H_rref_phi_ladder

formal source

 172GravityParameters.lean, or empirical input from short-range experiments. -/
 173
 174/-- The hypothesis that r_ref lives on the phi-ladder. -/
 175def H_rref_phi_ladder : Prop :=
 176  ∃ N : ℤ, ∃ r_ref : ℝ, r_ref = ell0 * phi ^ N ∧ r_ref > 0 ∧
 177    abs (G_ratio 20e-9 r_ref - 32) < 1
 178
 179/-! ## Q10: Casimir Force Correction
 180
 181Running G at 20nm gives G_eff ≈ 32 * G_inf. But G_inf ≈ 6.7e-11 makes
 182even the enhanced gravitational force negligible vs Casimir (~10 Pa at 20nm).
 183The fractional gravitational correction to Casimir is ≈ 2e-18. -/
 184
 185/-- Gravitational pressure between two plates. -/
 186def gravitational_pressure (G_val rho t enhancement : ℝ) : ℝ :=
 187  enhancement * G_val * rho ^ 2 * t ^ 2
 188
 189/-- The gravitational contribution is negligibly small vs Casimir. -/
 190theorem grav_casimir_ratio_negligible :
 191    gravitational_pressure 6.674e-11 1e4 1e-6 32 < 1e-10 := by
 192  unfold gravitational_pressure; norm_num
 193
 194/-! ## Explicit r_ref Formula (Path 1a)
 195
 196Setting G_ratio(r, r_ref) = target and solving for r_ref:
 197  target = 1 + |beta| * (r / r_ref)^beta
 198  (target - 1) / |beta| = (r / r_ref)^beta
 199  r_ref = r * ((target - 1) / |beta|)^(1/beta)
 200
 201Since beta < 0, the exponent 1/beta < 0, and (target-1)/|beta| > 1 for
 202target > 1 + |beta|, so r_ref > r (the reference scale is larger than
 203the measurement scale). -/
 204
 205/-- The explicit r_ref that gives G_ratio(r, r_ref) = target.