gravArea_inv_sq

formal source

  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
  83        demandedRate(M, T_dyn(a)) = bandwidth(gravArea(a))
  84
  85    At a < a_sat, the system is bandwidth-saturated and ILG activates.
  86    At a > a_sat, Newtonian gravity suffices.
  87
  88    In RS-native units (G = φ⁵, ℓ_P = 1, τ₀ = 1, k_R = ln φ):
  89        a_sat = π / (2 · ln(φ))  -/
  90noncomputable def saturationAccel : ℝ :=