{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r06-gradient-vector-0083", "pilot_number": "R0047-24408", "number": 24408, "family_id": "qf-mod44-gradient-boundary-counterexample", "title": { "zh-Hant": "第 24408 題：梯度向量", "en": "Question 24408: Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r06-gradient-vector-0083", "batch_id": "batch-20261015-advance-20261113-r06-gradient-vector-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-gradient-vector" ], "title_zh": "梯度向量", "stem_zh": "standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)", "explanation_zh": "Wave6第七層核對=(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "梯度向量" ] }, "variant": { "family_id": "qf-mod44-gradient-boundary-counterexample", "seed": 202611136083, "parameters": { "mode": "saddle_stationary", "scale": 8, "domain": "Exact_polynomial_gradient_vector", "operation": "gradient-boundary-counterexample", "semantic_template": "qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[8,1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261113-r06-gradient-vector-0083", "batch_id": "batch-20261015-advance-20261113-r06-gradient-vector-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4fa325f679adfa8b9a60780982ae2582ff935af94c4c50cf6371440304c410b0", "family_id": "qf-mod44-gradient-boundary-counterexample", "seed": 202611136083, "semantic_template": "qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[8,1]}" }, "fields": { "title": "Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts", "stem": "Use mode=saddle_stationary, positive scale=8, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).", "answer": { "kind": "expression", "value": "(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)", "explanation": "incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)." }, "skills": [ "Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts" ], "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": "standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=saddle_stationary, positive scale=8, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [gradient-boundary-counterexample]", "after": "derived = [[mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。", "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [saddle_stationary, 8]", "after": "derived = [saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate]", "relation": "implies", "rule_id": "Gradient_mode_specific_counterexample_with_actual_hypotheses", "conditions": [], "explanation": { "zh-Hant": "梯度[2sx,−2sy]在0為0；任意小(h,0)值正而(0,h)值負，因此不是局部極大也不是極小。", "en": "The gradient[2sx,−2sy] vanishes at0,but arbitrarily small(h,0) gives positive values and(0,h) negative values,so neither local extremum exists." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "saddle_stationary", "f(x,y)=8*(x^2-y^2)", [ 0, 0 ], [ 0, 0 ], true, false, "zero_gradient_at_a_saddle_is_not_an_extremum_certificate" ] }, "conclusion": { "canonical_value": "(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)", "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-20261015-advance-20261113-r06-gradient-vector-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r06-gradient-vector-foundations/questions.jsonl", "batch_sha256": "2126db15a137e63c9415addb2bbac655ac3a34f6aa471dc44c138067bd9af76d", "original_line": 83, "original_line_sha256": "a0a8f774e7b015ed49a9cfc628c9c562a476020b63808992d5aae1d220c7216b", "question_compact_sha256": "4fa325f679adfa8b9a60780982ae2582ff935af94c4c50cf6371440304c410b0", "english_row_compact_sha256": "bc9b572e878ebc6284f42d86db501ac0785dc43aa31eb5dc509a07d4aee18b33", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r06-gradient-vector-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r06-gradient-vector-foundations.en.jsonl" }, "mko_bindings": { "mko-gradient-vector": { "sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa", "english_sha256": "6ba53da50d59e50b734006c70a0c1e5cb7e4986a4a57e6d0b1f27a8ae84bd363", "original_path": "public/data/mko/gradient-vector.json", "english_path": "public/data/i18n/editions/mko-gradient-vector.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68", "original_candidate_sha256": "0771436d89777939e60a99c55eac565824af0b3a072dd10bf31c426b72ebe8e9", "original_candidate_line_sha256": "0771436d89777939e60a99c55eac565824af0b3a072dd10bf31c426b72ebe8e9" }, "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-20261026-009-p48", "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386", "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "62b49756ba428a6c8436cc03cf326c06389fe662aea0b31ef9f71c3ac7ec9a3d", "arithmetic_steps": 0 }, "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." } }