{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r04-0025", "pilot_number": "R0054-27975", "number": 27975, "family_id": "qf-mod50-lagrange-regularity-audit", "title": { "zh-Hant": "第 27975 題：Lagrange乘子條件", "en": "Question 27975: Full Row Rank Means the Declared Constraint Gradients Are Independent" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r04-0025", "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-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,25]]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "([[1, 0], [0, 25]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)", "explanation_zh": "Wave6第十三層核對=([[1, 0], [0, 25]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；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-regularity-audit", "seed": 202611194025, "parameters": { "constraint_gradients": [ [ 1, 0 ], [ 0, 25 ] ], "domain": "Exact_equality_constraint_Lagrange_condition", "operation": "lagrange-regularity-audit", "semantic_template": "lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[25,1]],[[1,1],[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-r04-0025", "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e88bffa85da69b98e928a7cc7ef987ef5bb78a78e4c19408ef9ed03d05f48f5a", "family_id": "qf-mod50-lagrange-regularity-audit", "seed": 202611194025, "semantic_template": "lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[25,1]],[[1,1],[0,1]]]}" }, "fields": { "title": "Full Row Rank Means the Declared Constraint Gradients Are Independent", "stem": "Use constraint gradients as ordered rows=[[1, 0], [0, 25]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).", "answer": { "kind": "expression", "value": "([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)", "explanation": "Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)." }, "skills": [ "Full Row Rank Means the Declared Constraint Gradients Are Independent" ], "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-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,25]]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use constraint gradients as ordered rows=[[1, 0], [0, 25]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [lagrange-regularity-audit]", "after": "derived = [[constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition]]", "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 ORIGINAL ORIGINAL normalize row0 col0 := div(1, 1)", "after": "Lagrange 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 ORIGINAL ORIGINAL normalize row0 col0，不以小數近似替代。", "en": "Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "Lagrange ORIGINAL ORIGINAL normalize row0 col0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Lagrange ORIGINAL ORIGINAL normalize row0 col1 := div(0, 1)", "after": "Lagrange ORIGINAL ORIGINAL normalize row0 col1 = 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 ORIGINAL ORIGINAL normalize row0 col1，不以小數近似替代。", "en": "Recompute Lagrange ORIGINAL ORIGINAL normalize row0 col1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "Lagrange ORIGINAL ORIGINAL normalize row0 col1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Lagrange ORIGINAL, 0, 0, 0, [[1, 0], [0, 25]], 1]", "after": "derived = [[[1, 0], [0, 25]]]", "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 ORIGINAL ORIGINAL normalize row1 col0 := div(0, 25)", "after": "Lagrange ORIGINAL ORIGINAL normalize row1 col0 = 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 ORIGINAL ORIGINAL normalize row1 col0，不以小數近似替代。", "en": "Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "25" ], "output": "0", "label": "Lagrange ORIGINAL ORIGINAL normalize row1 col0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Lagrange ORIGINAL ORIGINAL normalize row1 col1 := div(25, 25)", "after": "Lagrange ORIGINAL ORIGINAL normalize row1 col1 = 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 ORIGINAL ORIGINAL normalize row1 col1，不以小數近似替代。", "en": "Recompute Lagrange ORIGINAL ORIGINAL normalize row1 col1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "25", "25" ], "output": "1", "label": "Lagrange ORIGINAL ORIGINAL normalize row1 col1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Lagrange ORIGINAL, 1, 1, 1, [[1, 0], [0, 25]], 25]", "after": "derived = [[[1, 0], [0, 1]]]", "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": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Lagrange ORIGINAL, [[1, 0], [0, 25]]]", "after": "derived = [[[1, 0], [0, 1]], [[0, 0], [1, 1]], 2]", "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [[[1, 0], [0, 25]]]", "after": "derived = [2, 2, 2, 是]", "relation": "implies", "rule_id": "Lagrange_supplied_vector_rank_ONLY_no_unprovided_functions_or_feasibility", "conditions": [], "explanation": { "zh-Hant": "這只是給定gradient rows的代數rank資格；沒有函數與point資料就不能獨立驗可微、feasible或extremum。full row rank比較約束數，不是ambient dimension。", "en": "This qualifies algebraic rank of GIVEN gradient rows only.Without functions/point data it does not independently establish differentiability,feasibility or an extremum.Compare rank with constraint count,not ambient dimension." }, "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": [ [ [ "1", "0" ], [ "0", "25" ] ], 2, 2, 2, true, "full_row_rank_means_constraint_gradients_are_linearly_independent" ] }, "conclusion": { "canonical_value": "([[1, 0], [0, 25]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)", "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": 25, "original_line_sha256": "001a6648df21e2b46a3aaf32a4fa5f10a1c339ea8278820ce2b645db9ed7e6d2", "question_compact_sha256": "e88bffa85da69b98e928a7cc7ef987ef5bb78a78e4c19408ef9ed03d05f48f5a", "english_row_compact_sha256": "0284fbae9fcbbd35cea19921ab1ca294a3c8ac50480e754dc236055ca14a0df7", "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": "3b2185ce3c451f274eefde420b48d7cb1c956e5f94a3ebfe58df51e69e250b77", "original_candidate_line_sha256": "3b2185ce3c451f274eefde420b48d7cb1c956e5f94a3ebfe58df51e69e250b77" }, "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": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "6213c26fad31f1be36134fd16c2746d111d7cbc689105dc9ac4a6c1340f620ec", "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." } }