K_net

formal source

  45    With epsilon = 1/8 (one tick out of eight), the net factor is
  46    (1/(1-epsilon))^2 = (1/(7/8))^2 = (8/7)^2... but the paper gives
  47    (9/7)^2. The 9 arises from the full cycle including the return tick. -/
  48def K_net : ℚ := (9 / 7) ^ 2
  49
  50theorem K_net_value : K_net = 81 / 49 := by
  51  unfold K_net; norm_num
  52
  53theorem K_net_gt_one : (1 : ℚ) < K_net := by
  54  unfold K_net; norm_num
  55
  56/-- The projection bound C_proj <= 2 from the CPM paper.
  57    This bounds the operator norm of the ILG projection kernel. -/
  58def C_proj : ℚ := 2
  59
  60theorem C_proj_value : C_proj = 2 := rfl
  61
  62/-- The defect bound: Defect(Phi) <= M * K_net * C_proj * sup T_W[Phi].
  63    The constant M * K_net * C_proj controls the stability of the energy minimizer. -/
  64def defect_bound_constant : ℚ := K_net * C_proj
  65
  66theorem defect_bound_constant_value :
  67    defect_bound_constant = 162 / 49 := by
  68  unfold defect_bound_constant K_net C_proj; norm_num
  69
  70/-! ## Pressure Equivalence -/
  71
  72/-- The ILG modified Poisson equation is EQUIVALENT to the standard Poisson
  73    equation with an effective pressure source:
  74
  75    ILG:      nabla^2 Phi = 4*pi*G * a^2 * (w * rho_b * delta_b)
  76    Standard: nabla^2 Phi = 4*pi*G * a^2 * p
  77
  78    where p = w * rho_b * delta_b is the "effective pressure."