units_self_consistent

formal source

 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" ++