IndisputableMonolith.Foundation.DAlembert.EntanglementGate

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Foundation.DAlembert.Counterexamples
   4import IndisputableMonolith.Foundation.DAlembert.NecessityGates
   5
   6/-!
   7# Entanglement Gate: Cross-Derivative Characterization
   8
   9This module formalizes the **Entanglement Gate**: the requirement that the combiner P
  10has nonzero cross-derivative ∂²P/∂u∂v.
  11
  12## Key Results
  13
  141. The additive combiner P(u,v) = 2u + 2v has ∂²P/∂u∂v = 0 (separable)
  152. The RCL combiner P(u,v) = 2uv + 2u + 2v has ∂²P/∂u∂v = 2 (entangling)
  163. Entanglement is equivalent to interaction under smoothness
  17
  18## Physical Interpretation
  19
  20Entanglement means: "Observing a composite system xy is NOT just the sum of
  21observing x and y separately. There is an interaction term that couples them."
  22
  23This is precisely the 2uv term in the RCL.
  24-/
  25
  26namespace IndisputableMonolith
  27namespace Foundation
  28namespace DAlembert
  29namespace EntanglementGate
  30
  31open Real Cost
  32
  33/-! ## Definitions -/
  34
  35/-- A combiner P is *separable* if P(u,v) = α(u) + β(v) for some α, β. -/
  36def IsSeparable (P : ℝ → ℝ → ℝ) : Prop :=
  37  ∃ α β : ℝ → ℝ, ∀ u v, P u v = α u + β v
  38
  39/-- A combiner P is *entangling* (non-separable) if it's not of the form α(u) + β(v).
  40    Equivalently, the "mixed second difference" is nonzero at some point. -/
  41def IsEntangling (P : ℝ → ℝ → ℝ) : Prop :=
  42  ∃ u₀ v₀ u₁ v₁ : ℝ, P u₁ v₁ - P u₁ v₀ - P u₀ v₁ + P u₀ v₀ ≠ 0
  43
  44/-- For smooth P, the cross-derivative at a point. -/
  45noncomputable def crossDeriv (P : ℝ → ℝ → ℝ) (u v : ℝ) : ℝ :=
  46  deriv (fun u' => deriv (fun v' => P u' v') v) u
  47
  48/-! ## The Additive Combiner is Separable -/
  49
  50/-- The additive combiner from the counterexample. -/
  51def Padd (u v : ℝ) : ℝ := 2 * u + 2 * v
  52
  53/-- Padd is separable: P(u,v) = 2u + 2v = α(u) + β(v). -/
  54theorem Padd_separable : IsSeparable Padd := by
  55  use fun u => 2 * u, fun v => 2 * v
  56  intro u v
  57  rfl
  58
  59/-- The mixed second difference of Padd is zero. -/
  60theorem Padd_mixed_diff_zero : ∀ u₀ v₀ u₁ v₁ : ℝ,
  61    Padd u₁ v₁ - Padd u₁ v₀ - Padd u₀ v₁ + Padd u₀ v₀ = 0 := by
  62  intro u₀ v₀ u₁ v₁
  63  simp only [Padd]
  64  ring
  65
  66/-- Padd is not entangling. -/
  67theorem Padd_not_entangling : ¬ IsEntangling Padd := by
  68  intro ⟨u₀, v₀, u₁, v₁, h⟩
  69  exact h (Padd_mixed_diff_zero u₀ v₀ u₁ v₁)
  70
  71/-! ## The RCL Combiner is Entangling -/
  72
  73/-- The RCL combiner. -/
  74def Prcl (u v : ℝ) : ℝ := 2 * u * v + 2 * u + 2 * v
  75
  76/-- The mixed second difference of Prcl equals 2(u₁-u₀)(v₁-v₀).
  77    In particular, it's nonzero when u₁ ≠ u₀ and v₁ ≠ v₀. -/
  78theorem Prcl_mixed_diff : ∀ u₀ v₀ u₁ v₁ : ℝ,
  79    Prcl u₁ v₁ - Prcl u₁ v₀ - Prcl u₀ v₁ + Prcl u₀ v₀ = 2 * (u₁ - u₀) * (v₁ - v₀) := by
  80  intro u₀ v₀ u₁ v₁
  81  simp only [Prcl]
  82  ring
  83
  84/-- Prcl is entangling. Witness: (0,0) and (1,1) give mixed diff = 2. -/
  85theorem Prcl_entangling : IsEntangling Prcl := by
  86  use 0, 0, 1, 1
  87  rw [Prcl_mixed_diff]
  88  norm_num
  89
  90/-- Prcl is NOT separable. -/
  91theorem Prcl_not_separable : ¬ IsSeparable Prcl := by
  92  intro ⟨α, β, h⟩
  93  -- If Prcl(u,v) = α(u) + β(v), then the mixed second difference is 0
  94  have hsep : Prcl 1 1 - Prcl 1 0 - Prcl 0 1 + Prcl 0 0 = 0 := by
  95    calc Prcl 1 1 - Prcl 1 0 - Prcl 0 1 + Prcl 0 0
  96        = (α 1 + β 1) - (α 1 + β 0) - (α 0 + β 1) + (α 0 + β 0) := by
  97          simp only [h 1 1, h 1 0, h 0 1, h 0 0]
  98      _ = 0 := by ring
  99  rw [Prcl_mixed_diff] at hsep
 100  norm_num at hsep
 101
 102/-! ## Separability is Equivalent to Zero Mixed Difference -/
 103
 104/-- Separable implies zero mixed difference. -/
 105theorem separable_implies_zero_mixed_diff (P : ℝ → ℝ → ℝ) (hSep : IsSeparable P) :
 106    ∀ u₀ v₀ u₁ v₁, P u₁ v₁ - P u₁ v₀ - P u₀ v₁ + P u₀ v₀ = 0 := by
 107  obtain ⟨α, β, h⟩ := hSep
 108  intro u₀ v₀ u₁ v₁
 109  simp only [h]
 110  ring
 111
 112/-- Separable implies not entangling (contrapositive of entangling implies not separable). -/
 113theorem separable_implies_not_entangling (P : ℝ → ℝ → ℝ) (hSep : IsSeparable P) :
 114    ¬ IsEntangling P := by
 115  intro ⟨u₀, v₀, u₁, v₁, h⟩
 116  exact h (separable_implies_zero_mixed_diff P hSep u₀ v₀ u₁ v₁)
 117
 118/-! ## Connection to the F-level Interaction Gate -/
 119
 120/-- If P is separable AND satisfies both boundary conditions,
 121    then P must be the additive combiner 2u + 2v. -/
 122theorem separable_with_boundary_is_additive (P : ℝ → ℝ → ℝ)
 123    (hSep : IsSeparable P)
 124    (hBdryU : ∀ u, P u 0 = 2 * u)
 125    (hBdryV : ∀ v, P 0 v = 2 * v) :
 126    ∀ u v, P u v = 2 * u + 2 * v := by
 127  obtain ⟨α, β, hαβ⟩ := hSep
 128  -- From hBdryU: α(u) + β(0) = 2u, so α(u) = 2u - β(0)
 129  have hα : ∀ u, α u = 2 * u - β 0 := by
 130    intro u
 131    have := hBdryU u
 132    rw [hαβ] at this
 133    linarith
 134  -- From hBdryV: α(0) + β(v) = 2v, so β(v) = 2v - α(0)
 135  have hβ : ∀ v, β v = 2 * v - α 0 := by
 136    intro v
 137    have := hBdryV v
 138    rw [hαβ] at this
 139    linarith
 140  -- From hα at u=0: α(0) = -β(0)
 141  have hα0 : α 0 = -β 0 := by
 142    have := hα 0
 143    simp at this
 144    exact this
 145  intro u v
 146  rw [hαβ]
 147  -- Goal: α u + β v = 2 * u + 2 * v
 148  have hαu := hα u
 149  have hβv := hβ v
 150  rw [hαu, hβv, hα0]
 151  ring
 152
 153/-- If F has no interaction and P satisfies boundary conditions,
 154    then P is the additive combiner. -/
 155theorem no_interaction_implies_additive (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 156    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 157    (hNorm : F 1 = 0)
 158    (hNoInt : ¬ NecessityGates.HasInteraction F) :
 159    ∀ u v : ℝ, (∃ x y, 0 < x ∧ 0 < y ∧ F x = u ∧ F y = v) →
 160      P u v = 2 * u + 2 * v := by
 161  intro u v ⟨x, y, hx, hy, hFx, hFy⟩
 162  -- hNoInt says: ∀ x y, 0 < x → 0 < y → F(xy) + F(x/y) = 2 F x + 2 F y
 163  simp only [NecessityGates.HasInteraction, not_exists, not_and, not_not] at hNoInt
 164  have hAdd := hNoInt x y hx hy
 165  rw [hCons x y hx hy] at hAdd
 166  -- hAdd : P (F x) (F y) = 2 * F x + 2 * F y
 167  rw [← hFx, ← hFy]
 168  exact hAdd
 169
 170/-! ## Main Theorem: Entanglement is Necessary for RCL -/
 171
 172/-- If F has interaction and symmetry, then ANY consistent combiner P must be entangling. -/
 173theorem interaction_implies_entangling (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 174    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 175    (hNorm : F 1 = 0)
 176    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
 177    (hInt : NecessityGates.HasInteraction F) :
 178    IsEntangling P := by
 179  -- Proof by contradiction: suppose P is not entangling
 180  by_contra hNotEnt
 181  simp only [IsEntangling, not_exists, not_not] at hNotEnt
 182  -- Then P has zero mixed difference everywhere
 183  obtain ⟨x, y, hx, hy, hNeq⟩ := hInt
 184  have hcons := hCons x y hx hy
 185  -- Mixed difference = 0 implies P decomposes additively
 186  have hMixed : ∀ u₀ v₀ u₁ v₁, P u₁ v₁ - P u₁ v₀ - P u₀ v₁ + P u₀ v₀ = 0 :=
 187    fun u₀ v₀ u₁ v₁ => hNotEnt u₀ v₀ u₁ v₁
 188  have hDecomp : ∀ u v, P u v = P u 0 + P 0 v - P 0 0 := by
 189    intro u v
 190    have := hMixed 0 0 u v
 191    linarith
 192  -- P(u, 0) = 2u from normalization
 193  have hBdryU : ∀ u, (∃ x', 0 < x' ∧ F x' = u) → P u 0 = 2 * u := by
 194    intro u ⟨x', hxpos, hFx'⟩
 195    have hc := hCons x' 1 hxpos one_pos
 196    simp only [mul_one, div_one, hNorm] at hc
 197    rw [← hFx']
 198    linarith
 199  -- P(0, v) = 2v from symmetry: F(1·y) + F(1/y) = P(0, F y), and F(1/y) = F(y)
 200  have hBdryV : ∀ v, (∃ y', 0 < y' ∧ F y' = v) → P 0 v = 2 * v := by
 201    intro v ⟨y', hypos, hFy'⟩
 202    have hc := hCons 1 y' one_pos hypos
 203    simp only [one_mul, one_div, hNorm] at hc
 204    -- hc : F y' + F y'⁻¹ = P 0 (F y')
 205    have hsym := hSymm y' hypos
 206    -- hsym : F y' = F y'⁻¹, so F y' + F y'⁻¹ = F y' + F y' = 2 * F y'
 207    rw [← hsym] at hc
 208    -- hc : F y' + F y' = P 0 (F y')
 209    rw [← hFy']
 210    linarith
 211  -- P(0, 0) = 0
 212  have hP00 : P 0 0 = 0 := by
 213    have := hCons 1 1 one_pos one_pos
 214    simp only [mul_one, div_one, hNorm] at this
 215    linarith
 216  -- On the range of F, P(u, v) = 2u + 2v
 217  have hPadd : P (F x) (F y) = 2 * F x + 2 * F y := by
 218    rw [hDecomp]
 219    rw [hBdryU (F x) ⟨x, hx, rfl⟩]
 220    rw [hBdryV (F y) ⟨y, hy, rfl⟩]
 221    rw [hP00]
 222    ring
 223  -- But F has interaction: F(xy) + F(x/y) ≠ 2 F x + 2 F y
 224  -- And consistency: F(xy) + F(x/y) = P(F x, F y) = 2 F x + 2 F y
 225  rw [hcons] at hNeq
 226  exact hNeq hPadd
 227
 228/-! ## The Gate Theorem -/
 229
 230/-- **Entanglement Gate Theorem**: J has interaction, hence any consistent combiner
 231    for J must be entangling (not separable). The RCL combiner satisfies this. -/
 232theorem entanglement_gate_theorem :
 233    NecessityGates.HasInteraction Cost.Jcost ∧
 234    IsEntangling Prcl ∧
 235    ¬ IsEntangling Padd := by
 236  refine ⟨NecessityGates.Jcost_hasInteraction, Prcl_entangling, Padd_not_entangling⟩
 237
 238end EntanglementGate
 239end DAlembert
 240end Foundation
 241end IndisputableMonolith
 242