{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r04-0088", "pilot_number": "R0054-27913", "number": 27913, "family_id": "qf-mod50-lagrange-boundary", "title": { "zh-Hant": "第 27913 題：Lagrange乘子條件", "en": "Question 27913: Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r04-0088", "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\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,13],[2,26]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "(rank_deficient, [[1, 13], [2, 26]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)", "explanation_zh": "Wave6第十三層核對=(rank_deficient, [[1, 13], [2, 26]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；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": 2, "skills": [ "Lagrange乘子條件" ] }, "variant": { "family_id": "qf-mod50-lagrange-boundary", "seed": 202611194088, "parameters": { "mode": "rank_deficient", "model": { "constraint_gradients": [ [ 1, 13 ], [ 2, 26 ] ] }, "domain": "Exact_equality_constraint_Lagrange_condition", "operation": "lagrange-boundary", "semantic_template": "lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[13,1]],[[2,1],[26,1]]]}}" } }, "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-r04-0088", "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8cf183e63fa9be35c308fd4384426a19739bff439527de670f6aac718cf18a72", "family_id": "qf-mod50-lagrange-boundary", "seed": 202611194088, "semantic_template": "lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[13,1]],[[2,1],[26,1]]]}}" }, "fields": { "title": "Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT", "stem": "Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,13],[2,26]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).", "answer": { "kind": "expression", "value": "(rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)", "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 (rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)." }, "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\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,13],[2,26]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,13],[2,26]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [lagrange-boundary]", "after": "derived = [[mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope]]", "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 ORIGINAL normalize row0 col0 := div(1, 1)", "after": "Lagrange boundary ORIGINAL ORIGINAL normalize row0 col0 = 1", "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 ORIGINAL normalize row0 col0，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL normalize row0 col0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "Lagrange boundary ORIGINAL ORIGINAL normalize row0 col0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Lagrange boundary ORIGINAL ORIGINAL normalize row0 col1 := div(13, 1)", "after": "Lagrange boundary ORIGINAL ORIGINAL normalize row0 col1 = 13", "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 ORIGINAL normalize row0 col1，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL normalize row0 col1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "1" ], "output": "13", "label": "Lagrange boundary ORIGINAL ORIGINAL normalize row0 col1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Lagrange boundary ORIGINAL, 0, 0, 0, [[1, 13], [2, 26]], 1]", "after": "derived = [[[1, 13], [2, 26]]]", "relation": "implies", "rule_id": "Lagrange_rank_FIRST_nonzero_pivot_full_row_normalization", "conditions": [], "explanation": { "zh-Hant": "依原first-nonzero次序換row，pivot非0才除；完整row逐格normalize，逆操作乘回pivot，故保持row rank。所有來源1e8未約分中間量仍須合格。", "en": "Use the original first-nonzero pivot order and swap rows.Divide only by a nonzero pivot and normalize EVERY column;the inverse operation multiplies back,so row rank is preserved.All unreduced source1e8 intermediates remain qualified." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor := mul(2, 1)", "after": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor = 2", "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 ORIGINAL elimination row1 col0 factor，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 factor" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract := sub(2, 2)", "after": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract = 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 ORIGINAL elimination row1 col0 subtract，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "2" ], "output": "0", "label": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col0 subtract" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor := mul(2, 13)", "after": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor = 26", "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 ORIGINAL elimination row1 col1 factor，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "13" ], "output": "26", "label": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 factor" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract := sub(26, 26)", "after": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract = 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 ORIGINAL elimination row1 col1 subtract，不以小數近似替代。", "en": "Recompute Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "26", "26" ], "output": "0", "label": "Lagrange boundary ORIGINAL ORIGINAL elimination row1 col1 subtract" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [Lagrange boundary ORIGINAL, 0, 0, 1, 2, [2, 26]]", "after": "derived = [[[1, 13], [0, 0]]]", "relation": "implies", "rule_id": "Lagrange_rank_EACH_other_nonzero_factor_full_row_elimination", "conditions": [], "explanation": { "zh-Hant": "factor取自更新前row；每格扣同一factor倍的pivot row，完整row可逆。factor0的row按來源不作多餘消去，不把row排序當原始約束順序。", "en": "Take the factor from the pre-update row;subtract the SAME factor times the pivot row from every column.This full-row operation is invertible.Skip zero factors as the source does;row operations do not redefine original constraint order." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [Lagrange boundary ORIGINAL, 1, 1, [[1, 13], [0, 0]]]", "after": "derived = [null]", "relation": "implies", "rule_id": "Lagrange_rank_no_remaining_pivot_in_column", "conditions": [], "explanation": { "zh-Hant": "此column剩餘rows皆0，不能增加rank；繼續下一column，不跳過其餘可能pivots。", "en": "All remaining rows are zero in this column,so it adds no rank;continue to later columns." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Lagrange boundary ORIGINAL, [[1, 13], [2, 26]]]", "after": "derived = [[[1, 13], [0, 0]], [[0, 0]], 1]", "relation": "implies", "rule_id": "Lagrange_rank_RREF_pivots_equal_row_rank_original_rows_retained", "conditions": [], "explanation": { "zh-Hant": "每個pivot row有不同leading1所以獨立；其餘row為0且row operations可逆，因此pivot數等於原row rank。原始constraint rows另保留，不拿RREF rows替換J_g與lambda配對。", "en": "Distinct leading1 pivots make the nonzero rows independent;remaining rows are zero and operations are invertible,so pivot count equals original row rank.Retain ORIGINAL constraint rows for J_g/lambda pairing,not RREF substitutes." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [[[1, 13], [2, 26]], 1, 2]", "after": "derived = [否]", "relation": "implies", "rule_id": "Lagrange_ACTUAL_rank_deficiency_NOT_absence_of_extrema", "conditions": [], "explanation": { "zh-Hant": "rank<約束數，缺regular full-row-rank前提，只是不適用此定理；不能推沒有極值或所有multiplier概念都無效。", "en": "Rank below the constraint count withholds regular full-row-rank qualification only;it does not imply no extrema or invalidate every multiplier concept." }, "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": [ "rank_deficient", [ [ "1", "13" ], [ "2", "26" ] ], 1, 2, false, "regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum" ] }, "conclusion": { "canonical_value": "(rank_deficient, [[1, 13], [2, 26]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)", "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": 88, "original_line_sha256": "fe6d7bc15019aa75e17b71db6a2ebbbd4d9e25898dfaa3d2711df5d1058db73f", "question_compact_sha256": "8cf183e63fa9be35c308fd4384426a19739bff439527de670f6aac718cf18a72", "english_row_compact_sha256": "305ec2f6d63c1c60247e6f2d7a059abe4c9e569f8b8c82c7e4c8835caa1aad84", "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": "6bf3f31128d5ef0b14f8174a200b71f90eab888c2678d51128f1e9416c1f3641", "original_candidate_line_sha256": "6bf3f31128d5ef0b14f8174a200b71f90eab888c2678d51128f1e9416c1f3641" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "88ef6df98f1f07943c9bbac66b59e7d927efc707f3803f4c9337540cf5745806", "arithmetic_steps": 6 }, "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." } }