gravArea_pos

formal source

  49noncomputable def gravArea (G_N M a : ℝ) : ℝ :=
  50  4 * Real.pi * (G_N * M / a) ^ 2
  51
  52theorem gravArea_pos {G_N M a : ℝ} (hG : 0 < G_N) (hM : 0 < M) (ha : 0 < a) :
  53    0 < gravArea G_N M a := by
  54  unfold gravArea
  55  apply mul_pos
  56  · exact mul_pos (by linarith) Real.pi_pos
  57  · exact sq_pos_of_pos (div_pos (mul_pos hG hM) ha)
  58
  59/-- Gravitational area scales as 1/a². -/
  60theorem gravArea_inv_sq (G_N M a c : ℝ) (_hc : 0 < c) (_ha : 0 < a) :
  61    gravArea G_N M (c * a) = gravArea G_N M a / c ^ 2 := by
  62  unfold gravArea
  63  ring
  64
  65/-! ## §2. Dynamical Time and Demanded Rate -/
  66
  67/-- Keplerian dynamical time at radius r for mass M:
  68    T_dyn = 2π√(r³/(GM)). -/
  69noncomputable def dynamicalTime (G_N M r : ℝ) : ℝ :=
  70  2 * Real.pi * Real.sqrt (r ^ 3 / (G_N * M))
  71
  72/-- Newtonian acceleration at radius r: a = GM/r². -/
  73noncomputable def newtonAccel (G_N M r : ℝ) : ℝ :=
  74  G_N * M / r ^ 2
  75
  76/-! ## §3. The Saturation Condition -/
  77
  78/-- **DEFINITION**: Saturation acceleration a_sat.
  79
  80    The acceleration at which the demanded recognition rate equals the
  81    holographic bandwidth of the gravitational area:
  82