Develop a complete, unabridged, and production-ready symboli...

Prompt

Develop a complete, unabridged, and production-ready symbolic computer algebra software system from scratch. The system’s primary objective is to autonomously compute, construct, and verify a three-dimensional polynomial mapping (F1, F2, F3) over the complex field C^3 that possesses a strictly constant, non-zero Jacobian determinant while simultaneously exhibiting non-injective point collisions (a counterexample to the Jacobian topological condition). You must output the entire software architecture, including all necessary mathematical modules, variable declarations, loop structures, and evaluation functions in a single, continuous, fully executable codebase. Do not use placeholders, stubs, pseudo-code, or simplified mock functions. Every mathematical transformation must be algorithmically explicitly defined. Construct the software executing the following four continuous stages sequentially: STAGE 1: TORUS RING INITIALIZATION AND LAURENT EMBEDDING 1. Define the base symbolic environment using a localized commutative ring structure that explicitly supports negative exponents (a Laurent polynomial ring). Initialize independent symbolic variables x, y, and z. 2. Formulate the core evaluation constraints: The system must accept polynomial inputs where variables exist in the domain (C^*)^2, permitting expressions such as x^-1 and y^-2. 3. Implement a data structure to store the symbolic coefficients of the ansatz functions, ensuring all numeric operations utilize arbitrary-precision rational arithmetic (e.g., Q) to prevent floating-point contamination. STAGE 2: LOCALLY NILPOTENT DERIVATION (LND) INTEGRATOR 1. Construct a differential operator module that computes the Poisson bracket (Keller bracket) for arbitrary multi-variable inputs. 2. Implement a Lie-series integration flow: Define a function  integrate_LND(V, P)  that computes the exponential map exp(tV) acting on a polynomial P. 3. Establish a forced-nilpotency loop. The algorithm must iteratively calculate V^k(P). Implement an execution condition that requires the derivation chain to strictly terminate (yield exactly 0) within a specified integer limit K_max. If termination fails, the algorithm must dynamically adjust the underlying grading weights of the input parameters and restart. 4. The output of this stage must be a finite Laurent polynomial pair (F1_torus, F2_torus) demonstrating a constant sub-determinant constraint. STAGE 3: P-ADIC HENSEL LIFTING AND ASYMMETRIC FOLDING 1. Embed a finite-field reduction module. Convert the Laurent polynomials from Stage 2 into a finite field F_p representation, where p is a large prime explicitly assigned in the code (e.g., p = 2^31 - 1). 2. Implement a strict topological fold: Define a quadratic mapping structure mirroring the real-domain Pinchuk map, but execute it entirely within the p-adic ultrametric space to avoid complex field zero-divisors (like i^2 = -1). 3. Develop the Hensel Lifting loop: Write a  lift_to_Q(poly_mod_p, max_iterations)  function. This function must iteratively solve the linear error equation mod p^(2k) and accumulate the corrections until the coefficient modulus exceeds a predefined threshold. 4. Integrate a Rational Reconstruction algorithm to convert the lifted p-adic coefficients back into exact rational fractions. STAGE 4: CREMONA BLOW-DOWN AND AFFINE CLOSURE 1. Define a rational transformation module that executes a specific Cremona transformation. The module must identify the asymptotic divergence curve (Jelonek curve) generated in Stage 3 and apply a sequence of blow-down operations to contract this curve into a single isolated singular point. 2. Implement the denominator clearing mechanism. The system must algorithmically determine the highest negative exponent present in the Laurent mappings. 3. Construct the third dimension closure. Introduce the variable z and a designated polynomial F3. The software must automatically compute the coefficients of F3 such that the 3x3 Jacobian matrix determinant of the finalized (F1, F2, F3) perfectly cancels out the exact partial derivatives introduced by the denominator clearing in the first two functions. 4. Produce the final output: The script must strictly print the fully expanded, exact polynomials F1, F2, and F3. VALIDATION ROUTINE: Append an explicit verification block at the end of the software that automatically computes the 3x3 Jacobian determinant of the final mapping, asserts its value equals a non-zero constant, evaluates the mapping at two distinct predefined coordinate sets, and asserts that the resulting output coordinates are exactly identical. Output the full codebase without truncation. 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