{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r07-partial-derivative-0072", "pilot_number": "R0066-33572", "number": 33572, "family_id": "qf-mod25-partialdeep-partial-y-slice-derivative", "title": { "zh-Hant": "第 33572 題：偏導密度", "en": "Question 33572: A Fixed-x Slice and Its y-Derivative" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r07-partial-derivative-0072", "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-slice-derivative；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。", "answer": { "kind": "expression", "value": "([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)", "explanation_zh": "收束波核對=([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)；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": 3, "skills": [ "偏導密度" ] }, "variant": { "family_id": "qf-mod25-partialdeep-partial-y-slice-derivative", "seed": 202610257072, "parameters": { "coefficients": [ 9, -1, -3, 2, -1, -1 ], "fixed_x": -1, "y": -3, "domain": "Polynomial_partial_derivative_density", "operation": "partialdeep-partial-y-slice-derivative", "semantic_template": "qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}" } }, "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-0072", "batch_id": "batch-20261013-advance-20261025-r07-partial-derivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b526641b28afd652df7e1404abceb1e8dbf66cadc606d47adbc6f947eef32602", "family_id": "qf-mod25-partialdeep-partial-y-slice-derivative", "seed": 202610257072, "semantic_template": "qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}" }, "fields": { "title": "A Fixed-x Slice and Its y-Derivative", "stem": "Use the quadratic candidate coefficients=[9, -1, -3, 2, -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. fixed x=-1, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).", "answer": { "kind": "expression", "value": "([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)", "explanation": "Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)." }, "skills": [ "A Fixed-x Slice and Its y-Derivative" ], "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-slice-derivative；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。", "en": "Use the quadratic candidate coefficients=[9, -1, -3, 2, -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. fixed x=-1, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod25-partialdeep-partial-y-slice-derivative, [coefficients, domain, fixed_x, operation, y]]", "after": "derived = [partialdeep-partial-y-slice-derivative, 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 = [[9, -1, -3, 2, -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": "Slice ORIGINAL c_x times fixed x := mul(-1, -1)", "after": "Slice ORIGINAL c_x times fixed x = 1", "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 Slice ORIGINAL c_x times fixed x，不以小數近似替代。", "en": "Recompute Slice ORIGINAL c_x times fixed x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "Slice ORIGINAL c_x times fixed x" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Slice ORIGINAL fixed x squared := mul(-1, -1)", "after": "Slice ORIGINAL fixed x squared = 1", "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 Slice ORIGINAL fixed x squared，不以小數近似替代。", "en": "Recompute Slice ORIGINAL fixed x squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "Slice ORIGINAL fixed x squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Slice ORIGINAL c_xx times fixed x squared := mul(2, 1)", "after": "Slice ORIGINAL c_xx times fixed x squared = 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 Slice ORIGINAL c_xx times fixed x squared，不以小數近似替代。", "en": "Recompute Slice ORIGINAL c_xx times fixed x squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Slice ORIGINAL c_xx times fixed x squared" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Slice ORIGINAL constant terms sum := sum(9, 1, 2)", "after": "Slice ORIGINAL constant terms sum = 12", "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 Slice ORIGINAL constant terms sum，不以小數近似替代。", "en": "Recompute Slice ORIGINAL constant terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "1", "2" ], "output": "12", "label": "Slice ORIGINAL constant terms sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Slice ORIGINAL c_xy times fixed x := mul(-1, -1)", "after": "Slice ORIGINAL c_xy times fixed x = 1", "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 Slice ORIGINAL c_xy times fixed x，不以小數近似替代。", "en": "Recompute Slice ORIGINAL c_xy times fixed x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-1" ], "output": "1", "label": "Slice ORIGINAL c_xy times fixed x" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Slice ORIGINAL linear y terms sum := sum(-3, 1)", "after": "Slice ORIGINAL linear y terms sum = -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 Slice ORIGINAL linear y terms sum，不以小數近似替代。", "en": "Recompute Slice ORIGINAL linear y terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-3", "1" ], "output": "-2", "label": "Slice ORIGINAL linear y terms sum" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Slice ORIGINAL twice c_yy := mul(2, -1)", "after": "Slice ORIGINAL twice c_yy = -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 Slice ORIGINAL twice c_yy，不以小數近似替代。", "en": "Recompute Slice ORIGINAL twice c_yy with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-1" ], "output": "-2", "label": "Slice ORIGINAL twice c_yy" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [[9, -1, -3, 2, -1, -1], -1]", "after": "derived = [[12, -2, -1], [-2, -2]]", "relation": "implies", "rule_id": "Partial_FIXED_x_whole_y_polynomial_THEN_derivative", "conditions": [], "explanation": { "zh-Hant": "先固定x並逐項合併成s0+s1*y+s2*y²，之後對y微分成s1+2*s2*y；不可先把兩個座標都代完變常數才微分。", "en": "FIRST fix x and collect the WHOLE y-polynomial s0+s1*y+s2*y².THEN differentiate to s1+2*s2*y.Do not substitute BOTH coordinates into a constant before differentiating." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "Slice ORIGINAL derivative linear term at y := mul(-2, -3)", "after": "Slice ORIGINAL derivative linear term at y = 6", "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 Slice ORIGINAL derivative linear term at y，不以小數近似替代。", "en": "Recompute Slice ORIGINAL derivative linear term at y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-3" ], "output": "6", "label": "Slice ORIGINAL derivative linear term at y" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Slice ORIGINAL full derivative value := sum(-2, 6)", "after": "Slice ORIGINAL full derivative value = 4", "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 Slice ORIGINAL full derivative value，不以小數近似替代。", "en": "Recompute Slice ORIGINAL full derivative value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "6" ], "output": "4", "label": "Slice ORIGINAL full derivative value" } } ], "math_model": { "result": [ [ 9, -1, -3, 2, -1, -1 ], -1, [ 12, -2, -1 ], [ -2, -2 ], -3, 4 ] }, "conclusion": { "canonical_value": "([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)", "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": 72, "original_line_sha256": "1be724c8fca99e3efc08bfde432746143996bead540af64c23485203c15a294b", "question_compact_sha256": "b526641b28afd652df7e1404abceb1e8dbf66cadc606d47adbc6f947eef32602", "english_row_compact_sha256": "461c540b05be16c3a917f90ce06c16eeb892402140bd101d352fff21a0ffd55a", "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": "57f3d3c913e869b64fb62fc1ebc8bcb138d9d17663147f8d9dccd898260069e4", "original_candidate_line_sha256": "57f3d3c913e869b64fb62fc1ebc8bcb138d9d17663147f8d9dccd898260069e4" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "19ac8edb36ab4444782cdaaf1a5a9ccea6885bf7625260696ea73bd40b1911cc", "arithmetic_steps": 9 }, "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." } }