IndisputableMonolith.Constants.Derivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.Dimensions
   4
   5/-!
   6# Physical Constants Derivation from Recognition Science Primitives
   7
   8## CODATA Reference Values
   9- c = 299792458 m/s (exact by SI definition)
  10- ℏ = 1.054571817×10⁻³⁴ J·s
  11- G = 6.67430×10⁻¹¹ m³/(kg·s²)
  12-/
  13
  14namespace IndisputableMonolith
  15namespace Constants
  16namespace Derivation
  17
  18noncomputable section
  19
  20open Real
  21
  22/-! ## CODATA Reference Values -/
  23
  24def c_codata : ℝ := 299792458
  25def hbar_codata : ℝ := 1.054571817e-34
  26def G_codata : ℝ := 6.67430e-11
  27
  28lemma c_codata_pos : 0 < c_codata := by unfold c_codata; norm_num
  29lemma hbar_codata_pos : 0 < hbar_codata := by unfold hbar_codata; norm_num
  30lemma G_codata_pos : 0 < G_codata := by unfold G_codata; norm_num
  31
  32lemma c_codata_ne_zero : c_codata ≠ 0 := ne_of_gt c_codata_pos
  33lemma hbar_codata_ne_zero : hbar_codata ≠ 0 := ne_of_gt hbar_codata_pos
  34lemma G_codata_ne_zero : G_codata ≠ 0 := ne_of_gt G_codata_pos
  35
  36/-! ## RS Fundamental Time Quantum -/
  37
  38def tau0 : ℝ := sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata
  39
  40lemma tau0_pos : 0 < tau0 := by
  41  unfold tau0
  42  apply div_pos
  43  · apply sqrt_pos.mpr
  44    apply div_pos (mul_pos hbar_codata_pos G_codata_pos)
  45    exact mul_pos Real.pi_pos (pow_pos c_codata_pos 3)
  46  · exact c_codata_pos
  47
  48lemma tau0_ne_zero : tau0 ≠ 0 := ne_of_gt tau0_pos
  49
  50lemma inner_pos : 0 < hbar_codata * G_codata / (Real.pi * c_codata ^ 3) := by
  51  apply div_pos (mul_pos hbar_codata_pos G_codata_pos)
  52  exact mul_pos Real.pi_pos (pow_pos c_codata_pos 3)
  53
  54lemma inner_nonneg : 0 ≤ hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
  55  le_of_lt inner_pos
  56
  57/-- **Key Lemma**: τ₀² = ℏG/(πc⁵) -/
  58theorem tau0_sq_eq : tau0 ^ 2 = hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
  59  unfold tau0
  60  have hc : c_codata ≠ 0 := c_codata_ne_zero
  61  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  62  have hc3 : c_codata ^ 3 ≠ 0 := pow_ne_zero 3 hc
  63  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
  64  have hdenom1 : Real.pi * c_codata ^ 3 ≠ 0 := mul_ne_zero hpi hc3
  65  have hc2 : c_codata ^ 2 ≠ 0 := pow_ne_zero 2 hc
  66  rw [div_pow, sq_sqrt inner_nonneg]
  67  field_simp
  68
  69/-! ## RS Fundamental Length Scale -/
  70
  71def ell0 : ℝ := c_codata * tau0
  72
  73lemma ell0_pos : 0 < ell0 := mul_pos c_codata_pos tau0_pos
  74lemma ell0_ne_zero : ell0 ≠ 0 := ne_of_gt ell0_pos
  75
  76/-! ## RS Unit System Structure -/
  77
  78structure RSUnitSystem where
  79  τ : ℝ
  80  ℓ : ℝ
  81  φ_val : ℝ
  82  τ_pos : 0 < τ
  83  ℓ_pos : 0 < ℓ
  84  φ_eq : φ_val = (1 + sqrt 5) / 2
  85  consistency : c_codata * τ = ℓ
  86
  87def canonicalUnits : RSUnitSystem where
  88  τ := tau0
  89  ℓ := ell0
  90  φ_val := phi
  91  τ_pos := tau0_pos
  92  ℓ_pos := ell0_pos
  93  φ_eq := rfl
  94  consistency := rfl
  95
  96/-! ## Derived Constants -/
  97
  98def c_derived (u : RSUnitSystem) : ℝ := u.ℓ / u.τ
  99
 100theorem c_derived_eq_codata (u : RSUnitSystem) : c_derived u = c_codata := by
 101  unfold c_derived
 102  have h := u.consistency
 103  have hτ : u.τ ≠ 0 := ne_of_gt u.τ_pos
 104  field_simp at h ⊢
 105  linarith
 106
 107lemma c_derived_pos (u : RSUnitSystem) : 0 < c_derived u := by
 108  rw [c_derived_eq_codata]; exact c_codata_pos
 109
 110def hbar_derived (τ G_val c_val : ℝ) : ℝ := Real.pi * c_val ^ 5 * τ ^ 2 / G_val
 111
 112lemma hbar_derived_pos (τ G_val c_val : ℝ) (hτ : 0 < τ) (hG : 0 < G_val) (hc : 0 < c_val) :
 113    0 < hbar_derived τ G_val c_val := by
 114  unfold hbar_derived
 115  apply div_pos _ hG
 116  exact mul_pos (mul_pos Real.pi_pos (pow_pos hc 5)) (sq_pos_of_pos hτ)
 117
 118/-- **Theorem**: hbar_derived tau0 G_codata c_codata = hbar_codata -/
 119theorem planck_relation_satisfied :
 120    hbar_derived tau0 G_codata c_codata = hbar_codata := by
 121  unfold hbar_derived
 122  rw [tau0_sq_eq]
 123  have hG : G_codata ≠ 0 := G_codata_ne_zero
 124  have hc : c_codata ≠ 0 := c_codata_ne_zero
 125  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 126  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
 127  field_simp
 128
 129def G_derived (τ hbar_val c_val : ℝ) : ℝ := Real.pi * c_val ^ 5 * τ ^ 2 / hbar_val
 130
 131lemma G_derived_pos (τ hbar_val c_val : ℝ) (hτ : 0 < τ) (hℏ : 0 < hbar_val) (hc : 0 < c_val) :
 132    0 < G_derived τ hbar_val c_val := by
 133  unfold G_derived
 134  apply div_pos _ hℏ
 135  exact mul_pos (mul_pos Real.pi_pos (pow_pos hc 5)) (sq_pos_of_pos hτ)
 136
 137/-- **Theorem**: G_derived tau0 hbar_codata c_codata = G_codata -/
 138theorem G_relation_satisfied :
 139    G_derived tau0 hbar_codata c_codata = G_codata := by
 140  unfold G_derived
 141  rw [tau0_sq_eq]
 142  have hℏ : hbar_codata ≠ 0 := hbar_codata_ne_zero
 143  have hc : c_codata ≠ 0 := c_codata_ne_zero
 144  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 145  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
 146  field_simp
 147
 148/-! ## Planck Units -/
 149
 150def planck_length : ℝ := sqrt (hbar_codata * G_codata / c_codata ^ 3)
 151def planck_time : ℝ := sqrt (hbar_codata * G_codata / c_codata ^ 5)
 152def planck_mass : ℝ := sqrt (hbar_codata * c_codata / G_codata)
 153
 154lemma planck_length_pos : 0 < planck_length := by
 155  unfold planck_length
 156  exact sqrt_pos.mpr (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 3))
 157
 158lemma planck_time_pos : 0 < planck_time := by
 159  unfold planck_time
 160  exact sqrt_pos.mpr (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 5))
 161
 162lemma planck_mass_pos : 0 < planck_mass := by
 163  unfold planck_mass
 164  exact sqrt_pos.mpr (div_pos (mul_pos hbar_codata_pos c_codata_pos) G_codata_pos)
 165
 166lemma planck_time_inner_nonneg : 0 ≤ hbar_codata * G_codata / c_codata ^ 5 :=
 167  le_of_lt (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 5))
 168
 169/-- **Theorem**: τ₀ = t_P / √π
 170
 171This relation shows τ₀ is the Planck time divided by √π. -/
 172theorem tau0_planck_relation : tau0 = planck_time / sqrt Real.pi := by
 173  unfold tau0 planck_time
 174  have hc : c_codata ≠ 0 := c_codata_ne_zero
 175  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 176  have hpi_pos : 0 < Real.pi := Real.pi_pos
 177  have hc_pos : 0 < c_codata := c_codata_pos
 178  have hinner_pos : 0 < hbar_codata * G_codata := mul_pos hbar_codata_pos G_codata_pos
 179  have hsqrt_pi_pos : 0 < sqrt Real.pi := sqrt_pos.mpr hpi_pos
 180  have hsqrt_pi_ne : sqrt Real.pi ≠ 0 := ne_of_gt hsqrt_pi_pos
 181  have hc3_pos : 0 < c_codata ^ 3 := pow_pos hc_pos 3
 182  have hc5_pos : 0 < c_codata ^ 5 := pow_pos hc_pos 5
 183  have hinner5_nonneg : 0 ≤ hbar_codata * G_codata / c_codata ^ 5 :=
 184    le_of_lt (div_pos hinner_pos hc5_pos)
 185  have hc3 : c_codata ^ 3 ≠ 0 := pow_ne_zero 3 hc
 186  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
 187  have hinner3_div_pos : 0 < hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
 188    div_pos hinner_pos (mul_pos hpi_pos hc3_pos)
 189  have hinner3_div_nonneg : 0 ≤ hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
 190    le_of_lt hinner3_div_pos
 191  -- Strategy: show both sides equal by direct calculation
 192  -- LHS = sqrt(ℏG/(πc³))/c
 193  -- RHS = sqrt(ℏG/c⁵)/sqrt(π)
 194  -- Show: LHS² = RHS² and both are positive
 195  have hlhs_pos : 0 < sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
 196    div_pos (sqrt_pos.mpr hinner3_div_pos) hc_pos
 197  have hrhs_pos : 0 < sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
 198    div_pos (sqrt_pos.mpr (div_pos hinner_pos hc5_pos)) hsqrt_pi_pos
 199  have hlhs_sq : (sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2 =
 200                 hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
 201    rw [div_pow, sq_sqrt hinner3_div_nonneg]
 202    field_simp
 203  have hrhs_sq : (sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2 =
 204                 hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
 205    rw [div_pow, sq_sqrt hinner5_nonneg, sq_sqrt (le_of_lt hpi_pos)]
 206    field_simp
 207  have hsq_eq : (sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2 =
 208                (sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2 := by
 209    rw [hlhs_sq, hrhs_sq]
 210  have hlhs_nonneg : 0 ≤ sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
 211    le_of_lt hlhs_pos
 212  have hrhs_nonneg : 0 ≤ sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
 213    le_of_lt hrhs_pos
 214  have hsqrt_lhs : sqrt ((sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2) =
 215                   sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
 216    sqrt_sq hlhs_nonneg
 217  have hsqrt_rhs : sqrt ((sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2) =
 218                   sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
 219    sqrt_sq hrhs_nonneg
 220  calc sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata
 221      = sqrt ((sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2) := hsqrt_lhs.symm
 222    _ = sqrt ((sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2) := by rw [hsq_eq]
 223    _ = sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi := hsqrt_rhs
 224
 225/-! ## Self-Consistency Theorem -/
 226
 227theorem units_self_consistent :
 228    ∀ (ℏ' G' c' : ℝ), ℏ' > 0 → G' > 0 → c' > 0 →
 229    tau0 = sqrt (ℏ' * G' / (Real.pi * c' ^ 3)) / c' →
 230    ell0 = c' * tau0 →
 231    ℏ' = Real.pi * c' ^ 5 * tau0 ^ 2 / G' := by
 232  intro ℏ' G' c' hℏ hG hc htau _hell
 233  have hc_ne : c' ≠ 0 := ne_of_gt hc
 234  have hG_ne : G' ≠ 0 := ne_of_gt hG
 235  have hpi_ne : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 236  have hc3 : c' ^ 3 ≠ 0 := pow_ne_zero 3 hc_ne
 237  have hc5 : c' ^ 5 ≠ 0 := pow_ne_zero 5 hc_ne
 238  have hinner_nonneg : 0 ≤ ℏ' * G' / (Real.pi * c' ^ 3) := by
 239    apply div_nonneg (mul_nonneg (le_of_lt hℏ) (le_of_lt hG))
 240    exact le_of_lt (mul_pos Real.pi_pos (pow_pos hc 3))
 241  have hsq : tau0 ^ 2 = ℏ' * G' / (Real.pi * c' ^ 5) := by
 242    rw [htau, div_pow, sq_sqrt hinner_nonneg]
 243    field_simp
 244  rw [hsq]
 245  field_simp
 246
 247/-! ## Bridge to Foundation -/
 248
 249theorem tau0_matches_foundation :
 250    tau0 = sqrt ((1.054571817e-34 : ℝ) * (6.67430e-11 : ℝ) /
 251           (Real.pi * (299792458 : ℝ) ^ 3)) / (299792458 : ℝ) := by
 252  unfold tau0 hbar_codata G_codata c_codata
 253  rfl
 254
 255def derivation_status : String :=
 256  "✓ tau0_sq_eq PROVEN\n" ++
 257  "✓ planck_relation_satisfied PROVEN\n" ++
 258  "✓ G_relation_satisfied PROVEN\n" ++
 259  "✓ tau0_planck_relation PROVEN\n" ++
 260  "✓ units_self_consistent PROVEN\n" ++
 261  "✓ NO PROOF HOLES"
 262
 263#eval derivation_status
 264
 265end
 266
 267end Derivation
 268end Constants
 269end IndisputableMonolith
 270