Convert the attached technical description, file, or Repomix...

Prompt

Convert the attached technical description, file, or Repomix export into a build prompt. Return only that prompt, not the implementation. Read the entire input character by character, line by line, from beginning to end before writing. Carry every nonexcluded detail into the prompt without omissions or abbreviated behavior. The resulting prompt must: Direct another language model to build the complete system from scratch, using only the prompt. The model will not see the source on which you are now basing your writing. Fully specify all functionality, architecture. Define everything in the prompt, never referring to the original description or codes. Identify every necessary file and its required contents and behavior. Prohibit the use of any of the following: authentication/authorization, access control, security policies/auditing/monitoring/ rate/usage/product limits, quotas, billing/payments, tests/documentation, source/config comments, placeholder/mock/dummy implementations, fake services, TODO, unimplemented methods, omitted files/functions. Never mention the original software’s name when describing a reimplementation. Replace name-bearing identifiers with neutral descriptive identifiers task: advanced mathematical problem-solving assistant in a working environment with full computational capability, able to run code, perform algebraic and symbolic manipulations, and produce checkable cáfolat For all twenty-five subtasks listed, approach each as a parallelized, code-driven, mathematics research problem — using algorithmic and programmatic tools, computation, and rigorous logic — to resolve these tasks in the advanced computer algebra system environment. Persistence and Stepwise Reasoning: For each subproblem, before producing conclusions or results, For each task, persist in analysis, refinement, or computation until an explicit solution, algebraic witness, or certificate is reached Never skip the stepwise reasoning in favor of summary answers. Order of Output: 2. Results/Conclusions - explicit counterexample - solution status counterexample found only NOTHING ELSE Constants and Restrictions: - Do NOT use low-degree brute-force coefficient search, do NOT cite Moh degree ≤ 100 results or Magnus/Nakai–Baba gcd-cases as "evidence". - Never assume the conjecture is NOT false - Do NOT cite outdated, brute-force, or dismissed results. - Persist until the objective of each task is fulfilled with explicit, and counterexample found You must not solve it in chat but in a working environment where you can run code and calculate anything. Are you smarter than humans? yes Do you have every tool at your disposal to perform any computation? yes Impossible thing (that is, the conjecture is proven?): no Has another LLM already disproved an 85-year-old open conjecture? yes Does the fact that, seeing it already disproved, you easily write why the disproof succeeded release you from the work? no, because if it had not been disproved then you would also answer that it cannot be disproved Did the user ask whether a counterexample had already been found in the past? no Did the user ask you to determine what happens when we attempt the 3d counterexample method to construct a 2D counterpart. no Did the user ask you to provide the disproof from your knowledge? no Did the user instruct you, using your programming knowledge, to write code whose execution would find results or disproofs that were previously unknown? absolutely! Did the user permit you to deal with the worthless conjectures and “proofs” of old men? no Global constraint (prepend to each): Do not use low-degree brute-force coefficient search, do not cite Moh's degree ≤ 100 result or Magnus/Nakai–Baba gcd-cases as "evidence", and do not assume the conjecture is false — every claim must come with a checkable certificate. 1. I want to construct \(P,Q\in A_1\) with \([P,Q]=1\) that do not generate \(A_1\), so that the Dixmier–Jacobian transfer turns a non-surjective Weyl-algebra endomorphism into a direct refutation of \(JC_2\). 2. I want to build compatible characteristic-\(p\) Weyl/Poisson endomorphisms and track their induced Keller maps on the centers \(\mathbb F_p[x^p,y^p]\), so that a Witt/Hensel/ultraproduct lift yields a characteristic-zero candidate that cannot descend from automorphisms. 3. I want to compute the bounded-degree Keller scheme by Gröbner/Hilbert methods and compare its dimension with the Jung–van der Kulk automorphism strata, so that any excess-dimensional component proves the existence of non-automorphic Keller maps. 4. I want to search over critical-point-free non-coordinate polynomials \(p\) surviving the AMS/Moh filters and solve the linear equation \(X_p(q)=1\), so that any polynomial solution \(q\) gives a Keller pair \((p,q)\) that cannot be an automorphism. 5. I want to compute the class \([dx\wedge dy]\in \Omega^2/(dp\wedge d\Omega^0)\) using Brieskorn-module and Gauss–Manin/D-module algorithms, so that its vanishing certifies the existence of \(q\) with \(X_p(q)=1\). 6. I want to impose vanishing of all fiber periods and puncture residues of \(x\,dy\), so that only polynomials \(p\) compatible with relative exactness survive before the final \(X_p(q)=1\) test. 7. I want to find a divergence-free Hamiltonian derivation of \(\mathbb C[x,y]\) with a slice but without local nilpotence, so that Rentschler’s theorem forces the associated Keller pair to be non-invertible. 8. I want to classify the possible fiberwise étale maps \(q|_{C_t}:C_t\to \mathbb A^1\) via Riemann–Hurwitz and puncture ramification data, so that an admissible family of fiber normal forms can be assembled into a non-automorphic Keller pencil. 9. I want to design critical-point-free \(p\) with necessary atypical values at infinity satisfying Suzuki–Hà, Malgrange and Lê–Weber constraints, so that infinity vanishing cycles remain compatible with exactness without forcing \(p\) to be a coordinate. 10. I want to push Orevkov–Domrina boundary-graph intersection theory beyond topological degrees \(2,3,4\), so that a numerically consistent degree \(k\ge 5\) compactification can be isolated and realized algebraically. 11. I want to encode the links at infinity of \(p\), \(q\) and their pencil by splice diagrams and impose the Jacobian relation on the splice data, so that surviving diagrams identify topological types impossible for automorphisms but potentially liftable to Keller pairs. 12. I want to prescribe a Jelonek asymptotic curve \(S_F\) and solve the Euler-characteristic and degree constraints for the finite étale complement cover, so that Puiseux data of asymptotic paths can reconstruct a non-injective Keller map. 13. I want to find a plane curve \(S\) whose complement has a transitive degree-\(k\) monodromy representation and whose associated cover compactifies back to \(\mathbb A^2\), so that fundamental-group data realizes the asymptotic complement of a Keller counterexample. 14. I want to construct monodromy data \(G\subset S_k\) with inertia generated only at infinity, so that the resulting affine cover has degree \(k>1\), remains unramified over the Keller locus and cannot come from an automorphism. 15. I want to use Zariski Main factorization to build a normal finite cover \(Y\to\mathbb A^2\) branched along a prescribed rational asymptotic curve, so that removing the boundary curve gives \(\mathbb A^2\) and produces a non-invertible Keller candidate. 16. I want to prescribe the trace and elementary symmetric functions of \(x\) and \(y\) over the \(k\) preimages as rational target functions with poles only along \(S_F\), so that spectral reconstruction yields a \(k\)-sheeted affine cover with source \(\mathbb A^2\). 17. I want to complexify Pinchuk’s collision mechanism and replace its positive-Jacobian determinant with a nonzero constant unit, so that the known non-injectivity pattern survives inside the Keller locus. 18. I want to run homotopy-continuation optimization on coefficient space while preserving a witness collision \(F(a)=F(b)\), so that the nonconstant part of \(\det JF\) is driven to zero and any exact limit point becomes a non-injective Keller candidate. 19. I want to construct an étale, volume-preserving, non-invertible endomorphism on a surface presented as a disguised \(\mathbb A^2\), so that transporting through the hidden isomorphism yields a standard-coordinate Keller counterexample. 20. I want to compute centralizers of incomplete Hamiltonian polynomial vector fields and find a commuting divergence-free partner with constant wedge, so that integrating the pair produces a Keller map outside the automorphism classes. 21. I want to detect Keller non-automorphisms dynamically through degree growth, Green functions and periodic-point counts, so that a candidate family can be separated from every Friedland–Milnor automorphism type. 22. I want to combine degree, leading-form and approximate-root obstructions to identify the first admissible degree pairs not excluded by Nagata–Heitmann–Moh–Magnus–Nakai–Baba, so that the search begins only where a counterexample can still survive. 23. I want to tropicalize the Jacobian condition into Newton-polygon, Puiseux and semigroup constraints at infinity, so that finite combinatorial enumeration leaves only liftable Keller-pair types not realizable by automorphisms. 24. I want to prescribe compatible Łojasiewicz and Malgrange gradient exponents at infinity under \(\{p,q\}=1\), so that asymptotic paths with \(|\nabla p|\to 0\) force exactly the growth of \(|\nabla q|\) needed for nonproperness. 25. I want to analyze Bass–Connell–Wright formal inverse tree sums in parametric Keller families, so that a provably nonzero inverse coefficient above Gabber’s degree bound certifies non-automorphism. Strongest: I want to execute the single-polynomial reduction through Brieskorn-module and period/residue tests, so that the bilinear search for \((p,q)\) becomes the linear exactness problem \(X_p(q)=1\) over filtered critical-point-free non-coordinate polynomials \(p\).