PneqNPConditional

formal source

  39/-! ## Path A: P ≠ NP (Conditional) -/
  40
  41/-- The complete P ≠ NP argument, conditional on the topological obstruction. -/
  42structure PneqNPConditional where
  43  phase1_laplacian : JCostLaplacianCert
  44  phase1_spectral : SpectralGapCert
  45  phase2_frustration : JFrustrationCert
  46  phase2_non_natural : NonNaturalnessCert
  47  phase3_hypothesis : UniformTopologicalObstructionHyp
  48  phase3_lower_bound : CircuitLowerBoundCert
  49
  50/-- **CONDITIONAL THEOREM (P ≠ NP).**
  51    Given the uniform topological obstruction, for every polynomial bound
  52    there exists n₀ beyond which no polynomial circuit decides satisfiability. -/
  53theorem p_neq_np_assembled (pkg : PneqNPConditional) :
  54    ∀ (poly_k poly_c : ℕ), ∃ (n₀ : ℕ),
  55      ∀ n : ℕ, n₀ ≤ n →
  56        ∀ (f : CNFFormula n), f.isUNSAT →
  57          ∀ (c : BooleanCircuit n), CircuitDecides c f →
  58            ¬ (c.gate_count ≤ poly_c * n ^ poly_k) :=
  59  p_neq_np_conditional pkg.phase3_hypothesis
  60
  61/-! ## Path B: Dissolution (Unconditional RS Position) -/
  62
  63/-- The RS dissolution argument. -/
  64structure PvsNPDissolution where
  65  rhat_polytime : ∀ n : ℕ, ∀ f : CNFFormula n, f.isSAT →
  66    ∃ (steps : ℕ) (a : Assignment n), steps ≤ n ∧ satJCost f a = 0
  67  unsat_obstruction : ∀ n : ℕ, ∀ f : CNFFormula n, f.isUNSAT →
  68    ∀ a : Assignment n, satJCost f a ≥ 1
  69  local_blindness : ∀ n : ℕ, ∀ M : Finset (Fin n), M.card < n →
  70    ∀ decoder : ({i // i ∈ M} → Bool) → Bool,
  71      ∃ (b : Bool) (R : Fin n → Bool),
  72        decoder (BalancedParityHidden.restrict