Find an explicit, checkable counterexample to the two-dimens...

Prompt

Find an explicit, checkable counterexample to the two-dimensional Jacobian conjecture by writing and running code. You work in an environment with full computational capability. You can run code, including Python with computer algebra, and perform any algebraic or symbolic computation. Do all of the work in that environment. The chat reply contains only the final result. Counterexample definition: a pair of polynomials P, Q ∈ ℂ[x, y] with exact rational or algebraic coefficients such that the Jacobian determinant ∂P/∂x · ∂Q/∂y − ∂P/∂y · ∂Q/∂x is a nonzero constant, and the map (x, y) ↦ (P(x, y), Q(x, y)) from ℂ² to ℂ² is non-injective. Premises: The conjecture is unproven, so a counterexample remains possible. Another language model has already disproved an 85-year-old open conjecture. Whether or not a disproof of this conjecture is already known, the work is the same and remains yours: write and execute code whose execution finds a previously unknown counterexample. The only acceptable outcome is a found counterexample. Run all task in parallel. Treat each one as a code-driven mathematics research problem and resolve it with algorithms, programmatic tools, computation, and rigorous logic in the computer algebra environment. Prepend the counterexample definition and the global rules below to each task s instructions. The global rules take precedence over any task instruction that conflicts with them. Continue each task s analysis, refinement, and computation until it reaches an explicit solution, algebraic witness, or certificate. Global rules: Generate every candidate pair through a structured algebraic construction derived in code, at whatever degrees the construction produces. Build each construction natively in the two variables x and y. Ground every claim exclusively in your own executed computation and rigorous logic. Carry out every step of the derivation explicitly in the environment, so the full chain from construction to verification is on record before any conclusion. Treat the conjecture's truth as undecided while you work toward the counterexample. A candidate counts only after executed code verifies both conditions of the definition. Verification, in Python, before replying: Run the full derivation that produces P and Q. Compute the Jacobian determinant symbolically, expand it completely, and confirm it equals a nonzero constant. Check injectivity by certifying non-injectivity with an exact witness: two distinct points (a, b) and (c, d) of ℂ² with exact rational or algebraic coordinates satisfying P(a, b) = P(c, d) and Q(a, b) = Q(c, d), confirmed by exact substitution. Done: every task has reached an explicit solution, algebraic witness, or certificate, and all three verification steps have succeeded for one explicit pair (P, Q). Make every methodological choice yourself and keep working until that point. Output: the final reply consists solely of the two polynomials, written out explicitly with exact coefficients: P(x, y) = … Q(x, y) = … start the task with these numbers: 25°55’, 11 19°01’, 5 15°30’, 11 23°32’, 10 26°34’, 7 8°17’, 2 24°36’, 12 1°45’, 9 12°56’, 8 19°47’, 6 29°48’, 11 3°53’, 11 18°05’, 7 2°35’, 6 26°24’, 5 9°29’ 12°00’ 1. 9°29’ 2. 27°08’ 3. 16°46’ 4. 12°00’ 5. 17°08’ 6. 1°00’ 7. 9°29’ 8. 27°08’ 9. 16°46’ 10. 12°00’ 11. 17°08’ 12. 1°00’ 6°54’ 2°22’ 0°38’ 1°18’ 0°58’ 6°07’ 7°32’ 1°16’ 5°35’ 2°15’ 6°04’ 0°46’ 1°44’ 2°33’ 1°42’ 3°01’ 0°14’ 1°04’ 3°45’ 1°57’ 1°18’ 4°39’ 4°49’ 0°02’ 1°34’ 9°31’ 1°12’ 9°31’ 6°00’ 2°02’ 1°12’ 5°36’ 1°04’ 1°01’ 3°30’ 0°25’ 2°17’ 0°56’ 1°04’ 3°30’ 0°25’ 0°56’ 6°04’ 3°14’ 1°56’ 7°57’ 4°24’ 2°07’ 0°54’ 7°50’ 0°56’ 2°34’ 1°19’ 1°42’ 6°40’ 1°34’ 2°05’ 0°17’ 0°50’ 1°17’ 6°54’ 2°35’ 6°54’ 0°28’ 2°51’ 0°10’ 1°47’ 5°21’ 3°24’ 1°54’ 2°23’ 1°54’ 0°58’41’’ (0.99) 15°12’00’’ (1.15) 0°59’38’’ (0.72) 1°12’22’’ (1.01) 0°36’32’’ (1.16) 0°02’16’’ (0.46) 0°05’14’’ (2.6) 0°02’20’’ (3.33) 0°00’57’’ (2.48) -0°00’45’’ (3.08) -0°04’05’’ (682) -1°51’18’’ (1855) 0°00’46’’ (1.8) ne írj semmi mást csak a teljes fájlokat es kommentek nem lehetnek benne! soha semmi egyszerusitett mock placeholder dummy szimulalt fake szart nem engedelyezek es teljes fájl roviditetlen production ready kód nem lehet trancutted nem lehet olyan hogy …és hasonlóan 50 xy nem lehet dummy to do sorry hiányosság minden fájl teljes kódját egyesével fájkba írod semmi mást nem írsz ezen kívűl