eight_tick_step_computable

formal source

 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)),
 149      ∀ p : Phase, ∃ p' : Phase, (p, p') ∈ table ∧ step p = p' :=
 150  finite_function_is_computable (α := Phase) (β := Phase) step
 151
 152/-! ## VI. RS Complexity Classes -/
 153
 154/-- **THEOREM IC-003.12**: φ is irrational, so RS dynamics involving φ-ladders
 155    cannot be exactly computed by finite rational algorithms.
 156    This places exact RS computations in the class of "real number computations"
 157    (beyond classical Turing machines for exact values). -/
 158theorem rs_dynamics_beyond_rational : ¬ ∃ q : ℚ, (q : ℝ) = phi :=
 159  fun ⟨q, hq⟩ => no_exact_phi_computation q hq
 160
 161/-- **THEOREM IC-003.13**: However, RS dynamics can be approximated to arbitrary
 162    precision by rational arithmetic (since ℝ is the completion of ℚ).
 163    This places approximate RS computations within Turing-machine computation. -/
 164theorem rs_dynamics_approximable : ∀ ε > 0, ∃ q : ℚ, |phi - (q : ℝ)| < ε := by
 165  intro ε hε
 166  obtain ⟨q, hq1, hq2⟩ := exists_rat_btwn (show phi - ε < phi + ε by linarith)
 167  exact ⟨q, by rw [abs_lt]; exact ⟨by linarith, by linarith⟩⟩
 168
 169/-! ## Summary Certificate -/
 170
 171def ic003_certificate : String :=
 172  "═══════════════════════════════════════════════════════════\n" ++
 173  "  IC-003: CHURCH-TURING PHYSICS — STATUS: DERIVED\n" ++
 174  "═══════════════════════════════════════════════════════════\n" ++
 175  "✓ phase_space_finite:          |Phase| = 8\n" ++
 176  "✓ phase_functions_finite:      |Phase→Phase| = 8^8\n" ++
 177  "✓ ledger_state_space_finite:   |DiscreteLedger| = 2^8\n" ++