`alpha_from_phi`

definition

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module: IndisputableMonolith.Relativity.GRLimit.Parameters
domain: Relativity
line: 24 · github
papers citing: none yet

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IndisputableMonolith.Relativity.GRLimit.Parameters on GitHub at line 24.

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formal source

  21namespace GRLimit
  22
  23/-- ILG exponent α from RS: α = (1 - 1/φ)/2 ≈ 0.191 -/
  24noncomputable def alpha_from_phi : ℝ :=
  25  (1 - 1 / Constants.phi) / 2
  26
  27/-- ILG lag constant C_lag from RS: C_lag = φ^(-5) ≈ 0.090 -/
  28noncomputable def cLag_from_phi : ℝ :=
  29  Constants.phi ^ (-5 : ℝ)
  30
  31/-- PROVEN: Both parameters are positive. -/
  32theorem rs_params_positive :
  33  0 < alpha_from_phi ∧ 0 < cLag_from_phi := by
  34  constructor
  35  · unfold alpha_from_phi
  36    have hφ_pos : 0 < Constants.phi := Constants.phi_pos
  37    have hφ_gt_one : 1 < Constants.phi := Constants.one_lt_phi
  38    have : 0 < 1 - 1 / Constants.phi := by
  39      have : 1 / Constants.phi < 1 := (div_lt_one hφ_pos).mpr hφ_gt_one
  40      linarith
  41    linarith
  42  · unfold cLag_from_phi
  43    exact Real.rpow_pos_of_pos Constants.phi_pos _
  44
  45/-- PROVEN: α < 1 (straightforward from φ > 1). -/
  46theorem alpha_lt_one : alpha_from_phi < 1 := by
  47  unfold alpha_from_phi
  48  have hφ_pos : 0 < Constants.phi := Constants.phi_pos
  49  have : 1 - 1 / Constants.phi < 1 := by
  50    have : 0 < 1 / Constants.phi := div_pos (by norm_num) hφ_pos
  51    linarith
  52  have : (1 - 1 / Constants.phi) / 2 < 1 / 2 := by
  53    exact div_lt_div_of_pos_right this (by norm_num)
  54  linarith

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