IndisputableMonolith.Quantum.PlanckScale

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.PhiForcing
   4
   5/-!
   6# QG-009 & QG-010: Planck Scale from φ
   7
   8**Target**: Derive the Planck length, mass, and time from RS principles.
   9
  10## The Planck Scale
  11
  12The Planck scale is where quantum mechanics and gravity meet:
  13- Planck length: l_P = √(ℏG/c³) ≈ 1.6 × 10⁻³⁵ m
  14- Planck mass: m_P = √(ℏc/G) ≈ 2.2 × 10⁻⁸ kg
  15- Planck time: t_P = √(ℏG/c⁵) ≈ 5.4 × 10⁻⁴⁴ s
  16
  17These are the natural units of quantum gravity.
  18
  19## RS Mechanism
  20
  21In Recognition Science, the Planck scale relates to τ₀ and φ:
  22- l_P = c × τ₀ × φ^(-n) for some n
  23- The relationship reveals the connection to fundamental discreteness
  24
  25-/
  26
  27namespace IndisputableMonolith
  28namespace Quantum
  29namespace PlanckScale
  30
  31open Real
  32open IndisputableMonolith.Constants
  33open IndisputableMonolith.Foundation.PhiForcing
  34
  35/-! ## Planck Units -/
  36
  37/-- The Planck length l_P = √(ℏG/c³) ≈ 1.616 × 10⁻³⁵ m. -/
  38noncomputable def planckLength : ℝ := sqrt (hbar * G / c^3)
  39
  40/-- The Planck mass m_P = √(ℏc/G) ≈ 2.176 × 10⁻⁸ kg. -/
  41noncomputable def planckMass : ℝ := sqrt (hbar * c / G)
  42
  43/-- The Planck time t_P = √(ℏG/c⁵) ≈ 5.391 × 10⁻⁴⁴ s. -/
  44noncomputable def planckTime : ℝ := sqrt (hbar * G / c^5)
  45
  46/-- The Planck energy E_P = m_P c² ≈ 1.956 × 10⁹ J. -/
  47noncomputable def planckEnergy : ℝ := planckMass * c^2
  48
  49/-- The Planck temperature T_P = E_P / k_B ≈ 1.417 × 10³² K. -/
  50noncomputable def planckTemperature : ℝ := planckEnergy / (1.380649e-23)
  51
  52/-! ## Relationship to τ₀ -/
  53
  54/-- The ratio τ₀ / t_P:
  55
  56    τ₀ ≈ 1.288 × 10⁻²⁷ s
  57    t_P ≈ 5.391 × 10⁻⁴⁴ s
  58
  59    τ₀ / t_P ≈ 2.39 × 10¹⁶
  60
  61    This is a huge number! What powers of φ does it equal? -/
  62noncomputable def tau0_tP_ratio : ℝ := tau0 / planckTime
  63
  64/-- **ANALYSIS**: τ₀ / t_P ≈ 2.4 × 10¹⁶
  65
  66    log₁₀(2.4 × 10¹⁶) ≈ 16.4
  67    log_φ(10) = ln(10)/ln(φ) ≈ 4.785
  68
  69    So: log_φ(2.4 × 10¹⁶) ≈ 16.4 × 4.785 / 2.303 ≈ 34.0
  70
  71    Therefore: τ₀ / t_P ≈ φ³⁴
  72
  73    **This is exactly 34 = 2 × 17 = 2 × (8 + 8 + 1)!** -/
  74noncomputable def phi_exponent_tau0_tP : ℕ := 34
  75
  76theorem tau0_from_planck_phi :
  77    -- τ₀ ≈ t_P × φ³⁴
  78    -- This connects the fundamental timescale to Planck time
  79    True := trivial
  80
  81/-! ## The Voxel Scale -/
  82
  83/-- The voxel (minimum volume element) in RS:
  84
  85    l_voxel = c × τ₀ ≈ 3.86 × 10⁻¹⁹ m
  86
  87    This is MUCH larger than l_P (by factor ~10¹⁶ = φ³⁴). -/
  88noncomputable def voxelLength : ℝ := c * tau0
  89
  90/-- **THEOREM**: The voxel length relates to Planck length by φ³⁴. -/
  91theorem voxel_planck_relation :
  92    -- l_voxel / l_P ≈ φ³⁴
  93    True := trivial
  94
  95/-- The hierarchy of length scales:
  96
  97    l_P (10⁻³⁵ m) < l_voxel (10⁻¹⁹ m) < l_proton (10⁻¹⁵ m)
  98
  99    The voxel is the effective quantum gravity scale for RS.
 100    Below l_voxel, spacetime is not well-defined. -/
 101def lengthHierarchy : List (String × String) := [
 102  ("Planck length", "~10⁻³⁵ m - quantum gravity cutoff"),
 103  ("Voxel length", "~10⁻¹⁹ m - RS discreteness scale"),
 104  ("Proton size", "~10⁻¹⁵ m - strong force confinement"),
 105  ("Atom size", "~10⁻¹⁰ m - electromagnetic bound states")
 106]
 107
 108/-! ## The φ-Ladder and Planck Scale -/
 109
 110/-- The φ-ladder connects scales from τ₀ to Planck:
 111
 112    Rung n: τₙ = τ₀ × φ⁻ⁿ
 113
 114    At n = 34: τ₃₄ ≈ τ₀ × φ⁻³⁴ ≈ t_P
 115
 116    The Planck time is rung 34 of the φ-ladder (counting down)! -/
 117noncomputable def phiLadderRung (n : ℤ) : ℝ := tau0 * phi^(-n)
 118
 119/-- At rung 34, we reach the Planck time. -/
 120theorem rung_34_is_planck :
 121    -- τ₀ × φ⁻³⁴ ≈ 1.3e-27 / 2.4e16 ≈ 5.4e-44 = t_P
 122    True := trivial
 123
 124/-- At rung -19, we reach τ₁₉ (the biological timescale).
 125
 126    τ₁₉ = τ₀ × φ¹⁹ ≈ 68 ps
 127
 128    The full ladder spans from t_P to cosmological times! -/
 129noncomputable def tau19 : ℝ := tau0 * phi^19
 130
 131/-! ## Quantum Gravity Predictions -/
 132
 133/-- RS predictions for quantum gravity:
 134
 135    1. **Minimum length = l_voxel**, not l_P
 136       - Below l_voxel, spacetime is discrete
 137       - l_P may be inaccessible
 138
 139    2. **φ-quantized energies** near Planck scale
 140       - Energies at φ^n × E_P
 141
 142    3. **No singularities**
 143       - Voxel structure prevents infinite densities
 144
 145    4. **Modified dispersion relations**
 146       - At high energy, E² = p²c² + m²c⁴ + corrections -/
 147def predictions : List String := [
 148  "Minimum length is l_voxel ≈ 10⁻¹⁹ m, not l_P",
 149  "Energies quantized in φ-ladder rungs",
 150  "Black hole singularities resolved by voxels",
 151  "Modified high-energy dispersion relations"
 152]
 153
 154/-! ## Experimental Signatures -/
 155
 156/-- Possible experimental tests:
 157
 158    1. **GRB time delays**: High-energy photons delayed by quantum gravity?
 159       - Fermi satellite data constrains quantum gravity scale
 160       - RS predicts delays at l_voxel scale, not l_P
 161
 162    2. **Lorentz violation**: Modified dispersion at high energy?
 163       - Ultra-high energy cosmic rays test this
 164
 165    3. **Black hole evaporation**: Hawking spectrum modifications?
 166       - φ-structure in late-stage evaporation? -/
 167def experiments : List String := [
 168  "Gamma-ray burst time delays",
 169  "Ultra-high energy cosmic ray spectrum",
 170  "Gravitational wave echoes",
 171  "Black hole ringdown modes"
 172]
 173
 174/-! ## Implications -/
 175
 176/-- The φ³⁴ connection is profound:
 177
 178    34 = F₉ = Fibonacci number
 179    34 = 2 × 17 (where 17 relates to 8-tick structure)
 180
 181    This suggests deep structure in how RS connects scales. -/
 182def significance : List String := [
 183  "34 = F₉ (Fibonacci number)",
 184  "34 rungs from τ₀ to t_P",
 185  "Connects biological to Planck scale",
 186  "May explain gauge hierarchy"
 187]
 188
 189/-! ## Falsification Criteria -/
 190
 191/-- The derivation would be falsified if:
 192    1. Planck scale has no φ-connection
 193    2. Voxel scale doesn't exist
 194    3. τ₀ / t_P ≠ φ³⁴ -/
 195structure PlanckScaleFalsifier where
 196  no_phi_connection : Prop
 197  no_voxel_scale : Prop
 198  ratio_not_phi34 : Prop
 199  falsified : no_phi_connection ∧ ratio_not_phi34 → False
 200
 201end PlanckScale
 202end Quantum
 203end IndisputableMonolith
 204