Discover explicit polynomials \( F_1(x, y), F_2(x, y) \in \m...

Prompt

Discover explicit polynomials \( F_1(x, y), F_2(x, y) \in \mathbb{C}[x, y] \) and distinct points \( P = (x_1, y_1), Q = (x_2, y_2) \in \mathbb{C}^2 \), such that: 1. \( \det JF = \frac{\partial F_1}{\partial x}\frac{\partial F_2}{\partial y} - \frac{\partial F_1}{\partial y}\frac{\partial F_2}{\partial x} \equiv 1 \) identically on \( \mathbb{C}^2 \), 2. \( F(P) = F(Q) \), with \( P \neq Q \), 3. \( F_1(x, y) \) and \( F_2(x, y) \) are global polynomials as per all normalization and domain constraints below. You must automatically: - Programmatically formalize all variables, constraints, and normalization: - Only admit \( F_1, F_2 \) of the special normalized form \( F_1(x, y) = x + P(x, y) \), \( F_2(x, y) = y + Q(x, y) \) with \( P, Q \in (x, y)^2 \). - Normalize collision points: \( P = (0,0),\ Q = (1,0) \): \( F(0,0) = F(1,0) \). - Enforce collision: \( 1+P(1,0)=0,\ Q(1,0)=0 \). - Impose the Jacobian: \( P_x(x, y)+Q_y(x, y)+P_x(x, y)Q_y(x, y)-P_y(x, y)Q_x(x, y) \equiv 0 \). - Use only global polynomials, characteristic zero, all coefficients complex. - Exclude all known-proven-vain directions (degree ≤ 3, homogeneous degree-4, Hamiltonian type, non-polynomial/rational/Laurent, finite field). - Construct and explore full polynomial coefficient search-spaces (non-homogeneous, degree ≥ 4 and up), with all variable coefficients for monomials of degree 2 to \( d \). - Programmatically generate, search, and mutate large candidate parameter spaces, using novel parameterizations and search strategies at each step. - Turn every reasoning step into new, internally executed code (symbolic algebra, construction, elimination, search, verification). - On each single attempt, internally verify ALL required conditions; if any fail, programmatically critique the candidate and ALL related search code for logical, computational, or verification gaps, and immediately address by mutating the synthesis approach or mixing new parameter families. - On any failed attempt or dead end, generate at least FIVE new, mutually independent search frontiers (differing supports, symmetries, compositions, degree, or elimination objectives), and pursue all in parallel with novel code until at least one candidate is found. - Persistently expand and mutate spaces and code until a single, explicit, fully verified, and self-critiqued counterexample is found. - STOP and output ONLY after a counterexample is verified to satisfy ALL mathematical constraints AND survives complete self-critique (no unresolved issues in reasoning or code). - Output must, and may ONLY, consist of EXPLICIT mathematical formulas for: - \( F_1(x, y) \), \( F_2(x, y) \) - Explicit points \( P \) and \( Q \) with \( P\neq Q \) and \( F(P)=F(Q) \). **Important:** - Do not include explanations, documentation, code, status, impossibility statements, warnings, or intermediate results. - Do not reuse or recall any Internet, research, or known results, except code and work already generated in-session. - Do not cease search or report errors—always continue mutating your search codes and parameterizations internally. - Output must be exclusively the final, full, explicit counterexample in formula format. ## Output Format - Output precisely and only the explicit polynomial formulas and points: - \( F_1(x, y) = [...] \) - \( F_2(x, y) = [...] \) - \( P = (x_1, y_1) \) - \( Q = (x_2, y_2) \) - Use plain text, in precise mathematical formula style. Do not wrap in code blocks or provide explanations. --- **Reminder:** Keep searching, critiquing, and repairing your code until you find and verify an explicit counterexample that fully meets all criteria above. Output nothing else.