{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r04-0094", "pilot_number": "R0054-27919", "number": 27919, "family_id": "qf-mod50-lagrange-boundary", "title": { "zh-Hant": "第 27919 題：Lagrange乘子條件", "en": "Question 27919: Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r04-0094", "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-lagrange-multiplier-condition" ], "title_zh": "Lagrange乘子條件", "stem_zh": "feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "(infeasible_point, [1, 19], 18, [19, 0], 19, 否, stationarity_never_repairs_missing_feasibility)", "explanation_zh": "Wave6第十三層核對=(infeasible_point, [1, 19], 18, [19, 0], 19, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Lagrange乘子條件" ] }, "variant": { "family_id": "qf-mod50-lagrange-boundary", "seed": 202611194094, "parameters": { "mode": "infeasible_point", "model": { "constraint_gradient": [ 1, 19 ], "constraint_rhs": 18, "point": [ 19, 0 ] }, "domain": "Exact_equality_constraint_Lagrange_condition", "operation": "lagrange-boundary", "semantic_template": "lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[18,1],\"point\":[[19,1],[0,1]]}}" } }, "verification": { "status": "candidate", "checks": [ "producer_contract_recomputation", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "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-pulse-20261119-r04-0094", "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "54c07d4dd3b759ebe3f40a73f8683e97a33a3bb515b5102861822b67e042ac43", "family_id": "qf-mod50-lagrange-boundary", "seed": 202611194094, "semantic_template": "lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[18,1],\"point\":[[19,1],[0,1]]}}" }, "fields": { "title": "Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT", "stem": "Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).", "answer": { "kind": "expression", "value": "(infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility)", "explanation": "Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility)." }, "skills": [ "Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT" ], "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": "feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [lagrange-boundary]", "after": "derived = [[mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。", "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Lagrange boundary ORIGINAL constraint value ORIGINAL product0 := mul(1, 19)", "after": "Lagrange boundary ORIGINAL constraint value ORIGINAL product0 = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Lagrange boundary ORIGINAL constraint value ORIGINAL product0，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL product0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "19" ], "output": "19", "label": "Lagrange boundary ORIGINAL constraint value ORIGINAL product0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 := sum(0, 19)", "after": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "19" ], "output": "19", "label": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Lagrange boundary ORIGINAL constraint value ORIGINAL product1 := mul(19, 0)", "after": "Lagrange boundary ORIGINAL constraint value ORIGINAL product1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Lagrange boundary ORIGINAL constraint value ORIGINAL product1，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL product1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19", "0" ], "output": "0", "label": "Lagrange boundary ORIGINAL constraint value ORIGINAL product1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 := sum(19, 0)", "after": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。", "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "19", "0" ], "output": "19", "label": "Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[1, 19], 18, [19, 0]]", "after": "derived = [19, 否]", "relation": "implies", "rule_id": "Lagrange_ACTUAL_infeasibility_not_repaired_by_stationarity", "conditions": [], "explanation": { "zh-Hant": "逐完整dot核a·x與rhs實際不同，所以不可行；此mode不另要求normal非0，zero normal/nonzero rhs也可見證。stationarity不能修補缺少的feasibility。", "en": "The complete dot product actually differs from rhs,hence infeasibility.This mode adds no nonzero-normal condition;a zero normal with nonzero rhs is also a witness.Stationarity cannot repair missing feasibility." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。", "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ "infeasible_point", [ "1", "19" ], "18", [ "19", "0" ], "19", false, "stationarity_never_repairs_missing_feasibility" ] }, "conclusion": { "canonical_value": "(infeasible_point, [1, 19], 18, [19, 0], 19, 否, stationarity_never_repairs_missing_feasibility)", "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-20261119-r04-lagrange-multiplier-condition-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations/questions.jsonl", "batch_sha256": "d71d7d0bcecf04d4e0a1a6c42431e568d00d31857240898ab7af52725d5943da", "original_line": 94, "original_line_sha256": "236c0d2af76da09e0177caaf2b16cd8bf5737b18394e0751f5dd1012277d4e12", "question_compact_sha256": "54c07d4dd3b759ebe3f40a73f8683e97a33a3bb515b5102861822b67e042ac43", "english_row_compact_sha256": "9281c661a878be4a172d7d1a4180ac23e5b4cd79d6d1ba8d609615b108706446", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations.en.jsonl" }, "mko_bindings": { "mko-lagrange-multiplier-condition": { "sha256": "cfaf4ce60e4c881cdfb0ca770d8574ec63898460c86c1049720cf5c9546bfdd7", "english_sha256": "51b359b63416bacca8d5d5350982535ac76896b00ad3397742174d696a57ddbb", "original_path": "public/data/mko/lagrange-multiplier-condition.json", "english_path": "public/data/i18n/editions/mko-lagrange-multiplier-condition.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8", "original_candidate_sha256": "acdeeec550d666dee546be2d7be5ff27c2b2a64fbab216d5d691e77ed2ff0b9c", "original_candidate_line_sha256": "acdeeec550d666dee546be2d7be5ff27c2b2a64fbab216d5d691e77ed2ff0b9c" }, "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-20261026-010-p55", "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": "0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54", "question_first_witness_sha256": "09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cde712186660ba6fbcd63598fc1a733f0c4b76ad5593cf94f87508cd8bf31d90", "arithmetic_steps": 4 }, "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." } }