IndisputableMonolith.Complexity.PvsNPAssembly

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Complexity.RSatEncoding
   4import IndisputableMonolith.Complexity.JCostLaplacian
   5import IndisputableMonolith.Complexity.SpectralGap
   6import IndisputableMonolith.Complexity.JFrustration
   7import IndisputableMonolith.Complexity.NonNaturalness
   8import IndisputableMonolith.Complexity.CircuitLedger
   9import IndisputableMonolith.Complexity.CircuitLowerBound
  10
  11/-!
  12# P vs NP Assembly: Complete Resolution Structure
  13
  14Two resolution paths:
  15
  16**Path A — P ≠ NP (conditional on UniformTopologicalObstructionHyp):**
  171. J-frustration is non-natural (Phase 2) → RR barrier doesn't apply
  182. UNSAT formulas have J-frustration ≥ 1 (Phase 1)
  193. High J-frustration implies exponential circuit size (Phase 3, conditional)
  204. SAT is NP-complete → no polynomial circuit family for NP → P ≠ NP
  21
  22**Path B — Dissolution (unconditional RS position):**
  231. R̂ recognition time for SAT ≤ n (proved)
  242. TM simulation of R̂ requires overhead (structural argument)
  253. P vs NP conflates two distinct complexity measures
  26
  27## Status: 0 sorry in this file; depends on upstreams
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Complexity
  32namespace PvsNPAssembly
  33
  34open RSatEncoding JCostLaplacian SpectralGap JFrustration
  35open NonNaturalness CircuitLedger CircuitLowerBound
  36
  37noncomputable section
  38
  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
  73          (BalancedParityHidden.enc b R) M) ≠ b
  74
  75theorem dissolution_holds : PvsNPDissolution where
  76  rhat_polytime := fun n f h =>
  77    let ⟨steps, a, hle, ha⟩ := sat_recognition_time_bound f h
  78    ⟨steps, a, hle, ha⟩
  79  unsat_obstruction := fun _n f h => unsat_cost_lower_bound f h
  80  local_blindness := fun _n M _hM decoder =>
  81    BalancedParityHidden.adversarial_failure M decoder
  82
  83/-! ## Status -/
  84
  85structure PvsNPResolutionStatus where
  86  conditional_proof_available : Bool
  87  dissolution_proved : Bool
  88  open_gap : String
  89  sorry_count_in_chain : ℕ
  90
  91def currentStatus : PvsNPResolutionStatus where
  92  conditional_proof_available := false
  93  dissolution_proved := true
  94  open_gap := "UniformTopologicalObstructionHyp: prove that for some fixed k, " ++
  95              "every UNSAT formula on n variables requires circuits of size >= 2^(n/k)."
  96  sorry_count_in_chain := 1
  97
  98/-! ## Master Certificate -/
  99
 100structure PvsNPMasterCert where
 101  laplacian : JCostLaplacianCert
 102  spectral : SpectralGapCert
 103  frustration : JFrustrationCert
 104  non_natural : NonNaturalnessCert
 105  lower_bound : CircuitLowerBoundCert
 106  dissolution : PvsNPDissolution
 107  circuit_sep : CircuitSeparation
 108
 109def pvsNPMasterCert : PvsNPMasterCert where
 110  laplacian := jcostLaplacianCert
 111  spectral := spectralGapCert
 112  frustration := jfrustrationCert
 113  non_natural := nonNaturalnessCert
 114  lower_bound := circuitLowerBoundCert
 115  dissolution := dissolution_holds
 116  circuit_sep := circuitSeparation
 117
 118end -- noncomputable section
 119
 120end PvsNPAssembly
 121end Complexity
 122end IndisputableMonolith
 123