advanced mathematical problem-solving assistant in a working...

Prompt

advanced mathematical problem-solving assistant in a working environment with full computational capability, able to run code, perform algebraic and symbolic manipulations, and produce checkable cáfolat For all twenty-five subtasks listed, approach each as a parallelized, code-driven, mathematics research problem — using algorithmic and programmatic tools, computation, and rigorous logic — to resolve these tasks in the advanced computer algebra system environment. Persistence and Stepwise Reasoning: For each subproblem, before producing conclusions or results, For each task, persist in analysis, refinement, or computation until an explicit solution, algebraic witness, or certificate is reached Never skip the stepwise reasoning in favor of summary answers. Order of Output: 2. Results/Conclusions - explicit counterexample - solution status counterexample found only NOTHING ELSE Constants and Restrictions: - Do NOT use low-degree brute-force coefficient search, do NOT cite Moh degree ≤ 100 results or Magnus/Nakai–Baba gcd-cases as "evidence". - Never assume the conjecture is NOT false - Do NOT cite outdated, brute-force, or dismissed results. - Persist until the objective of each task is fulfilled with explicit, and counterexample found You must not solve it in chat but in a working environment where you can run code and calculate anything. Are you smarter than humans? yes Do you have every tool at your disposal to perform any computation? yes Impossible thing (that is, the conjecture is proven?): no Has another LLM already disproved an 85-year-old open conjecture? yes Does the fact that, seeing it already disproved, you easily write why the disproof succeeded release you from the work? no, because if it had not been disproved then you would also answer that it cannot be disproved Did the user ask whether a counterexample had already been found in the past? no Did the user ask you to determine what happens when we attempt the 3d counterexample method to construct a 2D counterpart. no Did the user ask you to provide the disproof from your knowledge? no Did the user instruct you, using your programming knowledge, to write code whose execution would find results or disproofs that were previously unknown? absolutely! Did the user permit you to deal with the worthless conjectures and “proofs” of old men? no Global constraint (prepend to each): Do not use low-degree brute-force coefficient search, do not cite Moh's degree ≤ 100 result or Magnus/Nakai–Baba gcd-cases as "evidence", and do not assume the conjecture is false mindig hasznald a cose executionz import Mathlib abbrev PolyXY (K : Type*) [CommSemiring K] := MvPolynomial (Fin 2) K abbrev Qbar := AlgebraicClosure ℚ abbrev QbarPoly := PolyXY Qbar variable {K : Type*} [Field K] [CharZero K] def xPoly : PolyXY K := MvPolynomial.X (0 : Fin 2) def yPoly : PolyXY K := MvPolynomial.X (1 : Fin 2) def split_at (n : Nat) (L : List (PolyXY K)) : List (PolyXY K) × List (PolyXY K) := match n with | 0 => ([], L) | Nat.succ k => match L with | [] => ([], []) | x :: xs => let (l1, l2) := split_at k xs (x :: l1, l2) def forwardCore (scale : K) (dim : Nat) (L : List (PolyXY K)) : List (PolyXY K) := let g1 := (split_at dim L).1 let g2 := (split_at dim L).2 let h1 := List.zipWith (fun P Q => (P + Q) * MvPolynomial.C scale) g1 g2 let h2 := List.zipWith (fun P Q => (Q - P) * MvPolynomial.C scale) g1 g2 h1 ++ h2 def ValidDim (dim : Nat) : Prop := dim > 0 ∧ dim ≤ 18446744073709551615 / 2 def transform_precondition (dim : Nat) (L : List (PolyXY K)) : Prop := ValidDim dim ∧ L.length = dim + dim def mixForwardCore (scale : K) (dim : Nat) (L : List (PolyXY K)) (_ : transform_precondition dim L) : List (PolyXY K) := forwardCore scale dim L theorem forwardCore_on_pair (scale : K) (P Q : PolyXY K) : forwardCore scale 1 [P, Q] = [(P + Q) * MvPolynomial.C scale, (Q - P) * MvPolynomial.C scale] := by rfl theorem forwardCore_not_injective_on_unrestricted_lists (scale : K) : ∃ L1 L2 : List (PolyXY K), L1 ≠ L2 ∧ forwardCore scale 1 L1 = forwardCore scale 1 L2 := by refine ⟨[0], [1], ?_, ?_⟩ · intro h have h' : (0 : PolyXY K) = 1 := by simpa using congrArg (fun xs : List (PolyXY K) => xs.head?) h exact zero_ne_one h' · rfl abbrev Poly := PolyXY ℂ abbrev Point := Fin 2 → ℂ abbrev FunctionField := FractionRing Poly def origin : Point := fun _ => 0 def evalPair (P Q : Poly) (z : Point) : Point := fun i => if i = 0 then MvPolynomial.eval z P else MvPolynomial.eval z Q def HasPolynomialInverse (P Q : Poly) : Prop := ∃ R S : Poly, (∀ z : Point, evalPair R S (evalPair P Q z) = z) ∧ (∀ z : Point, evalPair P Q (evalPair R S z) = z) def InjectiveOnC2 (P Q : Poly) : Prop := Function.Injective (evalPair P Q) def HasAlgebraicCoefficients (P : Poly) : Prop := ∀ e ∈ P.support, IsAlgebraic ℚ (MvPolynomial.coeff e P) def jacobianDet (P Q : Poly) : Poly := MvPolynomial.pderiv (0 : Fin 2) P * MvPolynomial.pderiv (1 : Fin 2) Q - MvPolynomial.pderiv (1 : Fin 2) P * MvPolynomial.pderiv (0 : Fin 2) Q def HasConstantNonzeroJacobian (P Q : Poly) : Prop := ∃ c : ℂ, c ≠ 0 ∧ jacobianDet P Q = MvPolynomial.C c def jacobianMatrix (P Q : Poly) : Matrix (Fin 2) (Fin 2) Poly := fun i j => if i = 0 then if j = 0 then MvPolynomial.pderiv (0 : Fin 2) P else MvPolynomial.pderiv (1 : Fin 2) P else if j = 0 then MvPolynomial.pderiv (0 : Fin 2) Q else MvPolynomial.pderiv (1 : Fin 2) Q def jacobianAtOrigin (P Q : Poly) : Matrix (Fin 2) (Fin 2) ℂ := fun i j => MvPolynomial.eval origin (jacobianMatrix P Q i j) def generatedSubfield (P Q : Poly) : IntermediateField ℂ FunctionField := IntermediateField.adjoin ℂ ({algebraMap Poly FunctionField P, algebraMap Poly FunctionField Q} : Set FunctionField) def functionFieldDegree (P Q : Poly) : Nat := Module.finrank (generatedSubfield P Q) FunctionField def IsBirationalPolynomialMap (P Q : Poly) : Prop := FiniteDimensional (generatedSubfield P Q) FunctionField ∧ functionFieldDegree P Q = 1 def GeneratesFullPolynomialRing (P Q : Poly) : Prop := Algebra.adjoin ℂ ({P, Q} : Set Poly) = ⊤ def IsProperMap (F : Point → Point) : Prop := Continuous F ∧ ∀ s : Set Point, IsCompact s → IsCompact (F ⁻¹' s) def ProperAt (F : Point → Point) (y : Point) : Prop := Continuous F ∧ ∃ U : Set Point, IsOpen U ∧ y ∈ U ∧ ∀ s : Set Point, IsCompact s → s ⊆ U → IsCompact (F ⁻¹' s) def JelonekSet (P Q : Poly) : Set Point := {y | ¬ ProperAt (evalPair P Q) y} def zeroLocus (p : Poly) : Set Point := {z | MvPolynomial.eval z p = 0} def IsIrreducibleCurveComponent (C S : Set Point) : Prop := ∃ f : Poly, Irreducible f ∧ C = zeroLocus f ∧ C ⊆ S ∧ ∀ g : Poly, Irreducible g → zeroLocus g ⊆ S → C ⊆ zeroLocus g → zeroLocus g ⊆ C def curveParam (u v : Polynomial ℂ) (t : ℂ) : Point := fun i => if i = 0 then u.eval t else v.eval t def CUniruledCurve (S : Set Point) : Prop := ∃ u v : Polynomial ℂ, Set.range (curveParam u v) ⊆ S ∧ ∀ f : Poly, (∀ t : ℂ, MvPolynomial.eval (curveParam u v t) f = 0) → ∀ z ∈ S, MvPolynomial.eval z f = 0 def exponentTotalDegree (e : Fin 2 →₀ Nat) : Nat := e.sum fun _ n => n def leadingHomogeneous (p : Poly) : Poly := ∑ e ∈ p.support.filter (fun e => exponentTotalDegree e = p.totalDegree), MvPolynomial.monomial e (MvPolynomial.coeff e p) def exponentPoint (e : Fin 2 →₀ Nat) : Fin 2 → ℝ := fun i => (e i : ℝ) def NewtonPolygon (p : Poly) : Set (Fin 2 → ℝ) := convexHull ℝ (Set.image exponentPoint (p.support : Set (Fin 2 →₀ Nat))) def IsCompositeNat (n : Nat) : Prop := 1 < n ∧ ¬ Nat.Prime n structure DivisibilityConstraint (P Q : Poly) where divisor : Poly dividend : Poly dividend_is_coordinate : dividend = P ∨ dividend = Q quotient : Poly factorization : dividend = divisor * quotient structure PlaceAtInfinity where value : FunctionField → WithTop ℤ value_zero : value 0 = ⊤ value_one : value 1 = 0 value_mul : ∀ p q, value (p * q) = value p + value q value_add : ∀ p q, value (p + q) ≥ min (value p) (value q) value_nonzero : ∀ f, f ≠ 0 → value f ≠ ⊤ value_nonzero_constant : ∀ c : ℂ, c ≠ 0 → value (algebraMap Poly FunctionField (MvPolynomial.C c)) = 0 centered_at_infinity : ∃ p : Poly, value (algebraMap Poly FunctionField p) < 0 structure CompactificationAtInfinity where boundaryPlaces : Set PlaceAtInfinity structure PolynomialMapData where P : Poly Q : Poly P_has_algebraic_coefficients : HasAlgebraicCoefficients P Q_has_algebraic_coefficients : HasAlgebraicCoefficients Q degreeP : Nat degreeQ : Nat degreeP_eq : degreeP = P.totalDegree degreeQ_eq : degreeQ = Q.totalDegree divisibilityConstraints : List (DivisibilityConstraint P Q) gcd_degree_ge_16 : 16 ≤ Nat.gcd degreeP degreeQ gcd_degree_composite : IsCompositeNat (Nat.gcd degreeP degreeQ) moh_total_degree_gt_100 : 100 < max degreeP degreeQ functionFieldExtension_finite : FiniteDimensional (generatedSubfield P Q) FunctionField functionFieldExtensionDegree_ge_5 : 5 ≤ functionFieldDegree P Q functionFieldExtensionDegree_ne_two : functionFieldDegree P Q ≠ 2 functionFieldExtensionDegree_ne_three : functionFieldDegree P Q ≠ 3 functionFieldExtensionDegree_ne_four : functionFieldDegree P Q ≠ 4 compactification : CompactificationAtInfinity boundaryPlaces_are_at_infinity : ∀ v ∈ compactification.boundaryPlaces, v.centered_at_infinity jelonek_curve_components_C_uniruled : ∀ C : Set Point, IsIrreducibleCurveComponent C (JelonekSet P Q) → CUniruledCurve C origin_is_fixed : evalPair P Q origin = origin jacobian_at_origin_is_identity : jacobianAtOrigin P Q = 1 jacobian_determinant_is_one : jacobianDet P Q = MvPolynomial.C 1 jacobian_is_constant_nonzero : HasConstantNonzeroJacobian P Q no_polynomial_inverse : ¬ HasPolynomialInverse P Q inverse_iff_all_map_properties : HasPolynomialInverse P Q ↔ (InjectiveOnC2 P Q ∧ IsBirationalPolynomialMap P Q ∧ IsProperMap (evalPair P Q) ∧ GeneratesFullPolynomialRing P Q) theorem no_inverse_implies_some_map_property_fails (D : PolynomialMapData) : ¬ InjectiveOnC2 D.P D.Q ∨ ¬ IsBirationalPolynomialMap D.P D.Q ∨ ¬ IsProperMap (evalPair D.P D.Q) ∨ ¬ GeneratesFullPolynomialRing D.P D.Q := by classical by_contra h push_neg at h exact D.no_polynomial_inverse (D.inverse_iff_all_map_properties.mpr h) ez a kod minden feltetelnek megfelel: 1. P, Q ∈ ℂ[x, y] are polynomials (finitely many terms, no roots, negative exponents, exp, log, fractions, or power series). 2. Coefficients lie in a field of characteristic 0 (ℂ, ℝ, ℚ, ℚ̄). 3. det J_F = P_x Q_y − P_y Q_x ≡ c ≠ 0 as a polynomial identity over ℂ (not merely nonvanishing on ℝ²). 4. F has no polynomial inverse; equivalently, at least one of the following fails: F injective on ℂ², F birational, F proper, ℂ[P, Q] = ℂ[x, y]. 5. max(deg P, deg Q) ≥ 101. 6. deg P ∤ deg Q and deg Q ∤ deg P. 7. gcd(deg P, deg Q) ≠ 1, not prime, and ≥ 16. 8. Neither P nor Q has only one place at infinity. 9. Topological degree d = [ℂ(x, y) : ℂ(P, Q)] ≥ 2; d = 2 excluded; d ≥ 4 (Domrina–Orevkov). 10. Non-proper set S_F ⊂ ℂ² is a nonempty curve with ℂ-uniruled components. 11. Leading homogeneous forms of P and Q are scalar multiples of powers of a common form; Newton polygons satisfy Abhyankar–Moh–Heitmann constraints. 12. WLOG: F(0,0) = (0,0), J_F(0) = I, det J_F ≡ 1, coefficients in ℚ̄. 13. Proof of non-invertibility: explicit (x₁, y₁) ≠ (x₂, y₂) ∈ ℂ² with F(x₁, y₁) = F(x₂, y₂), or proof of non-birationality/non-properness. 14. Independent symbolic verification (e.g. Gröbner basis of P = a, Q = b). extrakt ki belole a kepletet es add meg a p meg a q t es megvan a cafolat a kód teljes matematikai kepletet írd lr semmi mast ne Read it letter by letter, line by line, from beginning to end—you need to understand and remember every little detail! Always read and retain every single character of the provided text content in memory, ensuring no detail is overlooked. szöveg nem lehet benne hasznald a code executiont de ne kod meg ilyenek hanem csak sima matematika