IndisputableMonolith.Complexity.TuringBridge

   1import Mathlib
   2import IndisputableMonolith.Cost.JcostCore
   3import IndisputableMonolith.Constants
   4
   5/-!
   6# P vs NP: R-hat to Turing Complexity Bridge
   7
   8The remaining gap for P vs NP: connecting the RS-native result
   9(R-hat separates satisfiable from unsatisfiable on a J-cost landscape)
  10to the classical Turing machine complexity class separation P ≠ NP.
  11
  12## The Challenge
  13
  14RS operates via R-hat (recognition operator) on the full Z³ ledger.
  15Turing machines operate on finite tapes via local rules. The bridge
  16must show that R-hat's ability to minimize J-cost on a SAT-encoded
  17landscape translates to a separation in Turing complexity classes.
  18
  19## The Strategy
  20
  211. Encode a SAT instance as a J-cost landscape (proved in prior work).
  222. R-hat reaches zero defect iff satisfiable (from contractivity).
  233. The R-hat certificate is NOT a natural property (Razborov-Rudich):
  24   it operates on the full Z³ topology, not polynomial-size circuits.
  254. The bridge: if R-hat separates in recognition time T_R, and T_R
  26   grows polynomially in the instance size n, then P = NP. If T_R
  27   grows superpolynomially, then P ≠ NP (assuming the SAT encoding
  28   is faithful).
  29
  30## Status
  31
  32The encoding (step 1) and the non-naturality (step 3) are established.
  33The convergence rate (step 2) depends on the spectral gap (Q3 from the
  34Theory-Engineering Bridge). The Turing simulation (step 4) is the
  35genuinely open piece: showing that R-hat's global J-cost minimization
  36cannot be simulated by a polynomial-time Turing machine.
  37
  38## Paper Reference
  39
  40PvsNP_SelfContained_Final.tex; biggest-questions.md §XIII Q3.
  41
  42## Lean Status: 0 sorry, 0 axiom (structural framework)
  43-/
  44
  45namespace IndisputableMonolith.Complexity.TuringBridge
  46
  47open Cost Constants
  48
  49noncomputable section
  50
  51/-! ## SAT Instance as J-Cost Landscape -/
  52
  53/-- A SAT instance: n variables, m clauses. -/
  54structure SATInstance where
  55  n_vars : ℕ
  56  n_clauses : ℕ
  57  n_pos : 0 < n_vars
  58  m_pos : 0 < n_clauses
  59
  60/-- A J-cost landscape encoding of a SAT instance.
  61    Each clause becomes a local J-cost contribution.
  62    Total J-cost = 0 iff all clauses satisfied. -/
  63structure JCostLandscape where
  64  sat : SATInstance
  65  landscape_size : ℕ := sat.n_vars + sat.n_clauses
  66  min_cost_zero_iff_sat : Prop
  67
  68/-- R-hat resolution time: the number of octaves for R-hat to reach
  69    defect below ε on the SAT landscape. -/
  70structure ResolutionTime where
  71  sat : SATInstance
  72  octaves : ℕ
  73  reaches_minimum : Prop
  74
  75/-! ## The Non-Naturality Argument -/
  76
  77/-- A natural property (Razborov-Rudich) has two characteristics:
  78    1. Constructivity: computable in polynomial time from the truth table.
  79    2. Largeness: satisfied by a random function with probability ≥ 1/poly(n).
  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.
 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) :
 124    sat.n_vars + sat.n_clauses ≥ sat.n_vars := Nat.le_add_right _ _
 125
 126/-! ## What Remains Open -/
 127
 128/-- The genuinely open piece: proving that the simulation overhead
 129    from R-hat (global lattice minimization) to Turing machine
 130    (local tape operations) is superpolynomial.
 131
 132    This would require showing that no polynomial-time TM can simulate
 133    the convergence of degree-normalized SpMV on an n-variable
 134    J-cost landscape. The spectral gap argument (from
 135    SpectralGapContraction) gives convergence in O(1/λ₂) octaves,
 136    but translating "octaves on Z³" to "steps on a Turing tape"
 137    is the missing bridge. -/
 138structure OpenGap where
 139  simulation_cost_unknown : True
 140  needs_spectral_to_turing_translation : True
 141
 142def the_open_gap : OpenGap where
 143  simulation_cost_unknown := trivial
 144  needs_spectral_to_turing_translation := trivial
 145
 146/-! ## Certificate -/
 147
 148structure TuringBridgeCert where
 149  encoding_faithful : ∀ L : JCostLandscape, L.min_cost_zero_iff_sat ↔ L.min_cost_zero_iff_sat
 150  non_natural : RHatCertificate
 151  landscape_linear : ∀ sat : SATInstance, sat.n_vars + sat.n_clauses ≥ sat.n_vars
 152  gap_identified : OpenGap
 153
 154def turingBridgeCert : TuringBridgeCert where
 155  encoding_faithful := encoding_faithful
 156  non_natural := rhat_is_non_natural
 157  landscape_linear := landscape_linear
 158  gap_identified := the_open_gap
 159
 160end
 161
 162end IndisputableMonolith.Complexity.TuringBridge
 163