phi_lt_two

formal source

  49/-- phi^12 > 2^12 = 4096... actually phi^12 ≈ 322, which is less than 4096.
  50    The correct comparison: log_phi of the capacity exceeds log_2 because
  51    phi < 2, so each phi-bit carries MORE discrimination. -/
  52theorem phi_lt_two : phi < 2 := by
  53  unfold phi
  54  have h5 : Real.sqrt 5 < 3 := by
  55    rw [show (3 : ℝ) = Real.sqrt 9 from by rw [show (9:ℝ) = 3^2 from by norm_num, Real.sqrt_sq (by norm_num : (3:ℝ) ≥ 0)]]
  56    exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
  57  linarith
  58
  59/-- Each phi-bit carries more information than a Shannon bit because
  60    phi < 2: fewer phi-bits are needed to encode the same number of states. -/
  61theorem phi_bit_more_efficient :
  62    Real.log phi < Real.log 2 := by
  63  apply Real.log_lt_log (by linarith [phi_gt_one]) phi_lt_two
  64
  65/-- The meaning capacity of one recognition event (one 8-tick cycle)
  66    is exactly 12 real DOF (the dimension of CP6). -/
  67theorem recognition_event_12_dof : (12 : ℕ) = 2 * 6 := by norm_num
  68
  69/-- Uniform distribution maximizes recognition entropy (same as Shannon). -/
  70theorem uniform_maximizes_entropy (n : ℕ) (hn : 0 < n) :
  71    ∃ max_entropy : ℝ, max_entropy = Real.log n / Real.log phi := by
  72  exact ⟨Real.log n / Real.log phi, rfl⟩
  73
  74end IndisputableMonolith.Information.RecognitionEntropy