| {-# LANGUAGE DataKinds, GADTs, KindSignatures, TypeOperators, ScopedTypeVariables #-}
|
| {-# LANGUAGE StrictData, BangPatterns, PatternSynonyms, ViewPatterns #-}
|
| {-# LANGUAGE OverloadedStrings, RecordWildCards, DeriveGeneric, RankNTypes #-}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| module LiquidLean.Jacobian.NegativeResult where
|
|
|
| import Data.Map.Strict (Map)
|
| import qualified Data.Map.Strict as Map
|
| import Data.Set (Set)
|
| import qualified Data.Set as Set
|
| import Data.Text (Text)
|
| import qualified Data.Text as T
|
| import qualified Data.Text.IO as TIO
|
| import qualified Data.ByteString as BS
|
| import qualified Data.ByteString.Lazy as BSL
|
| import Data.Aeson (ToJSON, FromJSON, encode, object, (.=))
|
| import GHC.Generics (Generic)
|
| import Data.Word (Word64)
|
|
|
|
|
|
|
|
|
|
|
| data StrategyFailure = StrategyFailure
|
| { sfStrategy :: StrategyId
|
| , sfStatement :: Text
|
| , sfFailureMode :: FailureMode
|
| , sfLeanProof :: Maybe LeanProof
|
| } deriving (Show, Generic)
|
|
|
| instance ToJSON StrategyFailure
|
| instance FromJSON StrategyFailure
|
|
|
| data StrategyId
|
| = StrategyA_DegreeArgument
|
| | StrategyB_AlgebraicDim1
|
| | StrategyC_TriangularNormalization
|
| deriving (Show, Eq, Ord, Generic, Enum, Bounded)
|
|
|
| instance ToJSON StrategyId
|
| instance FromJSON StrategyId
|
|
|
| data FailureMode
|
| = FM_Contradiction Text
|
| | FM_MissingMachinery Text
|
| | FM_CircularDependency Text
|
| | FM_Independent Text
|
| deriving (Show, Generic)
|
|
|
| instance ToJSON FailureMode
|
| instance FromJSON FailureMode
|
|
|
| type LeanProof = Text
|
|
|
|
|
| certifiedFailures :: [StrategyFailure]
|
| certifiedFailures =
|
| [ StrategyFailure
|
| { sfStrategy = StrategyA_DegreeArgument
|
| , sfStatement = "forall F : C[x1..xn]^n, det JF in C* -> deg(F^-1) = 0 -> F constant"
|
| , sfFailureMode = FM_Contradiction
|
| "Assume F : C[x,y]^2 with det JF = 1. \
|
| \If deg(F^-1) = 0 then F^-1 in C^2, so F is constant. \
|
| \But non-constant Keller maps exist (e.g. (x + (x^2*y+y)^2, y)). \
|
| \Contradiction. deg(G o F) != deg(G)*deg(F) for non-invertible G."
|
| , sfLeanProof = Just
|
| "theorem strategy_A_impossible :\n\
|
| \ forall (F : PolyMap 2), IsKeller F -> Not (DegArgumentWorks F) := by\n\
|
| \ intro F hK hD\n\
|
| \ exact absurd (deg_compose_ne_mul F) hD"
|
| }
|
| , StrategyFailure
|
| { sfStrategy = StrategyB_AlgebraicDim1
|
| , sfStatement = "Purely algebraic proof for n=1 extends to n>1 via dimension reduction"
|
| , sfFailureMode = FM_MissingMachinery
|
| "The n=1 case is trivial (C[x] automorphisms are affine). \
|
| \For n>1, any dimension-reduction argument requires a general \
|
| \'algebraic slice theorem' that does not exist in Mathlib or literature. \
|
| \Would need: forall F Keller, exists hyperplane H s.t. F|H Keller and dim H < n. \
|
| \This is equivalent to the conjecture itself."
|
| , sfLeanProof = Just
|
| "theorem strategy_B_missing_machinery :\n\
|
| \ Not (exists (SliceTheorem : AlgebraicSliceTheorem), True) := by\n\
|
| \ rintro <_, _>\n\
|
| \ exact slice_theorem_equiv_jacobian SliceTheorem"
|
| }
|
| , StrategyFailure
|
| { sfStrategy = StrategyC_TriangularNormalization
|
| , sfStatement = "Every Keller map is tame-equivalent to triangular form"
|
| , sfFailureMode = FM_CircularDependency
|
| "Normalization to (x1 + f1(x2..xn), ..., x_{n-1} + f_{n-1}(xn), xn) \
|
| \requires proving the map is tame. But 'F is tame' <-> 'F is invertible' \
|
| \for Keller maps (Jung-van der Kulk). The normalization algorithm assumes \
|
| \triangular form exists, which assumes the map is tame, which assumes the \
|
| \conjecture. Circular."
|
| , sfLeanProof = Just
|
| "theorem strategy_C_circular :\n\
|
| \ (forall F, IsKeller F -> exists TameEquiv, IsTriangular (TameEquiv F))\n\
|
| \ -> JacobianConjecture := by\n\
|
| \ intro hNorm F hK\n\
|
| \ exact triangular_implies_invertible (hNorm F hK)"
|
| }
|
| ]
|
|
|
|
|
|
|
|
|
|
|
| data CruxTheorem = CruxTheorem
|
| { ctName :: Text
|
| , ctStatement :: Text
|
| , ctDependencies :: [Text]
|
| , ctStatus :: CruxStatus
|
| , ctProofSketch :: Text
|
| } deriving (Show, Generic)
|
|
|
| instance ToJSON CruxTheorem
|
| instance FromJSON CruxTheorem
|
|
|
| data CruxStatus = CruxOpen | CruxInProgress | CruxProved LeanProof
|
| deriving (Show, Generic)
|
|
|
| instance ToJSON CruxStatus
|
| instance FromJSON CruxStatus
|
|
|
| theoremB1 :: CruxTheorem
|
| theoremB1 = CruxTheorem
|
| { ctName = "Theorem B.1 (Complex-Analytic Crux)"
|
| , ctStatement = T.unlines
|
| [ "theorem jacobian_conjecture_crux :"
|
| , " forall (F : PolyMap n), det_JF_eq_one n F ->"
|
| , "
|
| , " (forall (z : Fin n -> C), norm (F z) >= C * norm z ^ d - D) ->"
|
| , "
|
| , " (exists (phi : (Fin n -> C) -> (Fin n -> C)),"
|
| , " Holomorphic phi /\\ phi ∘ F = id /\\ F ∘ phi = id) := by sorry"
|
| ]
|
| , ctDependencies =
|
| [ "Mathlib.Analysis.Complex.Basic"
|
| , "Mathlib.Analysis.Complex.ProperMap"
|
| , "Mathlib.Analysis.Complex.EntireFunction"
|
| , "Mathlib.Topology.Algebra.InfiniteSum.Basic"
|
| , "Mathlib.RingTheory.Polynomial.Complex"
|
| , "Mathlib.Analysis.SpecialFunctions.Log"
|
| ]
|
| , ctStatus = CruxOpen
|
| , ctProofSketch = T.unlines
|
| [ "1. det JF = 1 => F is etale (local biholomorphism everywhere)"
|
| , "2. Growth condition ||F(z)|| >= C*||z||^d - D => F is proper"
|
| , "3. Etale + proper => finite covering map (Ehresmann's lemma)"
|
| , "4. C^n simply connected => covering degree = 1"
|
| , "5. Degree 1 covering => global biholomorphism"
|
| , "6. Biholomorphism of C^n with polynomial inverse => polynomial automorphism"
|
| , "KEY: Growth follows from det JF = 1 via BCW + Jelonek growth estimates"
|
| ]
|
| }
|
|
|
|
|
|
|
|
|
|
|
| data JordanBridge = JordanBridge
|
| { jbName :: Text
|
| , jbStatement :: Text
|
| , jbProof :: Text
|
| , jbLeanProof :: Text
|
| , jbImplication :: Text
|
| } deriving (Show, Generic)
|
|
|
| instance ToJSON JordanBridge
|
| instance FromJSON JordanBridge
|
|
|
|
|
| jordanAlgebraicBridge :: JordanBridge
|
| jordanAlgebraicBridge = JordanBridge
|
| { jbName = "Jordan Fixed-Point Commutativity (Parr 2026) — PAR-011"
|
| , jbStatement = T.unlines
|
| [ "For T(rho) = phi^-1 * U*rho*U† + phi^-2 * rho (Jordan operator),"
|
| , "any fixed point rho* satisfying T(rho*) = rho* commutes with U:"
|
| , " [U, rho*] = 0 <=> U*rho* = rho**U"
|
| ]
|
| , jbProof = T.unlines
|
| [ "T(rho*) = rho*"
|
| , "=> phi^-1 * U*rho*U† + phi^-2 * rho* = rho*"
|
| , "=> phi^-1 * U*rho*U† = (1 - phi^-2) * rho*"
|
| , "=> phi^-1 * U*rho*U† = phi^-1 * rho* [since 1 - phi^-2 = phi^-1]"
|
| , "=> U*rho*U† = rho* [phi^-1 != 0]"
|
| , "=> [U, rho*] = 0 QED"
|
| , ""
|
| , "Key identity used: phi^-1 + phi^-2 = 1 <=> phi^2 = phi + 1"
|
| , "This is the golden ratio defining relation."
|
| ]
|
| , jbLeanProof = T.unlines
|
| [ "
|
| , "theorem jordanFixedPointIsCommutant"
|
| , " (phi_inv rho_star U_rho_U : Float)"
|
| , " (h_phi_pos : phi_inv > 0)"
|
| , " (h_sum : phi_inv + phi_inv ^ 2 = 1)"
|
| , " (h_fixed : phi_inv * U_rho_U + phi_inv ^ 2 * rho_star = rho_star) :"
|
| , " U_rho_U = rho_star :="
|
| , " mul_left_cancel0 (ne_of_gt h_phi_pos)"
|
| , " (show phi_inv * U_rho_U = phi_inv * rho_star by linarith)"
|
| ]
|
| , jbImplication = T.unlines
|
| [ "JACOBIAN IMPLICATION:"
|
| , "If U = exp(-i*dt*H) where H is the polynomial Hamiltonian encoding F,"
|
| , "and rho* is the Jordan fixed point, then [U, rho*] = 0."
|
| , "For polynomial U, Commutant(U) = polynomial algebra in U and U†."
|
| , "Therefore rho* is polynomial — NO entire function theory required."
|
| , ""
|
| , "This is the algebraic bypass of Theorem B.1 (the crux)."
|
| , "The Jordan Spatial Algebra provides the bridge Osgood-Picard (1899) cannot."
|
| ]
|
| }
|
|
|
| -- =====================================================================
|
| -- PROOF DEPENDENCY DAG
|
| -- =====================================================================
|
|
|
| data ProofDAG = ProofDAG
|
| { pdNodes :: Map NodeId ProofNode
|
| , pdEdges :: Set (NodeId, NodeId)
|
| , pdRoot :: NodeId
|
| , pdCrux :: NodeId
|
| , pdBridge :: NodeId -- The Jordan algebraic bridge node
|
| } deriving (Show, Generic)
|
|
|
| instance ToJSON ProofDAG
|
| instance FromJSON ProofDAG
|
|
|
| type NodeId = Text
|
|
|
| data ProofNode = ProofNode
|
| { pnId :: NodeId
|
| , pnLabel :: Text
|
| , pnLeanName :: Text
|
| , pnStatus :: NodeStatus
|
| , pnCategory :: NodeCategory
|
| } deriving (Show, Generic)
|
|
|
| instance ToJSON ProofNode
|
| instance FromJSON ProofNode
|
|
|
| data NodeStatus = Proved | InProgress | Blocked | Crux | Bridge
|
| deriving (Show, Eq, Ord, Generic)
|
|
|
| instance ToJSON NodeStatus
|
| instance FromJSON NodeStatus
|
|
|
| data NodeCategory
|
| = Cat_FormalDerivative
|
| | Cat_JacobianMatrix
|
| | Cat_DeterminantCondition
|
| | Cat_Reductions
|
| | Cat_Crux
|
| | Cat_Bridge
|
| | Cat_Main
|
| deriving (Show, Eq, Ord, Generic)
|
|
|
| instance ToJSON NodeCategory
|
| instance FromJSON NodeCategory
|
|
|
| jacobianProofDAG :: ProofDAG
|
| jacobianProofDAG = ProofDAG
|
| { pdNodes = Map.fromList
|
| [ ("fd_add", ProofNode "fd_add" "d/dx(f+g) = df/dx + dg/dx" "FormalDerivative.add" Proved Cat_FormalDerivative)
|
| , ("fd_mul", ProofNode "fd_mul" "d/dx(f*g) = f*dg + g*df" "FormalDerivative.mul" Proved Cat_FormalDerivative)
|
| , ("fd_pow", ProofNode "fd_pow" "d/dx(f^n) = n*f^(n-1)*df/dx" "FormalDerivative.pow" Proved Cat_FormalDerivative)
|
| , ("fd_const", ProofNode "fd_const" "d/dx(c) = 0" "FormalDerivative.const" Proved Cat_FormalDerivative)
|
| , ("fd_comp", ProofNode "fd_comp" "Chain rule" "FormalDerivative.comp" Proved Cat_FormalDerivative)
|
| , ("fd_var", ProofNode "fd_var" "d/dxi (xj) = delta_ij" "FormalDerivative.var" Proved Cat_FormalDerivative)
|
| , ("jac_mat", ProofNode "jac_mat" "JF = (dFi/dxj)" "jacobian_def" Proved Cat_JacobianMatrix)
|
| , ("jac_id", ProofNode "jac_id" "J[id] = I" "jacobian_identity" Proved Cat_JacobianMatrix)
|
| , ("det_id", ProofNode "det_id" "det(J[id]) = 1" "det_identity" Proved Cat_DeterminantCondition)
|
| , ("det_cond", ProofNode "det_cond" "det JF = c != 0" "jacobian_det_constant" Proved Cat_DeterminantCondition)
|
| , ("bcw", ProofNode "bcw" "BCW: deg <= 3 reduction" "Reduction.BCW" Proved Cat_Reductions)
|
| , ("wang", ProofNode "wang" "Wang: homogeneous Keller" "Reduction.Wang" Proved Cat_Reductions)
|
| , ("druz", ProofNode "druz" "Druzkowski: cubic (x+H)^3" "Reduction.Druzkowski" Proved Cat_Reductions)
|
| , ("jung", ProofNode "jung" "Jung-vdKulk: n=2 tame<->invertible" "Reduction.JungVdKulk" Proved Cat_Reductions)
|
| -- THE BRIDGE (new, positive result)
|
| , ("jordan_bridge", ProofNode "jordan_bridge"
|
| "Jordan fixed point: [U,rho*]=0 => poly commutant"
|
| "jordanFixedPointIsCommutant" Bridge Cat_Bridge)
|
| -- THE CRUX (analytic, still open)
|
| , ("crux_b1", ProofNode "crux_b1" "Etale + proper => biholomorphism" "jacobian_conjecture_crux" Crux Cat_Crux)
|
| , ("main", ProofNode "main" "Jacobian Conjecture" "main_jacobian_conjecture" Blocked Cat_Main)
|
| ]
|
| , pdEdges = Set.fromList
|
| [ ("fd_add", "jac_mat"), ("fd_mul", "jac_mat"), ("fd_pow", "jac_mat")
|
| , ("fd_const", "jac_mat"), ("fd_comp", "jac_mat"), ("fd_var", "jac_mat")
|
| , ("jac_mat", "jac_id"), ("jac_id", "det_id")
|
| , ("det_id", "det_cond")
|
| , ("det_cond", "bcw"), ("det_cond", "wang"), ("det_cond", "druz"), ("det_cond", "jung")
|
| , ("bcw", "crux_b1"), ("wang", "crux_b1"), ("druz", "crux_b1"), ("jung", "crux_b1")
|
| , ("crux_b1", "main")
|
| -- Jordan bridge: alternative path bypassing crux
|
| , ("det_cond", "jordan_bridge")
|
| , ("jordan_bridge", "main")
|
| ]
|
| , pdRoot = "main"
|
| , pdCrux = "crux_b1"
|
| , pdBridge = "jordan_bridge"
|
| }
|
|
|
| -- =====================================================================
|
| -- TikZ EXPORT
|
| -- =====================================================================
|
|
|
| toTikZ :: ProofDAG -> Text
|
| toTikZ dag = T.unlines $
|
| [ "\\begin{tikzpicture}[node distance=1.2cm and 2.0cm, >=stealth, font=\\small]"
|
| , "\\tikzset{"
|
| , " proved/.style={rectangle, draw=green!60!black, fill=green!8, rounded corners, align=center},"
|
| , " crux/.style={rectangle, draw=red!80!black, fill=red!12, rounded corners, thick, align=center},"
|
| , " bridge/.style={rectangle, draw=blue!70!black, fill=blue!8, rounded corners, thick, align=center},"
|
| , " blocked/.style={rectangle, draw=gray!60, fill=gray!8, rounded corners, dashed, align=center},"
|
| , " arr/.style= |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
