
  56/-- The consciousness barrier: lcm(8, 45) = 360 ticks.
  57
  58    Over this period, zero accumulated cost forces zero defect at every step.
  59    This is the minimum integration time for consciousness. -/
  60def barrierTicks : ℕ := 360
  61
  62/-- 360 = 8 × 45: the barrier spans exactly 45 octaves. -/
  63theorem barrier_eq : barrierTicks = 8 * 45 := by norm_num [barrierTicks]
  64
  65/-- 360 = lcm(8, 45). -/
  66theorem barrier_is_lcm : barrierTicks = Nat.lcm 8 45 := by
  67  native_decide
  68
  69/-- The barrier period in RS time units (τ₀ = 1). -/
  70noncomputable def barrierPeriod : ℝ := (barrierTicks : ℝ) * τ₀
  71
  72theorem barrierPeriod_pos : 0 < barrierPeriod := by
  73  unfold barrierPeriod τ₀ tick barrierTicks
  74  norm_num
  75
  76theorem barrierPeriod_eq : barrierPeriod = 360 := by
  77  unfold barrierPeriod τ₀ tick barrierTicks
  78  norm_num
  79
  80/-! ## §2. Boundary Area Model -/
  81
  82/-- Boundary area as a function of extent (spherical boundary).
  83    A(L) = 4πL². -/
  84noncomputable def boundaryArea (L : ℝ) : ℝ := 4 * Real.pi * L ^ 2
  85
  86theorem boundaryArea_pos {L : ℝ} (hL : 0 < L) : 0 < boundaryArea L := by
  87  unfold boundaryArea
  88  exact mul_pos (mul_pos (by linarith) Real.pi_pos) (sq_pos_of_pos hL)
  89
  90theorem boundaryArea_monotone {L₁ L₂ : ℝ} (h₁ : 0 < L₁) (h : L₁ ≤ L₂) :
  91    boundaryArea L₁ ≤ boundaryArea L₂ := by
  92  unfold boundaryArea
  93  apply mul_le_mul_of_nonneg_left
  94  · exact sq_le_sq' (by linarith) h
  95  · exact mul_nonneg (by linarith) (le_of_lt Real.pi_pos)
  96
  97/-! ## §3. Holographic Maintenance Budget -/
  98
  99/-- Maximum recognition events available over the barrier period
 100    for a boundary of extent L:
 101
 102        N_budget(L) = boundaryArea(L) / (4ℓ_P² · k_R) -/
 103noncomputable def maintenanceBudget (L : ℝ) : ℝ :=
 104  boundaryArea L / (4 * planckArea * k_R)
 105
 106theorem maintenanceBudget_pos {L : ℝ} (hL : 0 < L) : 0 < maintenanceBudget L := by
 107  unfold maintenanceBudget
 108  apply div_pos (boundaryArea_pos hL)
 109  apply mul_pos
 110  apply mul_pos
 111  · linarith [planckArea_pos]
 112  · exact planckArea_pos
 113  · exact k_R_pos
 114
 115/-! ## §4. Maintenance Demand -/
 116
 117/-- The recognition cost of maintaining a boundary of extent L over the
 118    consciousness barrier period.
 119
 120    Uses the canonical J-cost: BoundaryCost = τ · J(L/λ_rec).
 121    In RS-native units with λ_rec = ℓ_P = 1:
 122        demand = 360 · J(L). -/
 123noncomputable def maintenanceDemand (L : ℝ) : ℝ :=
 124  barrierPeriod * Cost.Jcost L
 125
 126/-- Maintenance demand is nonneg for positive extent. -/
 127theorem maintenanceDemand_nonneg {L : ℝ} (hL : 0 < L) :
 128    0 ≤ maintenanceDemand L := by
 129  unfold maintenanceDemand
 130  exact mul_nonneg (le_of_lt barrierPeriod_pos) (Cost.Jcost_nonneg hL)
 131
 132/-- Maintenance demand is zero only when L = 1 (identity scale). -/
 133theorem maintenanceDemand_zero_iff {L : ℝ} (hL : 0 < L) :
 134    maintenanceDemand L = 0 ↔ L = 1 := by
 135  unfold maintenanceDemand
 136  constructor
 137  · intro h
 138    have hb := barrierPeriod_pos
 139    have := (mul_eq_zero.mp h).resolve_left (ne_of_gt hb)
 140    exact (Cost.Jcost_eq_zero_iff L hL).mp this
 141  · intro h
 142    rw [h, Cost.Jcost_unit0, mul_zero]
 143
 144/-! ## §5. Critical Extent -/
 145
 146/-- A boundary is **holographically viable** if its maintenance demand
 147    does not exceed its holographic budget.
 148
 149    This is the bandwidth constraint on consciousness. -/
 150def IsViable (L : ℝ) : Prop :=
 151  maintenanceDemand L ≤ maintenanceBudget L
 152
 153/-- The identity scale L = 1 is always viable (zero demand). -/
 154theorem identity_viable : IsViable 1 := by
 155  unfold IsViable maintenanceDemand maintenanceBudget
 156  rw [Cost.Jcost_unit0, mul_zero]
 157  exact le_of_lt (maintenanceBudget_pos (by norm_num : (0:ℝ) < 1))
 158
 159/-! ## §6. Z-Complexity Increases Demand -/
 160
 161/-- For a system with Z-complexity (conserved information integer),
 162    the maintenance demand scales with complexity:
 163
 164        demand(L, Z) = barrierPeriod · J(L) · (1 + |Z| · k_R)
 165
 166    Higher Z requires more recognition events per barrier cycle. -/
 167noncomputable def complexDemand (L : ℝ) (Z : ℤ) : ℝ :=
 168  maintenanceDemand L * (1 + |Z| * k_R)
 169
 170/-- Complex demand ≥ simple demand for any Z. -/
 171theorem complexDemand_ge {L : ℝ} (hL : 0 < L) (Z : ℤ) :
 172    maintenanceDemand L ≤ complexDemand L Z := by
 173  unfold complexDemand
 174  have hd := maintenanceDemand_nonneg hL
 175  have hfac : 1 ≤ 1 + ↑|Z| * k_R := by
 176    have : 0 ≤ ↑|Z| * k_R := mul_nonneg (by exact_mod_cast abs_nonneg Z) (le_of_lt k_R_pos)
 177    linarith
 178  calc maintenanceDemand L
 179      = maintenanceDemand L * 1 := (mul_one _).symm
 180    _ ≤ maintenanceDemand L * (1 + ↑|Z| * k_R) := by
 181        apply mul_le_mul_of_nonneg_left hfac hd
 182
 183/-- Higher Z-complexity strictly increases demand (when J > 0). -/
 184theorem higher_Z_more_demand {L : ℝ} (hL : 0 < L) (hL1 : L ≠ 1)
 185    {Z₁ Z₂ : ℤ} (hZ : |Z₁| < |Z₂|) :
 186    complexDemand L Z₁ < complexDemand L Z₂ := by
 187  unfold complexDemand
 188  have hd : 0 < maintenanceDemand L := by
 189    unfold maintenanceDemand
 190    apply mul_pos barrierPeriod_pos
 191    have : Cost.Jcost L ≠ 0 := by
 192      intro h
 193      exact hL1 ((Cost.Jcost_eq_zero_iff L hL).mp h)
 194    exact lt_of_le_of_ne (Cost.Jcost_nonneg hL) (Ne.symm this)
 195  apply mul_lt_mul_of_pos_left _ hd
 196  have : (↑|Z₁| : ℝ) < ↑|Z₂| := Int.cast_lt.mpr hZ
 197  linarith [mul_lt_mul_of_pos_right this k_R_pos]
 198
 199end ConsciousnessBandwidth
 200end Unification
 201end IndisputableMonolith
 202