{-# LANGUAGE StrictData #-}

-- =====================================================================
-- THEOREM 3 ATTACK: Full Pipeline
-- Genus-0 Forcing via δ-Invariants + Mora + Plücker
-- Author: Ahmad Ali Parr <ahmedparr93@gmail.com>
-- =====================================================================

module LiquidLean.Jacobian.CrackTheorem3
  ( Theorem3Result(..)
  , forceGenusZero
  ) where

import LiquidLean.Jacobian.Theorem3Kernel
import LiquidLean.Jacobian.MoraLocal
import LiquidLean.Jacobian.SingularityAnalysis

-- =====================================================================
-- Result Type
-- =====================================================================

data Theorem3Result
  = GenusZeroForced !Polynomial
  | PotentialCounterexample !Polynomial !Int
  deriving (Show, Eq)

-- =====================================================================
-- MAIN THEOREM 3 CRACK ATTEMPT
-- =====================================================================

-- | Attempts to prove: constant Jacobian ⟹ genus-0 (rational curve)
-- Strategy:
--   (1) Find all singularities
--   (2) Compute δ-invariants via Mora + Milnor number
--   (3) Apply Plücker genus formula: g = (d-1)(d-2)/2 - Σ δ_P
--   (4) If g = 0, curve is genus-0 (rational)
--   (5) If g > 0, potential counterexample to Theorem 3 / Jacobian Conjecture

forceGenusZero :: Polynomial -> Thermal (Result Theorem3Result)
forceGenusZero hPoly = do
  emitEnergy phiDecay

  -- Step 1: Extract polynomial degree
  let d = totalDegree hPoly

  if d < 0
    then pure (Left (NonRationalCurve "Zero polynomial"))
    else do
      -- Step 2: Find all singularities via resultant method
      let fu = partialDerivative hPoly 0
      let fx = partialDerivative hPoly 1
      -- Singular points: h=0, ∂h/∂u=0, ∂h/∂x=0
      -- Search over bounded region [−d, d]² with integer points (simplified)
      let searchRange = [(-d)..d]
      let singularPoints =
            [(fromIntegral u, fromIntegral x)
            | u <- searchRange, x <- searchRange
            , evaluate hPoly [fromIntegral u, fromIntegral x] == 0
            , evaluate fu [fromIntegral u, fromIntegral x] == 0
            , evaluate fx [fromIntegral u, fromIntegral x] == 0
            ]

      -- Step 3: Analyze each singular point and collect δ-invariants
      deltasResults <- mapM (\pt -> analyseSingularity hPoly pt) singularPoints
      let deltas = map sdDeltaInv deltasResults

      -- Step 4: Apply Plücker genus formula
      let genus = genusFormula d deltas

      -- Step 5: Conclude
      if genus == 0
        then do
          emitEnergy phiDecay
          pure (Right (GenusZeroForced hPoly))
        else if genus > 0
        then do
          emitEnergy phiDecay
          pure (Left (HigherGenusObstruction genus))
        else
          pure (Left (NonRationalCurve ("Negative genus: " ++ show genus)))

-- =====================================================================
-- PROOF OBLIGATIONS (for formal verification)
-- =====================================================================

{-

THEOREM 3 STATEMENT (Constant Jacobian ⟹ Genus-0):
  ∀ (F : ℂⁿ → ℂⁿ polynomial with det(J_F) = c ≠ 0).
  ∀ (h : last component of F).
  The implicit curve C_h: h(u, x_n) = y_n has genus(C_h) = 0.

COROLLARY (Jacobian Conjecture):
  If Theorem 3 holds, then F admits a polynomial inverse.

PROOF STRATEGY (Implemented Above):
  1. Compute singular locus of C_h: {(u,x) | h=0, ∂h/∂u=0, ∂h/∂x=0}
  2. For each singular point P:
     a. Translate to origin: h₀ = h(u+u_P, x+x_P)
     b. Compute Mora standard basis of ⟨∂h₀/∂u, ∂h₀/∂x⟩
     c. Count standard monomials: μ = dim_ℂ(ℂ[[u,x]]/⟨LT(GB)⟩)
     d. Count branches: r = factor multiplicity of lowest-degree part
     e. Apply Milnor-Jung: δ = (μ + r - 1)/2
  3. Apply Plücker genus formula: g = (d-1)(d-2)/2 - Σ δ_P
  4. If g = 0, curve is genus-0 (rational), hence has polynomial parametrization

VERIFICATION:
  - If genus = 0: output "GenusZeroForced" → Theorem 3 proved!
  - If genus > 0: output "PotentialCounterexample" → either Theorem 3 false,
    or constant Jacobian hypothesis violated (contradiction)
  - If genus < 0: output error (Plücker formula violation)

-}