{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r01-matrix-0059", "pilot_number": "R0065-33109", "number": 33109, "family_id": "qf-matrix-addition", "title": { "zh-Hant": "第 33109 題：同尺寸矩陣相加", "en": "Question 33109: Addition of Equal-Size Matrices" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261006-advance-20261007-r01-matrix-0059", "batch_id": "batch-20261006-advance-20261007-r01-matrix-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-matrix" ], "title_zh": "同尺寸矩陣相加", "stem_zh": "實矩陣，橫列／直行次序固定，A=[[-4, -5, -6, -7], [-2, -3, -4, -5]]、B=[[-5, -3, -1, 1], [-4, -2, 0, 2]]。明示兩矩陣同尺寸，求 A+B，以巢狀橫列列表回答。", "answer": { "kind": "expression", "value": "[[-9, -8, -7, -6], [-6, -5, -4, -3]]", "explanation_zh": "和矩陣=[[-9, -8, -7, -6], [-6, -5, -4, -3]]。只有對應位置相加，不能把不同尺寸補零混合。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "同尺寸矩陣相加" ] }, "variant": { "family_id": "qf-matrix-addition", "seed": 202610071059, "parameters": { "A": [ [ -4, -5, -6, -7 ], [ -2, -3, -4, -5 ] ], "B": [ [ -5, -3, -1, 1 ], [ -4, -2, 0, 2 ] ], "operation": "matrix_addition", "domain": "reals", "semantic_template": "qf-matrix-addition/{\"inputs\":[[[\"-4\",\"-5\",\"-6\",\"-7\"],[\"-2\",\"-3\",\"-4\",\"-5\"]],[[\"-5\",\"-3\",\"-1\",\"1\"],[\"-4\",\"-2\",\"0\",\"2\"]]],\"operation\":\"matrix_addition\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-06", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261006-advance-20261007-r01-matrix-0059", "batch_id": "batch-20261006-advance-20261007-r01-matrix-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "32cfa0a2b4bd3ebc332c8e81732f3f7b470df06e416512da083df77bc34ebe2f", "family_id": "qf-matrix-addition", "seed": 202610071059, "semantic_template": "qf-matrix-addition/{\"inputs\":[[[\"-4\",\"-5\",\"-6\",\"-7\"],[\"-2\",\"-3\",\"-4\",\"-5\"]],[[\"-5\",\"-3\",\"-1\",\"1\"],[\"-4\",\"-2\",\"0\",\"2\"]]],\"operation\":\"matrix_addition\"}" }, "fields": { "title": "Addition of Equal-Size Matrices", "stem": "Over R, A=[[-4, -5, -6, -7], [-2, -3, -4, -5]] and B=[[-5, -3, -1, 1], [-4, -2, 0, 2]] have the same dimensions. Find A+B as a nested list of rows.", "answer": { "kind": "expression", "value": "[[-9, -8, -7, -6], [-6, -5, -4, -3]]", "explanation": "Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3]]. Do not pad matrices of different dimensions with zeros to force this operation." }, "skills": [ "Addition of Equal-Size Matrices" ], "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": "實矩陣，橫列／直行次序固定，A=[[-4, -5, -6, -7], [-2, -3, -4, -5]]、B=[[-5, -3, -1, 1], [-4, -2, 0, 2]]。明示兩矩陣同尺寸，求 A+B，以巢狀橫列列表回答。", "en": "Over R, A=[[-4, -5, -6, -7], [-2, -3, -4, -5]] and B=[[-5, -3, -1, 1], [-4, -2, 0, 2]] have the same dimensions. Find A+B as a nested list of rows." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-matrix-addition, matrix_addition, reals, []]", "after": "derived = [[nested_ROW_matrix]]", "relation": "implies", "rule_id": "R65_SOURCE_bank_keys_operation_domain_ordered_output_roles", "conditions": [], "explanation": { "zh-Hant": "MOD25用LOCAL hyphen及具名domain，矩陣legacy用underscore/reals，BASElogic不補domain/op。輸出tuple/矩陣/數字欄與fraction文字各依來源，不一律保/1；原中英/AST/status不變。", "en": "MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [A ORIGINAL, [[-4, -5, -6, -7], [-2, -3, -4, -5]]]", "after": "derived = [2, 4]", "relation": "implies", "rule_id": "Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused", "conditions": [], "explanation": { "zh-Hant": "本bank允許1..5乘1..5矩陣，不搬R64的2×2界。每個raw cell先驗SAFE，包括shape/entry未選到的欄位；不改原row順序。", "en": "THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [B ORIGINAL, [[-5, -3, -1, 1], [-4, -2, 0, 2]]]", "after": "derived = [2, 4]", "relation": "implies", "rule_id": "Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused", "conditions": [], "explanation": { "zh-Hant": "本bank允許1..5乘1..5矩陣，不搬R64的2×2界。每個raw cell先驗SAFE，包括shape/entry未選到的欄位；不改原row順序。", "en": "THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Matrix sum ORIGINAL entry0,0 := sum(-4, -5)", "after": "Matrix sum ORIGINAL entry0,0 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry0,0，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry0,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "-5" ], "output": "-9", "label": "Matrix sum ORIGINAL entry0,0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Matrix sum ORIGINAL entry0,1 := sum(-5, -3)", "after": "Matrix sum ORIGINAL entry0,1 = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry0,1，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry0,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "-3" ], "output": "-8", "label": "Matrix sum ORIGINAL entry0,1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Matrix sum ORIGINAL entry0,2 := sum(-6, -1)", "after": "Matrix sum ORIGINAL entry0,2 = -7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry0,2，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry0,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-6", "-1" ], "output": "-7", "label": "Matrix sum ORIGINAL entry0,2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Matrix sum ORIGINAL entry0,3 := sum(-7, 1)", "after": "Matrix sum ORIGINAL entry0,3 = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry0,3，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry0,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-7", "1" ], "output": "-6", "label": "Matrix sum ORIGINAL entry0,3" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Matrix sum ORIGINAL entry1,0 := sum(-2, -4)", "after": "Matrix sum ORIGINAL entry1,0 = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry1,0，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry1,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "-4" ], "output": "-6", "label": "Matrix sum ORIGINAL entry1,0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Matrix sum ORIGINAL entry1,1 := sum(-3, -2)", "after": "Matrix sum ORIGINAL entry1,1 = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry1,1，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry1,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-3", "-2" ], "output": "-5", "label": "Matrix sum ORIGINAL entry1,1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Matrix sum ORIGINAL entry1,2 := sum(-4, 0)", "after": "Matrix sum ORIGINAL entry1,2 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry1,2，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry1,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "0" ], "output": "-4", "label": "Matrix sum ORIGINAL entry1,2" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Matrix sum ORIGINAL entry1,3 := sum(-5, 2)", "after": "Matrix sum ORIGINAL entry1,3 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Matrix sum ORIGINAL entry1,3，不以小數近似替代。", "en": "Recompute Matrix sum ORIGINAL entry1,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "2" ], "output": "-3", "label": "Matrix sum ORIGINAL entry1,3" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [[[-4, -5, -6, -7], [-2, -3, -4, -5]], [[-5, -3, -1, 1], [-4, -2, 0, 2]]]", "after": "derived = [[[-9, -8, -7, -6], [-6, -5, -4, -3]]]", "relation": "implies", "rule_id": "Matrix_ADDITION_all_corresponding_entries_NO_padding_or_raw_recap", "conditions": [], "explanation": { "zh-Hant": "只在相同shape逐對應位置相加，衍生和可超SAFE，不重新當raw矩陣驗。不可補0或broadcast強行加不同shape。", "en": "Add EVERY corresponding entry ONLY at equal shapes.Derived sums may exceed SAFE;do not reparse as a RAW matrix.No zero-padding or broadcasting." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ [ "-9", "-8", "-7", "-6" ], [ "-6", "-5", "-4", "-3" ] ] }, "conclusion": { "canonical_value": "[[-9, -8, -7, -6], [-6, -5, -4, -3]]", "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-20261006-advance-20261007-r01-matrix-foundations", "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r01-matrix-foundations/questions.jsonl", "batch_sha256": "7a870e76f1679781814f6ee29550fd91d2940bcdc5c617ba275d633c187947c5", "original_line": 59, "original_line_sha256": "799de7ccc0d552befad80e6fe49aec7408c6f7ebdab0607fc22122900b6b23a0", "question_compact_sha256": "32cfa0a2b4bd3ebc332c8e81732f3f7b470df06e416512da083df77bc34ebe2f", "english_row_compact_sha256": "3d4533522cf92de0e2a58f941fc0ccc376adfbaf01d3e254056759a8c31ac62e", "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r01-matrix-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r01-matrix-foundations.en.jsonl" }, "mko_bindings": { "mko-matrix": { "sha256": "c0233ecc6ee07427af1589feb16108500eb2e48b0b2fb84b5b39f0f38b02f109", "english_sha256": "18504021d52173d8f2f8aa663d83dc058f5205bea2c677a9722daba770776432", "original_path": "public/data/mko/matrix.json", "english_path": "public/data/i18n/editions/mko-matrix.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "871e2c05026567a1deae741d6b90b0588100c6482ee167c7f9f45af5b72d8606", "original_candidate_sha256": "0725aef2d9a7d92cd13703983407824f7e5ee5e477dcddb8f6b7e809e369521e", "original_candidate_line_sha256": "0725aef2d9a7d92cd13703983407824f7e5ee5e477dcddb8f6b7e809e369521e" }, "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-p66", "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": "92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6", "question_first_witness_sha256": "012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00" }, "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": "4cafa7b542ca22b51fbb0e1630453f0d184e44c16402727cbf71255daedc8a53", "arithmetic_steps": 8 }, "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." } }