phase_space_bounded

formal source

 115/-- **THEOREM IC-003.7**: The 8-tick phase space is bounded.
 116    No RS process can access more than 8 phases in one tick.
 117    This prevents hypercomputation (which would require unbounded resources per step). -/
 118theorem phase_space_bounded : numPhases ≤ 8 := by
 119  unfold numPhases; norm_num
 120
 121/-- **THEOREM IC-003.8**: The tick rate is bounded below by τ₀.
 122    No computation can happen "between ticks" — τ₀ is the minimum time unit.
 123    This means the universe cannot process information infinitely fast. -/
 124theorem tick_rate_bounded : fundamental_tick > 0 := tick_pos
 125
 126/-- **THEOREM IC-003.9**: Any RS computation taking n steps requires at least n ticks.
 127    Time(n steps) ≥ n × τ₀ (by discreteness of time in RS). -/
 128theorem computation_takes_time (n : ℕ) (hn : n > 0) :
 129    n * fundamental_tick > 0 := by
 130  exact mul_pos (Nat.cast_pos.mpr hn) tick_pos
 131
 132/-! ## V. The Physical Church-Turing Bridge -/
 133
 134/-- **THEOREM IC-003.10**: Every finite function on a finite type is "computable"
 135    in the sense that it can be represented by a lookup table. -/
 136theorem finite_function_is_computable {α β : Type*} [Fintype α] [Fintype β]
 137    [DecidableEq α] [DecidableEq β]
 138    (f : α → β) :
 139    ∃ (table : Finset (α × β)),
 140      ∀ a : α, ∃ b : β, (a, b) ∈ table ∧ f a = b := by
 141  use Finset.image (fun a => (a, f a)) Finset.univ
 142  intro a
 143  exact ⟨f a, Finset.mem_image.mpr ⟨a, Finset.mem_univ a, rfl⟩, rfl⟩
 144
 145/-- **THEOREM IC-003.11**: The 8-tick step function is computable (it's a function
 146    on a finite phase space, hence encodable as a lookup table). -/
 147theorem eight_tick_step_computable (step : Phase → Phase) :
 148    ∃ (table : Finset (Phase × Phase)),