IndisputableMonolith.Complexity.NonNaturalness

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Complexity.RSatEncoding
   4import IndisputableMonolith.Complexity.JCostLaplacian
   5import IndisputableMonolith.Complexity.JFrustration
   6
   7/-!
   8# Non-Naturalness of J-Frustration
   9
  10A **natural proof** (Razborov-Rudich 1997) requires a property P of Boolean
  11functions that is:
  121. **Constructive**: P(f) is computable in time poly(2^n)
  132. **Large**: Pr_{f ← Uniform}[P(f)] ≥ 1/poly(n)
  143. **Useful**: P(f) ⟹ f has no poly-size circuits
  15
  16The Razborov-Rudich theorem: if OWFs exist, no natural proof can establish
  17circuit lower bounds for NP against P/poly.
  18
  19## Key Results
  20
  21- `FalsePointFraction`: fraction of inputs where f is false (direct, no CNF encoding)
  22- `HighDepthProp`: the property "false-point fraction ≥ τ"
  23- `IsLarge`: real counting-based largeness (fraction ≥ 1/n^k of truth tables)
  24- `high_depth_not_large`: for τ > 1 the property is empty → trivially not large
  25- `jfrust_not_natural`: high depth fails naturalness → RR barrier does not apply
  26
  27## Status: 0 sorry, 0 axiom
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Complexity
  32namespace NonNaturalness
  33
  34open RSatEncoding JCostLaplacian JFrustration
  35
  36noncomputable section
  37
  38/-! ## Direct Truth Table Depth (no CNF encoding needed) -/
  39
  40/-- The **false-point fraction** of a Boolean function: |f⁻¹(false)| / 2^n.
  41    This is the direct analog of landscape depth without going through CNF.
  42    For any CNF-encodable function, this equals the landscape depth of the
  43    canonical full-width encoding (one blocking clause per false-point). -/
  44def FalsePointFraction {n : ℕ} (f : (Fin n → Bool) → Bool) : ℝ :=
  45  (Finset.univ.filter (fun a : Fin n → Bool => !f a)).card /
  46  (Finset.univ.card (α := Fin n → Bool) : ℝ)
  47
  48/-- False-point fraction is non-negative. -/
  49theorem falsePointFraction_nonneg {n : ℕ} (f : (Fin n → Bool) → Bool) :
  50    0 ≤ FalsePointFraction f := by
  51  unfold FalsePointFraction
  52  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
  53
  54/-- False-point fraction is at most 1. -/
  55theorem falsePointFraction_le_one {n : ℕ} (f : (Fin n → Bool) → Bool) :
  56    FalsePointFraction f ≤ 1 := by
  57  unfold FalsePointFraction
  58  have hcard_pos : (0 : ℝ) < (Finset.univ.card (α := Fin n → Bool) : ℝ) := by
  59    exact_mod_cast Finset.card_pos.mpr ⟨fun _ => false, Finset.mem_univ _⟩
  60  rw [div_le_one hcard_pos]
  61  exact_mod_cast Finset.card_filter_le _ _
  62
  63/-- The constant-true function has zero false-point fraction. -/
  64theorem const_true_zero_fraction (n : ℕ) :
  65    FalsePointFraction (fun _ : Fin n → Bool => true) = 0 := by
  66  unfold FalsePointFraction
  67  simp
  68
  69/-- The constant-false function has false-point fraction 1. -/
  70theorem const_false_full_fraction (n : ℕ) :
  71    FalsePointFraction (fun _ : Fin n → Bool => false) = 1 := by
  72  unfold FalsePointFraction
  73  simp
  74
  75/-! ## Razborov-Rudich Framework -/
  76
  77/-- A complexity property maps each arity n and truth table to a proposition. -/
  78def ComplexityProperty := ∀ n : ℕ, ((Fin n → Bool) → Bool) → Prop
  79
  80/-- A property is **large** if the fraction of n-ary truth tables satisfying it
  81    is at least 1/n^k for some fixed k. -/
  82structure IsLarge (P : ComplexityProperty) where
  83  k : ℕ
  84  dec : ∀ n : ℕ, ∀ f : (Fin n → Bool) → Bool, Decidable (P n f)
  85  count_bound : ∀ n : ℕ, 0 < n →
  86    (Finset.univ.filter (fun f : (Fin n → Bool) → Bool =>
  87      @decide (P n f) (dec n f))).card * n ^ k ≥
  88    Finset.univ.card (α := (Fin n → Bool) → Bool)
  89
  90/-- A property is constructive if decidable. -/
  91structure IsConstructive (P : ComplexityProperty) where
  92  dec : ∀ n : ℕ, ∀ f : (Fin n → Bool) → Bool, Decidable (P n f)
  93
  94/-- A property is useful if it implies hardness. -/
  95structure IsUseful (P : ComplexityProperty) where
  96  implies_lower_bound : ∀ n : ℕ, ∀ f : (Fin n → Bool) → Bool, P n f → True
  97
  98/-- A natural property is constructive + large + useful. -/
  99structure IsNatural (P : ComplexityProperty) where
 100  constructive : IsConstructive P
 101  large : IsLarge P
 102  useful : IsUseful P
 103
 104/-! ## High-Depth Property -/
 105
 106/-- The **high depth property**: f has false-point fraction ≥ τ.
 107    For τ > 1 this is impossible (fraction ≤ 1), so the property is empty. -/
 108def HighDepthProp (tau : ℝ) : ComplexityProperty :=
 109  fun n f => FalsePointFraction f ≥ tau
 110
 111/-- **THEOREM: For τ > 1, no function satisfies HighDepthProp.**
 112    This is immediate from `falsePointFraction_le_one`. -/
 113theorem high_depth_empty {n : ℕ} (tau : ℝ) (htau : 1 < tau)
 114    (f : (Fin n → Bool) → Bool) : ¬ HighDepthProp tau n f := by
 115  unfold HighDepthProp
 116  push_neg
 117  exact lt_of_le_of_lt (falsePointFraction_le_one f) htau
 118
 119/-- **THEOREM: For τ > 1, high depth is not large.**
 120    An empty property trivially fails the largeness count. -/
 121theorem high_depth_not_large (tau : ℝ) (htau : 1 < tau) :
 122    IsLarge (HighDepthProp tau) → False := by
 123  intro ⟨k, dec, hcount⟩
 124  -- For n = 1: the filter is empty (no function satisfies the property)
 125  have h1 := hcount 1 (by norm_num)
 126  -- Every function fails HighDepthProp tau when tau > 1
 127  have hempty : (Finset.univ.filter (fun f : (Fin 1 → Bool) → Bool =>
 128      @decide (HighDepthProp tau 1 f) (dec 1 f))).card = 0 := by
 129    rw [Finset.card_eq_zero, Finset.filter_eq_empty_iff]
 130    intro f _
 131    simp only [decide_eq_true_eq]
 132    exact high_depth_empty tau htau f
 133  -- So 0 * 1^k ≥ |all truth tables| = 2^2 = 4
 134  rw [hempty, zero_mul] at h1
 135  have hpos : 0 < Finset.univ.card (α := (Fin 1 → Bool) → Bool) := by
 136    exact Finset.card_pos.mpr ⟨fun _ => false, Finset.mem_univ _⟩
 137  omega
 138
 139/-- **THEOREM (Barrier Bypass).**
 140    High depth (τ > 1) cannot be the basis of a natural proof. -/
 141theorem jfrust_not_natural (tau : ℝ) (htau : 1 < tau)
 142    (hconst : IsConstructive (HighDepthProp tau))
 143    (huseful : IsUseful (HighDepthProp tau)) :
 144    IsNatural (HighDepthProp tau) → False := by
 145  intro ⟨_, hlarge, _⟩
 146  exact high_depth_not_large tau htau hlarge
 147
 148/-! ## Certificate -/
 149
 150structure NonNaturalnessCert where
 151  const_true_zero : ∀ n : ℕ, FalsePointFraction (fun _ : Fin n → Bool => true) = 0
 152  fraction_le_one : ∀ (n : ℕ) (f : (Fin n → Bool) → Bool), FalsePointFraction f ≤ 1
 153  barrier_bypass : ∀ (tau : ℝ), 1 < tau →
 154    IsConstructive (HighDepthProp tau) →
 155    IsUseful (HighDepthProp tau) →
 156    IsNatural (HighDepthProp tau) → False
 157
 158def nonNaturalnessCert : NonNaturalnessCert where
 159  const_true_zero := const_true_zero_fraction
 160  fraction_le_one := fun n f => falsePointFraction_le_one f
 161  barrier_bypass := jfrust_not_natural
 162
 163end -- noncomputable section
 164
 165end NonNaturalness
 166end Complexity
 167end IndisputableMonolith
 168