 154      .Wire
 155
 156/-- A full SAT instance encoded as CA tape -/
 157structure SATGadget where
 158  /-- Number of variables -/
 159  n : ℕ
 160  /-- Number of clauses -/
 161  m : ℕ
 162  /-- Clause gadgets -/
 163  clauses : List (ClauseGadget n)
 164  /-- Variable assignment cells -/
 165  var_positions : Fin n → ℤ
 166  /-- Output cell position -/
 167  output_position : ℤ
 168  /-- Total tape width used -/
 169  total_width : ℕ := n + 3 * m + 10
 170
 171/-- The evaluation time for a SAT instance with n variables and m clauses -/
 172noncomputable def sat_eval_time (n m : ℕ) : ℕ :=
 173  -- Depth of the clause evaluation tree: O(log m) for m clauses
 174  -- Width propagation: O(n^{1/3}) for n variables arranged optimally
 175  -- Total: O(n^{1/3} · log(n·m))
 176  Nat.ceil (Real.rpow n (1/3) * Real.log (n + m + 2))
 177
 178/-- **HYPOTHESIS**: SAT evaluation via Cellular Automata in sub-linear time.
 179
 180    STATUS: SCAFFOLD — The O(n^{1/3} log n) bound follows from the parallel
 181    propagation property of the CA on a 3D-embedded 1D tape, but the formal
 182    proof requires detailed counting of the dependency cone steps.
 183
 184    TODO: Formally prove the O(n^{1/3} log n) bound. -/
 185def H_SATCARuntime (sg : SATGadget) : Prop :=
 186  ∃ (t : ℕ), t ≤ sat_eval_time sg.n sg.m ∧
 187  -- After t steps, the output cell contains the SAT result

papers checked against this theorem

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