{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r07-partial-derivative-0034", "pilot_number": "R0066-33534", "number": 33534, "family_id": "qf-mod25-partialdeep-partial-y-polynomial", "title": { "zh-Hant": "第 33534 題：偏導密度", "en": "Question 33534: The Entire y-Partial Polynomial in the Same Coordinate Basis" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r07-partial-derivative-0034", "batch_id": "batch-20261013-advance-20261025-r07-partial-derivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-partial-derivative" ], "title_zh": "偏導密度", "stem_zh": "固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-4,1,-2,1,1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。", "answer": { "kind": "expression", "value": "([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])", "explanation_zh": "收束波核對=([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號消去h後才取極限；不以有限差分證偏導或推出全微分。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "偏導密度" ] }, "variant": { "family_id": "qf-mod25-partialdeep-partial-y-polynomial", "seed": 202610257034, "parameters": { "coefficients": [ -4, 1, -2, 1, 1, 1 ], "domain": "Polynomial_partial_derivative_density", "operation": "partialdeep-partial-y-polynomial", "semantic_template": "qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261025-r07-partial-derivative-0034", "batch_id": "batch-20261013-advance-20261025-r07-partial-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "42f1e10101ffc9456e792b6fca9ec642cba86280edee8c1282f3c471122581dc", "family_id": "qf-mod25-partialdeep-partial-y-polynomial", "seed": 202610257034, "semantic_template": "qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}" }, "fields": { "title": "The Entire y-Partial Polynomial in the Same Coordinate Basis", "stem": "Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).", "answer": { "kind": "expression", "value": "([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])", "explanation": "The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])." }, "skills": [ "The Entire y-Partial Polynomial in the Same Coordinate Basis" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-4,1,-2,1,1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。", "en": "Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y])." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod25-partialdeep-partial-y-polynomial, [coefficients, domain, operation]]", "after": "derived = [partialdeep-partial-y-polynomial, Polynomial_partial_derivative_density]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[-4, 1, -2, 1, 1, 1]]", "after": "derived = [1, x, y, x^2, x*y, y^2]", "relation": "implies", "rule_id": "Partial_FULL_polynomial_basis_and_fixed_other_coordinate", "conditions": [], "explanation": { "zh-Hant": "六係數依[1,x,y,x²,xy,y²]完整排列。偏導時另一座標固定；這個完整多項式支持符號恆等式，不從一個差分樣本猜導數或推一般函數全微分。", "en": "The SIX coefficients use the full[1,x,y,x²,xy,y²]basis.Hold the OTHER coordinate fixed.This supplied polynomial supports symbolic identities,not inference of a derivative from one difference sample or total differentiability of arbitrary functions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [y, -4, 1]", "after": "derived = [-4*(0)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [y, 1, x]", "after": "derived = [1*(0)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [y, -2, y]", "after": "derived = [-2*(1)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [y, 1, x^2]", "after": "derived = [1*(0)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [y, 1, x*y]", "after": "derived = [1*(x)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [y, 1, y^2]", "after": "derived = [1*(2*y)]", "relation": "implies", "rule_id": "Partial_EACH_monomial_power_product_and_constant_rules", "conditions": [], "explanation": { "zh-Hant": "逐原monomial套常數/冪法則或固定另一座標後的乘積規則，保留導數為0的項，再在[1,x,y]收集。", "en": "Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y]." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Partial ORIGINAL twice square coefficient := mul(2, 1)", "after": "Partial ORIGINAL twice square coefficient = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Partial ORIGINAL twice square coefficient，不以小數近似替代。", "en": "Recompute Partial ORIGINAL twice square coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Partial ORIGINAL twice square coefficient" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [y, [-4, 1, -2, 1, 1, 1], [[1, 0], [x, 0], [y, 1], [x^2, 0], [x*y, x], [y^2, 2*y]]]", "after": "derived = [[-2, 1, 2], [1,x,y]]", "relation": "implies", "rule_id": "Partial_COLLECT_all_coefficients_in_declared_basis", "conditions": [], "explanation": { "zh-Hant": "把全部導數項收集成固定basis係數，零係數不刪、x/y角色不互換；反向展開可核每個原項的導數貢獻。", "en": "Collect ALL derivative terms in the declared basis without deleting zero coefficients or swapping x/y.Re-expand to audit the derivative contribution of every original term." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ [ -4, 1, -2, 1, 1, 1 ], [ -2, 1, 2 ], "[1,x,y]" ] }, "conclusion": { "canonical_value": "([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261013-advance-20261025-r07-partial-derivative-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r07-partial-derivative-foundations/questions.jsonl", "batch_sha256": "6b2bb9d942c4d9785602fed035b922edb653a7317cb9355963393ee09ee97224", "original_line": 34, "original_line_sha256": "49ff7ae7db0c3a0305f2cab7d2edff2099817ac6c04c22907bba27d335941582", "question_compact_sha256": "42f1e10101ffc9456e792b6fca9ec642cba86280edee8c1282f3c471122581dc", "english_row_compact_sha256": "5b3eab680c2852c32ae1ed2ca8fa8bff21957c2c9a33a30f3ea5493db42ed432", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r07-partial-derivative-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r07-partial-derivative-foundations.en.jsonl" }, "mko_bindings": { "mko-partial-derivative": { "sha256": "7159526c8e30cc8cb9086f9690d279fd3852328abe371206b3441ecf6ef69ce8", "english_sha256": "11ec37b647f687ce00aba1864579424c7da8dccdbac8051d553f0a9763c182fe", "original_path": "public/data/mko/partial-derivative.json", "english_path": "public/data/i18n/editions/mko-partial-derivative.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310", "original_candidate_sha256": "a3eb1369416dd1b940e058b4a9528c628a632055861e437f8292c29af84f522d", "original_candidate_line_sha256": "a3eb1369416dd1b940e058b4a9528c628a632055861e437f8292c29af84f522d" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-012-p67", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267", "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 10, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "be9a9213a76b785003f46909285dbfba9443302765b6cb7044f1f4b22024d502", "arithmetic_steps": 1 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }