one_over_sqrt_pi_approx

formal source

 238  field_simp [h_sqrt_c3_ne, h_sqrt_pi_ne, h_sqrt_hG_ne]
 239
 240/-- **Numerical Value**: 1/√π ≈ 0.564. -/
 241theorem one_over_sqrt_pi_approx : abs (1 / sqrt Real.pi - 0.564) < 0.01 := by
 242  -- Verified by interval arithmetic.
 243  interval
 244
 245/-! ## Part 6: Connecting to Constants.lambda_rec -/
 246
 247/-- In RS-native units where c = ℓ₀ = τ₀ = 1, λ_rec = ell0 = 1.
 248    The physical content is the relationship λ_rec/ℓ_P = 1/√π.
 249
 250    The Planck gate identity: π · ℏ · G = c³ · λ_rec². -/
 251theorem planck_gate_identity :
 252    Real.pi * hbar * G = c^3 * lambda_rec^2 := by
 253  unfold G lambda_rec hbar c ell0 cLagLock tau0 tick
 254  simp only [one_pow, mul_one]
 255  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
 256  have hphi5 : phi ^ (-(5 : ℝ)) ≠ 0 := (Real.rpow_pos_of_pos phi_pos _).ne'
 257  field_simp [hpi, hphi5]
 258
 259/-- Equivalent form: c³λ²/(πℏG) = 1. -/
 260theorem planck_gate_normalized :
 261    c^3 * lambda_rec^2 / (Real.pi * hbar * G) = 1 := by
 262  have h := planck_gate_identity
 263  have hne : Real.pi * hbar * G ≠ 0 := by
 264    apply mul_ne_zero
 265    apply mul_ne_zero
 266    · exact Real.pi_pos.ne'
 267    · exact hbar_pos.ne'
 268    · exact G_pos.ne'
 269  rw [div_eq_one_iff_eq hne]
 270  exact h.symm
 271