{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r01-direct-sum-0077", "pilot_number": "R0036-18627", "number": 18627, "family_id": "qf-mod36-directsum-external-coordinate", "title": { "zh-Hant": "第 18627 題：子空間直和", "en": "Question 18627: External Direct-Sum Coordinates Live on a Product Carrier" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r01-direct-sum-0077", "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-direct-sum" ], "title_zh": "子空間直和", "stem_zh": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[1,2],\"right\":[3],\"other_left\":[4,3],\"other_right\":[5],\"scalar\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)", "explanation_zh": "Wave5第五層核對=([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "子空間直和" ] }, "variant": { "family_id": "qf-mod36-directsum-external-coordinate", "seed": 202611051077, "parameters": { "left": [ 1, 2 ], "right": [ 3 ], "other_left": [ 4, 3 ], "other_right": [ 5 ], "scalar": 2, "domain": "Exact_finite_dimensional_real_subspaces", "operation": "directsum-external-coordinate", "semantic_template": "qf-mod36-directsum-external-coordinate/{\"left\":[1,2],\"other_left\":[4,3],\"other_right\":[5],\"right\":[3],\"scalar\":2}" } }, "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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261105-r01-direct-sum-0077", "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "2db0c09ab6cfd8f7c87eb904b179538f0d1779b7ea869a6a6c159a22fa3f2dcb", "family_id": "qf-mod36-directsum-external-coordinate", "seed": 202611051077, "semantic_template": "qf-mod36-directsum-external-coordinate/{\"left\":[1,2],\"other_left\":[4,3],\"other_right\":[5],\"right\":[3],\"scalar\":2}" }, "fields": { "title": "External Direct-Sum Coordinates Live on a Product Carrier", "stem": "Use the pair (left,right) with left=[1, 2], right=[3], another pair other_left=[4, 3], other_right=[5], and scalar=2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary).", "answer": { "kind": "expression", "value": "([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)", "explanation": "Return the original pair, its coordinatewise sum with the other pair, and scalar times the original pair. This is a new pair carrier, not an unproved internal decomposition of an ambient space. The exact result, retaining the requested order, is ([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)." }, "skills": [ "External Direct-Sum Coordinates Live on a Product Carrier" ], "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": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-external-coordinate；資料={\"left\":[1,2],\"right\":[3],\"other_left\":[4,3],\"other_right\":[5],\"scalar\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the pair (left,right) with left=[1, 2], right=[3], another pair other_left=[4, 3], other_right=[5], and scalar=2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=directsum-external-coordinate;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[pair, sum, scaled_pair, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。", "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence." }, "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": "external left coordinate0 sum := sum(1, 4)", "after": "external left coordinate0 sum = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external left coordinate0 sum，不以小數近似替代。", "en": "Recompute external left coordinate0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "4" ], "output": "5", "label": "external left coordinate0 sum" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "external left coordinate1 sum := sum(2, 3)", "after": "external left coordinate1 sum = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external left coordinate1 sum，不以小數近似替代。", "en": "Recompute external left coordinate1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "3" ], "output": "5", "label": "external left coordinate1 sum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "external left coordinate0 scalar product := mul(2, 1)", "after": "external left coordinate0 scalar product = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external left coordinate0 scalar product，不以小數近似替代。", "en": "Recompute external left coordinate0 scalar product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "external left coordinate0 scalar product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "external left coordinate1 scalar product := mul(2, 2)", "after": "external left coordinate1 scalar product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external left coordinate1 scalar product，不以小數近似替代。", "en": "Recompute external left coordinate1 scalar product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "external left coordinate1 scalar product" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "external right coordinate0 sum := sum(3, 5)", "after": "external right coordinate0 sum = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external right coordinate0 sum，不以小數近似替代。", "en": "Recompute external right coordinate0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "5" ], "output": "8", "label": "external right coordinate0 sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "external right coordinate0 scalar product := mul(2, 3)", "after": "external right coordinate0 scalar product = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 external right coordinate0 scalar product，不以小數近似替代。", "en": "Recompute external right coordinate0 scalar product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "external right coordinate0 scalar product" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "original pair=([1, 2],[3]);block lengths=[2, 1];other pair=([4, 3],[5]);scalar=2", "after": "coordinatewise pair sum=[[5, 5], [8]];scaled pair=[[2, 4], [6]];new PRODUCT carrier,not assumed internal subspaces", "relation": "implies", "rule_id": "directsum_external_pair_not_ambient_internal_identification", "conditions": [], "explanation": { "zh-Hant": "兩個有標籤座標區塊保各自長度，連空區塊也合法。有限兩項的外直和在pair載體逐塊運算，不因長度巧合就把它宣稱某母空間內的唯一分解。", "en": "Retain both labeled coordinate blocks,including valid empty blocks.The two-term external direct sum uses coordinatewise operations on a pair carrier;matching lengths do not prove an internal decomposition of another ambient space." }, "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": [ [ [ "1", "2" ], [ "3" ] ], [ [ "5", "5" ], [ "8" ] ], [ [ "2", "4" ], [ "6" ] ], "external_product_carrier_not_internal_subspaces" ] }, "conclusion": { "canonical_value": "([[1, 2], [3]], [[5, 5], [8]], [[2, 4], [6]], external_product_carrier_not_internal_subspaces)", "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-20261014-advance-20261105-r01-direct-sum-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r01-direct-sum-foundations/questions.jsonl", "batch_sha256": "7e0295c5944e58b0de639be2d989049e9f65b8f21905ff6597dbf939a5f3be6d", "original_line": 77, "original_line_sha256": "0be8f44e2dcd48cdad9865637264d3085cf169d13d69ad505a53da33a55c3728", "question_compact_sha256": "2db0c09ab6cfd8f7c87eb904b179538f0d1779b7ea869a6a6c159a22fa3f2dcb", "english_row_compact_sha256": "7c17d3b00ef139c08aac3e24850ee88cf8efb6b78635d057ec4c1ac2b215e4c8", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.jsonl" }, "mko_bindings": { "mko-direct-sum": { "sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8", "english_sha256": "1fe3402da619d87b341bcf2c97aad834c0b1ea87391581c56651a5dc46bc3779", "original_path": "public/data/mko/direct-sum.json", "english_path": "public/data/i18n/editions/mko-direct-sum.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "01284e9e448f7c2b2bf032cf8a58c263c5bc34cf5b3063eecc5d297498003c32", "original_candidate_line_sha256": "01284e9e448f7c2b2bf032cf8a58c263c5bc34cf5b3063eecc5d297498003c32" }, "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-20261026-008-p37", "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301", "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "610173bff733c015c6e5d323d4dff51c478dbb8bcc83af4c3d7a039620418214", "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." } }