  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 -/
  81
  82/-- Spacetime is fundamentally discrete in RS.
  83
  84    The "voxel lattice" has spacing l₀ = c × τ₀.
  85
  86    On a lattice, momenta are bounded:
  87    - Maximum momentum: p_max = π ℏ / l₀
  88    - Brillouin zone: |p| < p_max -/
  89structure VoxelLattice where
  90  spacing : ℝ
  91  dimension : ℕ := 4  -- 3 space + 1 time
  92
  93noncomputable def fundamentalLattice : VoxelLattice := { spacing := l0 }
  94
  95/-- On a lattice with spacing a, momenta are periodic with period 2π/a.
  96
  97    The physical momentum range is the first Brillouin zone: |k| < π/a -/
  98noncomputable def brillouinCutoff (lattice : VoxelLattice) : ℝ :=
  99  Real.pi * hbar / lattice.spacing
 100
 101/-- **THEOREM**: The Brillouin zone cutoff equals π × p_max. -/
 102theorem brillouin_equals_pmax :

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