IndisputableMonolith.Foundation.DAlembert.TriangulatedProof

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Foundation.DAlembert.Counterexamples
   4import IndisputableMonolith.Foundation.DAlembert.NecessityGates
   5import IndisputableMonolith.Foundation.DAlembert.EntanglementGate
   6import IndisputableMonolith.Foundation.DAlembert.CurvatureGate
   7import IndisputableMonolith.Foundation.DAlembert.FourthGate
   8import IndisputableMonolith.Foundation.DAlembert.Unconditional
   9
  10/-!
  11# Triangulated Proof: Four Gates to Full Inevitability
  12
  13This module combines the four gates into a unified inevitability theorem.
  14
  15## The Four Gates
  16
  171. **Interaction Gate** (NecessityGates): F has interaction ⟺ F(xy) + F(x/y) ≠ 2F(x) + 2F(y) somewhere
  182. **Entanglement Gate** (EntanglementGate): P is entangling ⟺ mixed second difference ≠ 0
  193. **Curvature Gate** (CurvatureGate): G satisfies hyperbolic ODE ⟺ G'' = G + 1
  204. **d'Alembert Gate** (FourthGate): H = G+1 satisfies d'Alembert ⟺ H(t+u) + H(t-u) = 2H(t)H(u)
  21
  22## Why Four Gates?
  23
  24In the Option~A formulation used in the paper, Gate~3 is a \emph{normalized} closure:
  25the hyperbolic branch is stated directly as $G''=G+1$. Once the flat branch is
  26excluded by interaction/entanglement (and the spherical branch by calibration),
  27the solution is already forced to be $G=\cosh-1$, and the RCL combiner follows.
  28
  29In that formulation, the d'Alembert identity is therefore \emph{derived} (a signature
  30of $H=\cosh$) rather than an additional restriction. We keep it explicit because it
  31provides a compact classical viewpoint and a convenient certificate path in Lean.
  32
  33## Main Results
  34
  35- J has interaction (proved in NecessityGates)
  36- Interaction + symmetry ⟹ P is entangling (proved in EntanglementGate)
  37- RCL combiner is entangling (proved in EntanglementGate)
  38- J's log-lift satisfies hyperbolic ODE (proved in CurvatureGate)
  39- J has d'Alembert structure (proved in FourthGate)
  40- d'Alembert + calibration ⟹ G = cosh - 1 (proved in FourthGate)
  41- G = cosh - 1 ⟹ F = J (definition)
  42- F = J ⟹ P = RCL (proved unconditionally in Unconditional)
  43
  44## The Quadrilateral Structure
  45
  46```
  47                         F = J, P = RCL
  48                              ▲
  49              ┌───────────────┼───────────────┐
  50              │               │               │
  51         Interaction     Entangle        d'Alembert
  52           Gate            Gate            Gate
  53              │               │               │
  54              ▼               ▼               ▼
  55         F ≠ additive    P has cross    H(t+u)+H(t-u)
  56                           term          =2H(t)H(u)
  57                              │
  58                         Curvature
  59                           Gate
  60                              │
  61                              ▼
  62                         G'' = G+1
  63```
  64
  65In the Option~A formulation, the curvature gate is the decisive closure; the
  66d'Alembert identity is a derived cross-check and alternate characterization.
  67-/
  68
  69namespace IndisputableMonolith
  70namespace Foundation
  71namespace DAlembert
  72namespace TriangulatedProof
  73
  74open Real Cost NecessityGates EntanglementGate CurvatureGate Unconditional
  75
  76/-! ## The Core Classification -/
  77
  78/-- The fundamental trichotomy: under structural axioms, exactly one branch holds. -/
  79inductive CostBranch where
  80  | flat : CostBranch       -- Fquad: G = t²/2, P additive, no interaction
  81  | hyperbolic : CostBranch -- Jcost: G = cosh-1, P = RCL, has interaction
  82  deriving DecidableEq, Repr
  83
  84/-! ## Gate 1: Interaction Distinguishes the Branches -/
  85
  86/-- J is in the hyperbolic branch (has interaction). -/
  87theorem Jcost_is_hyperbolic : HasInteraction Cost.Jcost := Jcost_hasInteraction
  88
  89/-- Fquad is in the flat branch (no interaction). -/
  90theorem Fquad_is_flat : ¬ HasInteraction Counterexamples.Fquad := Fquad_noInteraction
  91
  92/-! ## Gate 2: Entanglement Characterizes the Combiner -/
  93
  94/-- RCL combiner is entangling. -/
  95theorem RCL_is_entangling : IsEntangling Prcl := Prcl_entangling
  96
  97/-- Additive combiner is not entangling. -/
  98theorem additive_not_entangling : ¬ IsEntangling Padd := Padd_not_entangling
  99
 100/-- Interaction forces entanglement (under symmetry). -/
 101theorem interaction_forces_entanglement (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 102    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 103    (hNorm : F 1 = 0)
 104    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
 105    (hInt : HasInteraction F) :
 106    IsEntangling P :=
 107  interaction_implies_entangling F P hCons hNorm hSymm hInt
 108
 109/-! ## Gate 3: Curvature Characterizes the ODE -/
 110
 111/-- J's log-lift satisfies the hyperbolic ODE. -/
 112theorem Jcost_hyperbolic_ODE : SatisfiesHyperbolicODE Gcosh := Gcosh_satisfies_hyperbolic
 113
 114/-- Fquad's log-lift satisfies the flat ODE. -/
 115theorem Fquad_flat_ODE : SatisfiesFlatODE Gquad := Gquad_satisfies_flat
 116
 117/-- The two ODEs are mutually exclusive. -/
 118theorem flat_not_hyperbolic : ¬ SatisfiesHyperbolicODE Gquad := Gquad_not_hyperbolic
 119
 120theorem hyperbolic_not_flat : ¬ SatisfiesFlatODE Gcosh := Gcosh_not_flat
 121
 122/-! ## The Bridge: Connecting the Gates -/
 123
 124/-- **Key Hypothesis**: Interaction + Structural Axioms forces the hyperbolic ODE.
 125
 126    This is the central bridge connecting the gates. It says:
 127    If F has interaction, symmetry, normalization, smoothness, and consistency,
 128    then the log-lift G satisfies G'' = G + 1.
 129
 130    This is NOT yet fully proved from first principles, but is strongly motivated by:
 131    1. The counterexample (no interaction) ⟹ flat ODE
 132    2. J (has interaction) ⟹ hyperbolic ODE
 133    3. Entanglement forces a specific functional form
 134
 135    We state it as an explicit hypothesis to make the logical structure clear.
 136-/
 137def InteractionForcesHyperbolicODE : Prop :=
 138  ∀ (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ),
 139    F 1 = 0 →
 140    (∀ x : ℝ, 0 < x → F x = F x⁻¹) →
 141    ContDiff ℝ 2 F →
 142    deriv (deriv (fun t => F (Real.exp t))) 0 = 1 →
 143    (∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) →
 144    HasInteraction F →
 145    SatisfiesHyperbolicODE (fun t => F (Real.exp t))
 146
 147/-! ## The Main Inevitability Theorem -/
 148
 149/-- **Full Inevitability Theorem (Triangulated Form)**
 150
 151    Under structural axioms + interaction, both F and P are uniquely forced.
 152
 153    This theorem makes the bridge hypothesis explicit, showing exactly what
 154    remains to be proved for full unconditional inevitability.
 155-/
 156theorem full_inevitability_triangulated
 157    (bridge : InteractionForcesHyperbolicODE)
 158    (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 159    (hNorm : F 1 = 0)
 160    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
 161    (hSmooth : ContDiff ℝ 2 F)
 162    (hCalib : deriv (deriv (fun t => F (Real.exp t))) 0 = 1)
 163    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 164    (hInt : HasInteraction F) :
 165    -- Part 1: P is entangling (unconditional from interaction)
 166    IsEntangling P ∧
 167    -- Part 2: G satisfies hyperbolic ODE (from bridge)
 168    SatisfiesHyperbolicODE (fun t => F (Real.exp t)) := by
 169  constructor
 170  -- Part 1: Interaction ⟹ Entanglement
 171  · exact interaction_forces_entanglement F P hCons hNorm hSymm hInt
 172  -- Part 2: Bridge hypothesis gives hyperbolic ODE
 173  · exact bridge F P hNorm hSymm hSmooth hCalib hCons hInt
 174
 175/-- If F = J, then P = RCL (unconditional, no bridge needed). -/
 176theorem P_forced_from_FJ (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 177    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 178    (hFJ : ∀ x : ℝ, 0 < x → F x = Cost.Jcost x) :
 179    ∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v := by
 180  have hJcons : ∀ x y : ℝ, 0 < x → 0 < y →
 181      Cost.Jcost (x * y) + Cost.Jcost (x / y) = P (Cost.Jcost x) (Cost.Jcost y) := by
 182    intro x y hx hy
 183    have h := hCons x y hx hy
 184    rw [hFJ x hx, hFJ y hy, hFJ (x * y) (mul_pos hx hy), hFJ (x / y) (div_pos hx hy)] at h
 185    exact h
 186  exact rcl_unconditional P hJcons
 187
 188/-! ## Summary: What Is Proved vs What Is Assumed -/
 189
 190/-- **Summary Theorem**: All four gates point to the same conclusion.
 191
 192    - Gate 1 (Interaction): Distinguishes J from Fquad
 193    - Gate 2 (Entanglement): Characterizes RCL vs additive combiner
 194    - Gate 3 (Curvature): Characterizes hyperbolic vs flat ODE
 195    - Gate 4 (d'Alembert): Forces λ = 1 in cosh(λt), completing the chain
 196
 197    All four gates are consistent: J passes all four, Fquad fails all four.
 198-/
 199theorem gates_consistent :
 200    -- J has all four properties
 201    HasInteraction Cost.Jcost ∧
 202    IsEntangling Prcl ∧
 203    SatisfiesHyperbolicODE Gcosh ∧
 204    FourthGate.HasDAlembert Cost.Jcost ∧
 205    -- Fquad/Padd have the opposite properties
 206    ¬ HasInteraction Counterexamples.Fquad ∧
 207    ¬ IsEntangling Padd ∧
 208    SatisfiesFlatODE Gquad ∧
 209    ¬ FourthGate.HasDAlembert Counterexamples.Fquad := by
 210  exact ⟨Jcost_hasInteraction, Prcl_entangling, Gcosh_satisfies_hyperbolic,
 211         FourthGate.Jcost_has_dAlembert_structure,
 212         Fquad_noInteraction, Padd_not_entangling, Gquad_satisfies_flat,
 213         FourthGate.Fquad_not_dAlembert_structure⟩
 214
 215/-! ## Complete Inevitability with Four Gates -/
 216
 217/-- **Full Inevitability with Four Gates**: d'Alembert structure completes the proof.
 218
 219    Unlike the three-gate version which required a bridge hypothesis,
 220    the four-gate version is fully proved:
 221
 222    d'Alembert structure + structural axioms ⟹ F = J ⟹ P = RCL
 223-/
 224theorem full_inevitability_four_gates (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 225    (hNorm : F 1 = 0)
 226    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
 227    (hSmooth : ContDiff ℝ 2 F)
 228    (hCalib : deriv (deriv (fun t => F (Real.exp t))) 0 = 1)
 229    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
 230    (hDA : FourthGate.HasDAlembert F) :
 231    -- Part 1: F = J
 232    (∀ x : ℝ, 0 < x → F x = Cost.Jcost x) ∧
 233    -- Part 2: P = RCL on [0,∞)²
 234    (∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v) := by
 235  constructor
 236  · -- Part 1: F = J from d'Alembert structure
 237    exact FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
 238  · -- Part 2: P = RCL from F = J
 239    have hFJ := FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
 240    exact P_forced_from_FJ F P hCons hFJ
 241
 242/-- The three gates are equivalent for distinguishing J from Fquad. -/
 243theorem gates_equivalent_for_Jcost :
 244    HasInteraction Cost.Jcost ↔
 245    (∃ P, (∀ x y, 0 < x → 0 < y →
 246      Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) ∧
 247      IsEntangling P) := by
 248  constructor
 249  · intro _
 250    use Prcl
 251    refine ⟨?_, Prcl_entangling⟩
 252    intro x y hx hy
 253    -- This follows from the J_computes_P lemma
 254    have h := J_computes_P x y hx hy
 255    -- h : J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y)
 256    -- Prcl u v = 2uv + 2u + 2v
 257    unfold Prcl
 258    linarith
 259  · intro ⟨_, _, _⟩
 260    exact Jcost_hasInteraction
 261
 262end TriangulatedProof
 263end DAlembert
 264end Foundation
 265end IndisputableMonolith
 266