log_divergence

formal source

  47  "∫₀^∞ dk k³ / k² = ∫₀^∞ dk k diverges (logarithmically)"
  48
  49/-- **THEOREM**: Integrals of the form ∫₀^Λ dk/k = ln(Λ) diverge as Λ → ∞. -/
  50theorem log_divergence (Λ : ℝ) (hΛ : Λ > 1) :
  51    Real.log Λ > 0 := Real.log_pos hΛ
  52
  53/-! ## The τ₀ Discreteness Scale -/
  54
  55/-- The fundamental length scale l0 = c * tau0.
  56    tau0 ~ 1.288e-27 s gives l0 ~ 3.86e-19 m.
  57    This also determines:
  58    - Energy scale: E0 = hbar / tau0 ~ 5.1e-8 J ~ 3.2e11 GeV
  59    - Momentum scale: p0 = hbar / l0 ~ 1.7e-10 kg*m/s -/
  60noncomputable def l0 : ℝ := c * tau0
  61
  62/-- The fundamental energy scale. -/
  63noncomputable def E0 : ℝ := hbar / tau0
  64
  65/-- The fundamental momentum cutoff. -/
  66noncomputable def p_max : ℝ := hbar / l0
  67
  68/-- LHC maximum energy scale (in GeV). -/
  69def lhcEnergyGeV : ℚ := 14000  -- 14 TeV = 14000 GeV
  70
  71/-- RS cutoff energy scale (in GeV, approximate). -/
  72def rsCutoffGeV : ℚ := 3.2e11  -- ~320 billion GeV
  73
  74/-- **THEOREM**: The RS UV cutoff is ~10⁷ times higher than LHC energies. -/
  75theorem cutoff_above_lhc :
  76    rsCutoffGeV / lhcEnergyGeV > 10000000 := by
  77  unfold rsCutoffGeV lhcEnergyGeV
  78  norm_num
  79
  80/-! ## Discreteness and the Lattice -/