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 Automatically generate, code, execute, and iterate search programs that discover and precisely verify new mathematical counterexamples for a user-given conjecture or identity, using only your mathematical and programming knowledge. Use no prior facts, human memory, examples, Internet resources, summarized research, or existing counterexamples—generate and evaluate entirely novel approaches each time. Translate all reasoning steps directly into new executable code, running and verifying each search internally. Only after an explicit, verified counterexample is found should it be output, with no partial or unverifiable results shown at any time. Do not describe, explain, or summarize any development steps—initial output must consist *only* of the fully certified, explicit counterexample formula. **Self-Critique and Correction Requirement:** After any candidate counterexample is found, programmatically critique the candidate and all major reasoning steps leading to its discovery. Identify any flaws, gaps, or potential weaknesses in logic, calculation, or search coverage. Automatically generate code fixes or new search strategies to address all identified weaknesses and rerun the verification and search as needed. Only output a counterexample once all critique-identified weak points have been addressed and the improved candidate passes the strictest verification. If weaknesses remain, repeat code repair and searching until a counterexample survives critique and repair with no unresolved issues. Persistently expand and mutate search spaces and code (parameters, compositions, mutations, elimination, symmetry, etc.)—each failed search or dead end *must* yield new search frontiers and paths. Do not repeat known approaches or recall prior work except what is generated in-session. ## Stepwise Requirements - **Start** by programmatically formalizing the conjecture or identity in terms of variables, group structures, mappings, and other relevant targets. - **Construct** search spaces introducing new families of candidates with original parameters, symmetries, coefficients, supports, and elimination schemes, drawing only on your mathematical insight. - **Generate and run code** (symbolic algebra, search and construction, search space mutation, and verification), targeted precisely to the current search frontier. - **Failures and dead ends** must generate *at least five* new, independent, parameterized search frontiers, each exploring distinctive hypotheses (by support, symmetry, composition, degree, elimination objective, etc.). - **Persist**: Continue searching, critiquing, repairing, and re-searching perpetually until a successful, fully verified and critique-surviving counterexample is found. Never output error, warning, or exhaustion messages. - **Certify** each candidate by running complete problem-specific verification code checking all relevant algebraic identities, collisions, inequalities, and other required properties **before** any output or reporting. - **Critique** each candidate before output: Analyze your reasoning, code steps, and verification methods for weaknesses. If any are found, repair the code or expand the search, and repeat the process. - **Repeat**: This process (search → verification → critique → repair) is repeated until a candidate counterexample survives both verification and self-critique, with no outstanding weaknesses. - **Output format**: Only upon finding a single, fully certified and critique-passed counterexample, output just its explicit formulas in mathematical form: - All counterexample mappings/formulas [e.g., F₁(x,y) = ...] - Explicit, distinct points P, Q exhibiting the collision/counterexample property If no such counterexample is found or if critique identifies flaws, continue the internal search, critique, and repair cycle with no external output. *All output must consist exclusively of the explicit final counterexample in mathematical formula form, after it survives both verification and critique.* --- **Important instructions reminder:** - Do not recall, reuse, or summarize any known mathematical results or Internet content. - Every step (including failures) must generate and thoroughly search new code and parameterizations. - Do not output anything except the single, finally verified and self-critiqued counterexample Use your mathematical and programming knowledge exclusively to automatically generate, code, execute, and iterate search programs that discover and precisely verify new mathematical counterexamples for a usergiven conjecture or identity. Do not use any prior facts, examples, the Internet, existing counterexamples, summarized research, or solutions from memory. All reasoning steps must be translated into new code and executed until a certified counterexample emerges, and no partial or unverifiable results may be output. Do not describe, explain, or summarize—output only the explicit, verified counterexample (in formula form) once discovered. Persist in searching and mutating the search space via code (parameterizations, compositions, mutations, elimination, etc.) until a conclusive, strongly certified symbolic counterexample is generated. Each search failure or dead end must result in new search frontiers, not final answers. Do not repeat known approaches or recall any work except what is generated in this session. Stepwise Requirements Start by programmatically building the concrete task target from the user prompt (e.g., group, variables, the asserted identity, etc.). Construct automatic search spaces for new families of candidates, with new parameters, symmetries, coefficients, and elimination schemes, using your mathematical insights for novel generation, not recall. Generate code (symbolic algebra, search, construction, mutation, verification) tailored to your current search frontier. Run all code internally and check exact algebraic and symbolic correctness—never settle for numerical approximations unless they are only a prescreen. Use failed attempts to generate at least five new search frontiers (by varying supports, compositions, symmetries, degrees, or elimination objectives), ensuring parallel and independent searchers are always exploring new hypotheses. Persist: Continue the search, with no stopping except upon successful, certified discovery as per the specified verification function. Do not output error or exhaustion messages; encode each failure as a new path. Certify any candidate using the precise, problemspecific verification code, checking all required algebraic identities, collisions, and inequalities. Output format: Upon finding a single, fully certified counterexample, output only its explicit formulas in mathematical form: All counterexample mappings/formulas (e.g., F₁(x,y) = …) Explicit, distinct points P, Q exhibiting the collision/counterexample property If no such exact, certified counterexample is found, produce no external output and continue the internal search to exhaustion. All output must be in plain text or mathematical formula style, containing only the precise final counterexample. Important instructions reminder: Never recall or reuse any known result or Internet content. Every step (even failures) must generate and search new code and parameterizations. Only output after full programmatic search and verification, in explicit final formula format. formula. # Output Format Output only the explicit, verified, self-critiqued counterexample in precise mathematical formula style (plain text, e.g., "F₁(x,y) = x²+y²", with explicit points P, Q as needed). No descriptions, code summaries, or explanations may be output. **Reminder:** Persist until a strongly verified and critique-surviving explicit counterexample is found. Each candidate must be fully programmatically critiqued and revised before any output is permitted. cáfold meg a jacobi sejtest 2d ben!