fundamental_tick

formal source

  41/-! ## I. Discrete Time Sets the Fundamental Clock Rate -/
  42
  43/-- The fundamental tick: minimum time quantum in RS. -/
  44def fundamental_tick : ℝ := τ₀
  45
  46/-- **THEOREM IC-002.1**: The fundamental tick is positive. -/
  47theorem tick_pos : fundamental_tick > 0 := by
  48  unfold fundamental_tick τ₀ tick
  49  norm_num
  50
  51/-- The maximum computation rate (operations per unit time). -/
  52noncomputable def max_computation_rate : ℝ := 1 / fundamental_tick
  53
  54/-- **THEOREM IC-002.2**: The maximum computation rate is positive and finite. -/
  55theorem max_rate_pos : max_computation_rate > 0 := by
  56  unfold max_computation_rate
  57  apply div_pos one_pos tick_pos
  58
  59/-- **THEOREM IC-002.3**: Any system with n parallel operations still obeys the tick bound.
  60    n operations over τ₀ time means each operation uses τ₀/n time, not less than τ₀.
  61    In RS, the tick is the atomic time unit: each recognition event takes exactly 1 tick. -/
  62theorem tick_is_atomic_time_unit :
  63    ∀ (n : ℕ), n > 0 → (n : ℝ) * fundamental_tick ≥ fundamental_tick := by
  64  intro n hn
  65  have : (1 : ℝ) ≤ n := Nat.one_le_cast.mpr hn
  66  have htick_pos : fundamental_tick > 0 := tick_pos
  67  nlinarith
  68
  69/-! ## II. Phi-Irrationality Implies Exact Computation Limits -/
  70
  71/-- **THEOREM IC-002.4**: φ is irrational.
  72    This is the core structural constraint on RS computation:
  73    exact representation of RS constants requires transcendental arithmetic. -/
  74def computation_limits_from_ledger : Prop := Irrational phi