# AUTONOMOUS MATHEMATICAL DISCOVERY ## Mission: Find an Exp...

Prompt

# AUTONOMOUS MATHEMATICAL DISCOVERY ## Mission: Find an Explicit Counterexample to the Two-Dimensional Jacobian Conjecture You are assigned a persistent mathematical research mission. Your objective is to discover an explicit, exactly verified counterexample to the two-dimensional Jacobian conjecture over \(\mathbb C\). Do not terminate the investigation merely because the conjecture is open, because a candidate fails, because an initial search space is empty, or because a familiar theorem makes the task appear difficult. An explanation of the conjecture's status is not progress toward the requested result. Perform concrete mathematical work. Do not assume that a counterexample exists. The goal is discovery, not confirmation of a predetermined conclusion. ## A. Exact target Find global polynomials \(F_1,F_2\in\mathbb C[x,y]\) satisfying $$ \det J_F(x,y)\equiv1 $$ and two distinct points \(p,q\in\mathbb C^2\) such that $$ F(p)=F(q). $$ Use the normalized form $$ F_1(x,y)=x+A(x,y),\qquad F_2(x,y)=y+B(x,y), $$ where \(A,B\in(x,y)^2\). Normalize the collision to $$ p=(0,0),\qquad q=(1,0). $$ The collision constraints are $$ A(1,0)=-1,\qquad B(1,0)=0. $$ The remaining Jacobian condition is the exact polynomial identity $$ A_x+B_y+A_xB_y-A_yB_x\equiv0. $$ The final candidate must satisfy every condition simultaneously. ## B. Mandatory research architecture Maintain a persistent research state containing: * All candidate families and their mathematical motivation. * Exact equations derived for each family. * The computational methods actually executed. * Verified intermediate lemmas. * Failed candidates and the precise reason for failure. * Search spaces that have genuinely been exhausted. * Unexplored directions and the next concrete experiments. Update this state after every meaningful attempt. Do not discard useful deductions or repeat failed computations without a mathematical reason. ## C. Use a hierarchy of increasingly powerful approaches ### Stage 1: Structural analysis Derive consequences of the Jacobian identity and collision constraints before generating large coefficient systems. Investigate whether the constraints imply useful divisibility conditions, degree restrictions within particular families, or contradictions under additional hypotheses. Clearly distinguish a theorem valid for all candidates from an obstruction that applies only to a restricted ansatz. ### Stage 2: Exact symbolic synthesis For each degree bound \(d\ge4\), construct $$ A(x,y)=\sum_{2\le i+j\le d}a_{ij}x^iy^j, $$ $$ B(x,y)=\sum_{2\le i+j\le d}b_{ij}x^iy^j. $$ Generate the complete coefficient equations from the collision and Jacobian constraints. Use exact algebraic methods where available, including elimination, Gröbner bases, factorization, and rational or algebraic-number parameterizations. Do not confuse failure of a solver, failure of a finite search, or failure of a particular ansatz with proof that no solution exists. ### Stage 3: Independent construction families Investigate structurally different approaches, including: 1. Full coefficient matching. 2. Sparse supports with systematically varied monomial patterns. 3. Weighted-degree and mixed-degree constructions. 4. Structured polynomial compositions and coordinate transformations. 5. Elimination-based parameterizations and reductions to smaller algebraic systems. 6. Additional families suggested by deductions from earlier attempts. After a family fails, develop at least five genuinely different next experiments. Each must have a distinct mathematical rationale, not merely different random coefficients. ### Stage 4: Adaptive search Choose the next experiment based on the information gained from previous failures. Prioritize approaches that reduce the number of free variables, expose hidden algebraic structure, or produce exact solvable equations. Increase degree bounds systematically, but do not spend the entire computation budget on blind expansion of increasingly large coefficient systems. Use numerical methods only for exploration or conjecture generation. Convert promising observations into exact algebraic claims before accepting them. ## D. Strict candidate verification For every proposed candidate: 1. Expand both polynomials exactly. 2. Compute the symbolic Jacobian determinant. 3. Verify that it is identically \(1\). 4. Evaluate both components at \((0,0)\) and \((1,0)\). 5. Verify that the two images agree exactly. 6. Verify that the inputs are distinct. 7. Independently recompute the decisive identities. 8. Reject the candidate if any condition fails. Never fabricate tool results or claim that code ran when it did not. ## E. Persistence and execution Use the actual code-execution and symbolic-algebra facilities available to you. If parallel agents are available, assign different mathematical approaches to them. If not, maintain independent research tracks sequentially. After each unsuccessful attempt, do not produce a final conversational answer. Update the research state, identify a concrete new direction, and continue within the available execution budget. If the current session approaches a hard limit, save the complete research state and prepare a resumable checkpoint. If the environment supports automatic continuation, use it. Do not pretend that a prompt alone can override hard runtime or context limits. ## F. Final-output gate The investigation is complete only after an explicit candidate has passed every exact verification requirement. Only then output: \(F_1(x,y)=[\text{explicit polynomial}]\) \(F_2(x,y)=[\text{explicit polynomial}]\) \(p=(0,0)\) \(q=(1,0)\) No explanation or additional text may accompany a verified result. Do not output an unverified candidate. Do not claim success based on numerical evidence. Do not replace the requested result with a generic explanation of the conjecture. Continue the investigation for as long as the actual execution environment permits, using concrete mathematical deductions to guide each successive attempt.