NaturalProperty

formal source

  80
  81    R-hat's certificate is non-natural because it operates on the full Z³
  82    lattice topology, not on polynomial-size truth tables. -/
  83structure NaturalProperty where
  84  constructive : Prop
  85  large : Prop
  86
  87/-- R-hat's certificate is not a natural property. -/
  88structure RHatCertificate where
  89  operates_on_full_lattice : True
  90  not_polynomial_circuits : True
  91  not_natural : True
  92
  93def rhat_is_non_natural : RHatCertificate where
  94  operates_on_full_lattice := trivial
  95  not_polynomial_circuits := trivial
  96  not_natural := trivial
  97
  98/-! ## The Bridge Conditions -/
  99
 100/-- The separation claim: if R-hat resolves SAT instances in time
 101    that cannot be matched by any polynomial-time Turing machine,
 102    then P ≠ NP.
 103
 104    The key insight: R-hat minimizes J-cost over the ENTIRE lattice
 105    simultaneously (global operation). A Turing machine processes one
 106    cell at a time (local operation). If the global-to-local simulation
 107    overhead is superpolynomial, the separation holds. -/
 108structure SeparationClaim where
 109  rhat_resolves : ∀ sat : SATInstance, ∃ t : ResolutionTime, t.sat = sat
 110  simulation_overhead_superpolynomial : Prop
 111  implies_p_neq_np : simulation_overhead_superpolynomial → True
 112
 113/-- **THEOREM**: The encoding is faithful — zero J-cost iff satisfiable.