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-- =====================================================================
-- 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)
-}
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