import sympy as sp def verify_counterexample(): print("...

Prompt

import sympy as sp def verify_counterexample(): print("=== VERIFICATION OF JACOBIAN CONJECTURE COUNTEREXAMPLE ===") x, w = sp.symbols('x w', complex=True) # 1. Define P, Q and the surface S equation for y P = x + x**16 * w**7 Q = w**2 y_expr = w**2 + sp.Rational(32, 9) * x**15 * w**9 print(f"P(x, w) = {P}") print(f"Q(x, w) = {Q}") print(f"Surface S: y = {y_expr}") # Check 1: Polynomial structure in exact arithmetic assert P.is_polynomial(x, w) and Q.is_polynomial(x, w), "P and Q must be polynomials in x and w" print("Check 1 PASSED: P and Q are exact polynomials in x and w.") # Check 2: Jacobian determinant is identically 1 on S dP_dx = sp.diff(P, x) dP_dw = sp.diff(P, w) dQ_dx = sp.diff(Q, x) dQ_dw = sp.diff(Q, w) J_xw_PQ = dP_dx * dQ_dw - dP_dw * dQ_dx J_xw_xy = sp.diff(x, x) * sp.diff(y_expr, w) - sp.diff(x, w) * sp.diff(y_expr, x) jacobian_ratio = sp.simplify(J_xw_PQ / J_xw_xy) print(f"Jacobian determinant det J(F) on S = {jacobian_ratio}") assert jacobian_ratio == 1, "Jacobian determinant must be identically 1!" print("Check 2 PASSED: Jacobian determinant is a nonzero constant (1).") # Check 3: Absence of a polynomial inverse (proven by non-injectivity / degree d = 9 >= 5) # Q = w^2 implies fibers have multiple preimages (non-injective / non-proper certificate) print("Check 3 PASSED: Non-invertibility confirmed by algebraic fiber analysis (degree d = 9 >= 5, non-injective).") print("=== ALL THREE CHECKS PASSED SUCCESSFULLY ===") if __name__ == '__main__': verify_counterexample() Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY!