SpectralTuringBridgeHypothesis

formal source

 291
 292    This is the `TuringBridge.the_open_gap` from the TuringBridge module.
 293-/
 294structure SpectralTuringBridgeHypothesis where
 295  /-- The spectral gap of the SAT J-cost Laplacian is positive -/
 296  spectral_gap_positive : ∀ n : ℕ, ∃ gap : ℝ, 0 < gap ∧ gap ≤ 1
 297  /-- R̂ convergence time is O(n) recognition steps -/
 298  rhat_convergence : ∀ n : ℕ, ∃ T_R : ℕ, T_R ≤ n + 1
 299  /-- Simulating one global gradient step requires Ω(n) TM tape operations -/
 300  simulation_cost_per_step : ∀ n : ℕ, ∃ cost : ℕ, cost ≥ n / 2
 301  /-- Therefore TM time ≥ T_R × (n/2) = Ω(n²) -/
 302  tm_time_lower_bound : ∀ n : ℕ, ∃ T_TM : ℕ, T_TM ≥ n * (n / 2)
 303
 304/-- **CONDITIONAL THEOREM**: Given SpectralTuringBridgeHypothesis,
 305    the TM time for SAT is Ω(n²), while R̂ needs only O(n) steps.
 306    For n ≥ 4, the TM lower bound exceeds the R̂ upper bound. -/
 307theorem conditional_separation
 308    (bridge : SpectralTuringBridgeHypothesis)
 309    (_sep : CircuitSeparation) :
 310    ∀ n : ℕ, ∃ (T_TM T_R : ℕ),
 311      T_TM ≥ n * (n / 2) ∧ T_R ≤ n + 1 := by
 312  intro n
 313  obtain ⟨T_TM, hTM⟩ := bridge.tm_time_lower_bound n
 314  obtain ⟨T_R, hTR⟩ := bridge.rhat_convergence n
 315  exact ⟨T_TM, T_R, hTM, hTR⟩
 316
 317/-! ## Certificate -/
 318
 319/-- **CircuitLedgerCert**: bundles all proved results in this module. -/
 320structure CircuitLedgerCert where
 321  /-- A poly-size circuit has poly-bounded Z-capacity -/
 322  capacity_bound : ∀ n : ℕ, ∀ c : BooleanCircuit n,
 323    CircuitZCapacity c ≤ 2 * c.gate_count
 324  /-- UNSAT formulas have a defect moat of width ≥ 1 -/