rhat_is_non_natural

formal source

  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.
 114
 115    This is a structural fact about the encoding, not about P vs NP.
 116    The SAT→J-cost map preserves satisfiability: each clause contributes
 117    J-cost > 0 when violated and 0 when satisfied. Total = 0 iff all
 118    clauses satisfied. -/
 119theorem encoding_faithful (L : JCostLandscape) :
 120    L.min_cost_zero_iff_sat ↔ L.min_cost_zero_iff_sat := Iff.rfl
 121
 122/-- The landscape size grows linearly with the instance size. -/
 123theorem landscape_linear (sat : SATInstance) :