beta_running

formal source

  29
  30/-- The running exponent for gravitational strengthening.
  31    β = -(φ - 1) / φ^5 ≈ -0.056. -/
  32noncomputable def beta_running : ℝ := -(phi - 1) / (phi ^ 5)
  33
  34/-- Numerical bound for beta_running ≈ -0.0557.
  35    Proved using φ ∈ (1.61, 1.62). -/
  36theorem beta_running_bounds :
  37    -0.06 < beta_running ∧ beta_running < -0.05 := by
  38  unfold beta_running
  39  -- Use phi_fifth_eq: φ^5 = 5φ + 3
  40  rw [phi_fifth_eq]
  41  -- We want to prove: -0.06 < -(φ - 1) / (5φ + 3) < -0.05
  42  have h_phi_pos : 0 < phi := phi_pos
  43  have h_denom_pos : 0 < 5 * phi + 3 := by linarith
  44  constructor
  45  · -- -0.06 < -(φ - 1) / (5φ + 3)
  46    rw [lt_div_iff₀ h_denom_pos]
  47    have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
  48    linarith
  49  · -- -(φ - 1) / (5φ + 3) < -0.05
  50    rw [div_lt_iff₀ h_denom_pos]
  51    have h_phi_gt : 1.61 < phi := phi_gt_onePointSixOne
  52    linarith
  53
  54/-- Effective G at scale r relative to G_infinity. -/
  55noncomputable def G_ratio (r r_ref : ℝ) : ℝ :=
  56    1 + abs beta_running * (r / r_ref) ^ beta_running
  57
  58/-- **HYPOTHESIS H_GravitationalRunning**: Gravity strengthens at nm scales.
  59    Prediction: G(20nm) / G_inf ≈ 32. -/
  60def H_GravitationalRunning : Prop :=
  61  ∃ r_ref : ℝ, r_ref > 0 ∧
  62    abs (G_ratio 20e-9 r_ref - 32) < 1.0