gamma_fixed_point

formal source

  43theorem gamma_pos : 0 < gamma := by unfold gamma; norm_num
  44
  45/-- The σ-conservation fixed-point equation: `(γ - 1)(γ - 2) = 2`. -/
  46theorem gamma_fixed_point :
  47    (gamma - 1) * (gamma - 2) = 2 := by
  48  unfold gamma; ring
  49
  50/-- `γ = 3` is the unique positive solution of `(γ - 1)(γ - 2) = 2`
  51with `γ > 2`. The other solution is `γ = 0`, which is non-physical. -/
  52theorem gamma_unique {x : ℝ} (hx : 2 < x) (h : (x - 1) * (x - 2) = 2) :
  53    x = gamma := by
  54  -- (x-1)(x-2) = x^2 - 3x + 2 = 2 ⇒ x^2 - 3x = 0 ⇒ x(x-3) = 0.
  55  -- Solutions: x = 0 or x = 3. With x > 2, forced x = 3 = gamma.
  56  have h_factored : x * (x - 3) = 0 := by nlinarith
  57  rcases mul_eq_zero.mp h_factored with h0 | h3
  58  · linarith
  59  · unfold gamma; linarith
  60
  61/-! ## §2. Small-world path-length scaling -/
  62
  63/-- Predicted average path length: `L(N) = log_φ N`. -/
  64def avgPathLength (N : ℝ) : ℝ := Real.log N / Real.log phi
  65
  66/-- For `N > 1`, average path length is positive. -/
  67theorem avgPathLength_pos {N : ℝ} (h : 1 < N) : 0 < avgPathLength N := by
  68  unfold avgPathLength
  69  have h_log_N_pos : 0 < Real.log N := Real.log_pos h
  70  have h_log_phi_pos : 0 < Real.log phi := Real.log_pos one_lt_phi
  71  exact div_pos h_log_N_pos h_log_phi_pos
  72
  73/-- Path length grows logarithmically in `N`. -/
  74theorem path_length_log_growth {N M : ℝ} (hN : 1 < N) (hM : N < M) :
  75    avgPathLength N < avgPathLength M := by
  76  unfold avgPathLength