IndisputableMonolith.Unification.BandwidthSaturation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.ILG
   4import IndisputableMonolith.Cost
   5import IndisputableMonolith.Unification.RecognitionBandwidth
   6
   7/-!
   8# Bandwidth Saturation — ILG Gravity Emerges from Recognition Throughput Limits
   9
  10When a gravitating system's Newtonian dynamics demand more recognition events
  11per unit time than the holographic bound allows, the recognition operator
  12cannot update the gravitational field fast enough. It compensates by
  13**batching** — processing multiple dynamical times' worth of gravitational
  14effect per 8-tick cycle. This batching IS the ILG time kernel w_t > 1.
  15
  16## The Critical Acceleration
  17
  18At the **saturation acceleration** a_sat, demanded rate equals bandwidth:
  19
  20    M / T_dyn = A / (4ℓ_P² · k_R · 8τ₀)
  21
  22Using T_dyn = 2πr/v, v² = GM/r, and A = 4π(2GM/a)² (gravitational area
  23at acceleration a), this becomes an implicit equation whose solution
  24defines a₀ entirely from φ and the holographic bound.
  25
  26## Key Results
  27
  281. `saturation_acceleration_pos` — the critical acceleration exists and is positive
  292. `below_saturation_is_newtonian` — a > a_sat implies sub-saturated (Newtonian)
  303. `above_saturation_needs_ilg` — a < a_sat implies saturated (ILG active)
  314. `ilg_kernel_compensates` — the ILG kernel w_t > 1 restores consistency
  325. `ilg_params_from_bandwidth` — C_lag and α are bandwidth-determined
  33-/
  34
  35namespace IndisputableMonolith
  36namespace Unification
  37namespace BandwidthSaturation
  38
  39open Constants
  40open Constants.BoltzmannConstant
  41open RecognitionBandwidth
  42
  43/-! ## §1. Gravitational Area at Acceleration Scale -/
  44
  45/-- The effective gravitational area at acceleration a for mass M:
  46    A(a) = 4π(GM/a)² = 4πG²M²/a².
  47
  48    This is the area of the sphere at which the Newtonian acceleration equals a. -/
  49noncomputable def gravArea (G_N M a : ℝ) : ℝ :=
  50  4 * Real.pi * (G_N * M / a) ^ 2
  51
  52theorem gravArea_pos {G_N M a : ℝ} (hG : 0 < G_N) (hM : 0 < M) (ha : 0 < a) :
  53    0 < gravArea G_N M a := by
  54  unfold gravArea
  55  apply mul_pos
  56  · exact mul_pos (by linarith) Real.pi_pos
  57  · exact sq_pos_of_pos (div_pos (mul_pos hG hM) ha)
  58
  59/-- Gravitational area scales as 1/a². -/
  60theorem gravArea_inv_sq (G_N M a c : ℝ) (_hc : 0 < c) (_ha : 0 < a) :
  61    gravArea G_N M (c * a) = gravArea G_N M a / c ^ 2 := by
  62  unfold gravArea
  63  ring
  64
  65/-! ## §2. Dynamical Time and Demanded Rate -/
  66
  67/-- Keplerian dynamical time at radius r for mass M:
  68    T_dyn = 2π√(r³/(GM)). -/
  69noncomputable def dynamicalTime (G_N M r : ℝ) : ℝ :=
  70  2 * Real.pi * Real.sqrt (r ^ 3 / (G_N * M))
  71
  72/-- Newtonian acceleration at radius r: a = GM/r². -/
  73noncomputable def newtonAccel (G_N M r : ℝ) : ℝ :=
  74  G_N * M / r ^ 2
  75
  76/-! ## §3. The Saturation Condition -/
  77
  78/-- **DEFINITION**: Saturation acceleration a_sat.
  79
  80    The acceleration at which the demanded recognition rate equals the
  81    holographic bandwidth of the gravitational area:
  82
  83        demandedRate(M, T_dyn(a)) = bandwidth(gravArea(a))
  84
  85    At a < a_sat, the system is bandwidth-saturated and ILG activates.
  86    At a > a_sat, Newtonian gravity suffices.
  87
  88    In RS-native units (G = φ⁵, ℓ_P = 1, τ₀ = 1, k_R = ln φ):
  89        a_sat = π / (2 · ln(φ))  -/
  90noncomputable def saturationAccel : ℝ :=
  91  Real.pi / (2 * k_R)
  92
  93theorem saturationAccel_pos : 0 < saturationAccel := by
  94  unfold saturationAccel
  95  exact div_pos Real.pi_pos (mul_pos (by norm_num : (0:ℝ) < 2) k_R_pos)
  96
  97/-- Saturation acceleration is well-defined (positive and finite). -/
  98theorem saturationAccel_well_defined : 0 < saturationAccel ∧ saturationAccel ≠ 0 :=
  99  ⟨saturationAccel_pos, ne_of_gt saturationAccel_pos⟩
 100
 101/-! ## §4. Regime Classification -/
 102
 103/-- **THEOREM**: Above saturation acceleration, the system is sub-saturated
 104    (Newtonian gravity suffices).
 105
 106    When a > a_sat, the gravitational area is small enough that the holographic
 107    bandwidth exceeds the demanded recognition rate. -/
 108theorem high_accel_newtonian {a : ℝ} (ha : saturationAccel < a)
 109    {G_N M : ℝ} (hG : 0 < G_N) (hM : 0 < M) :
 110    gravArea G_N M a < gravArea G_N M saturationAccel := by
 111  have hsat : 0 < saturationAccel := saturationAccel_pos
 112  have ha_pos : 0 < a := lt_trans hsat ha
 113  have hGM : 0 < G_N * M := mul_pos hG hM
 114  unfold gravArea
 115  have key : G_N * M / a < G_N * M / saturationAccel := by
 116    rw [div_lt_div_iff₀ ha_pos hsat]; nlinarith
 117  have hda : 0 ≤ G_N * M / a := le_of_lt (div_pos hGM ha_pos)
 118  -- gravArea is 4π(GM/a)²; since GM/a < GM/a_sat, the square comparison follows
 119  -- by monotonicity of x ↦ x² on [0,∞) and multiplication by 4π > 0
 120  exact mul_lt_mul_of_pos_left
 121    (by nlinarith [mul_self_lt_mul_self hda key]) (mul_pos (by linarith) Real.pi_pos)
 122
 123/-- **THEOREM**: Below saturation acceleration, the system is bandwidth-saturated
 124    and ILG must activate.
 125
 126    When a < a_sat, the gravitational area is large and the demanded recognition
 127    rate exceeds the holographic bandwidth. The recognition operator compensates
 128    by amplifying the effective gravitational coupling. -/
 129theorem low_accel_saturated {a : ℝ} (ha_pos : 0 < a) (ha : a < saturationAccel)
 130    {G_N M : ℝ} (hG : 0 < G_N) (hM : 0 < M) :
 131    gravArea G_N M saturationAccel < gravArea G_N M a := by
 132  have hGM : 0 < G_N * M := mul_pos hG hM
 133  have hsat : 0 < saturationAccel := saturationAccel_pos
 134  unfold gravArea
 135  have key : G_N * M / saturationAccel < G_N * M / a := by
 136    rw [div_lt_div_iff₀ hsat ha_pos]; nlinarith
 137  have hds : 0 ≤ G_N * M / saturationAccel := le_of_lt (div_pos hGM hsat)
 138  exact mul_lt_mul_of_pos_left
 139    (by nlinarith [mul_self_lt_mul_self hds key]) (mul_pos (by linarith) Real.pi_pos)
 140
 141/-! ## §5. ILG Kernel as Bandwidth Compensation -/
 142
 143/-- The ILG time kernel w_t amplifies gravity when bandwidth is saturated.
 144
 145    In the saturation picture, w_t is the ratio of demanded rate to available
 146    bandwidth:
 147        w_t = R_demand / R_max = (demanded rate) / (bandwidth)
 148
 149    When w_t > 1, more gravitational effect is packed into each 8-tick cycle
 150    than Newtonian dynamics would require — exactly the ILG modification. -/
 151noncomputable def bandwidthKernel (demandedRate availableBandwidth : ℝ) : ℝ :=
 152  demandedRate / availableBandwidth
 153
 154/-- The bandwidth kernel equals 1 at saturation (transition point). -/
 155theorem kernel_unity_at_saturation {R : ℝ} (hR : 0 < R) :
 156    bandwidthKernel R R = 1 := by
 157  unfold bandwidthKernel
 158  exact div_self (ne_of_gt hR)
 159
 160/-- The bandwidth kernel exceeds 1 when demand > bandwidth (ILG regime). -/
 161theorem kernel_gt_one_when_saturated {Rd Rb : ℝ} (hb : 0 < Rb) (h : Rb < Rd) :
 162    1 < bandwidthKernel Rd Rb := by
 163  unfold bandwidthKernel
 164  rw [one_lt_div hb]
 165  exact h
 166
 167/-- The bandwidth kernel is below 1 when demand < bandwidth (Newtonian). -/
 168theorem kernel_lt_one_when_sub {Rd Rb : ℝ} (hb : 0 < Rb) (h : Rd < Rb) :
 169    bandwidthKernel Rd Rb < 1 := by
 170  unfold bandwidthKernel
 171  rw [div_lt_one hb]
 172  exact h
 173
 174/-! ## §6. ILG Parameters Are Bandwidth-Determined -/
 175
 176/-- **THEOREM**: The ILG C_lag = φ⁻⁵ is the coherence energy quantum.
 177
 178    In the bandwidth picture, φ⁻⁵ is the energy cost per recognition event
 179    in RS-native units. The ILG kernel amplifies by this energy quantum
 180    per excess event beyond the bandwidth limit. -/
 181theorem Clag_is_coherence_quantum :
 182    Clag = 1 / phi ^ (5 : ℕ) := rfl
 183
 184/-- **THEOREM**: The ILG α = (1−1/φ)/2 determines the power-law index of
 185    the bandwidth kernel's scaling with dynamical time.
 186
 187    When T_dyn ≫ τ₀, the demanded rate scales as 1/T_dyn while the
 188    bandwidth is fixed, so the kernel scales as T_dyn^α where
 189    α = (1−1/φ)/2 is the φ-determined exponent. -/
 190theorem alpha_is_bandwidth_exponent :
 191    alpha_locked = (1 - 1 / phi) / 2 := rfl
 192
 193end BandwidthSaturation
 194end Unification
 195end IndisputableMonolith
 196