{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r02-dual-vector-space-0015", "pilot_number": "R0036-18765", "number": 18765, "family_id": "qf-mod36-dual-functional-evaluation", "title": { "zh-Hant": "第 18765 題：對偶向量空間", "en": "Question 18765: A Covector Evaluates a Vector but Is Not That Vector" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r02-dual-vector-space-0015", "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-dual-vector-space" ], "title_zh": "對偶向量空間", "stem_zh": "有限維R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[15,-1],\"vector\":[2,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([15, -1], [2, 17], 13, linear_functional_not_primal_vector)", "explanation_zh": "Wave5第五層核對=([15, -1], [2, 17], 13, linear_functional_not_primal_vector)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "對偶向量空間" ] }, "variant": { "family_id": "qf-mod36-dual-functional-evaluation", "seed": 202611052015, "parameters": { "coefficients": [ 15, -1 ], "vector": [ 2, 17 ], "domain": "Finite_dimensional_real_dual_coordinates", "operation": "dual-functional-evaluation", "semantic_template": "qf-mod36-dual-functional-evaluation/{\"coefficients\":[15,-1],\"vector\":[2,17]}" } }, "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-r02-dual-vector-space-0015", "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6909cad67f003ddaa2d94821a112430d52b5698fb521583d8da918d71af6bc0a", "family_id": "qf-mod36-dual-functional-evaluation", "seed": 202611052015, "semantic_template": "qf-mod36-dual-functional-evaluation/{\"coefficients\":[15,-1],\"vector\":[2,17]}" }, "fields": { "title": "A Covector Evaluates a Vector but Is Not That Vector", "stem": "Use row coefficients=[15, -1] and primal vector=[2, 17] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).", "answer": { "kind": "expression", "value": "([15, -1], [2, 17], 13, linear_functional_not_primal_vector)", "explanation": "Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([15, -1], [2, 17], 13, linear_functional_not_primal_vector)." }, "skills": [ "A Covector Evaluates a Vector but Is Not That Vector" ], "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R^n中的線性泛函與明示基底座標。操作=dual-functional-evaluation；資料={\"coefficients\":[15,-1],\"vector\":[2,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use row coefficients=[15, -1] and primal vector=[2, 17] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=dual-functional-evaluation;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[coefficients, vector, value, 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": "fixed original covector ROW=[15, -1];coordinate dimension=2", "after": "define phi(x)=sum_j row[j]*x[j] for EVERY real coordinate vector x;finite distributivity gives phi(alpha*u+beta*v)=alpha*phi(u)+beta*phi(v)", "relation": "implies", "rule_id": "dual_row_symbolic_universal_linearity_not_one_numeric_test", "conditions": [], "explanation": { "zh-Hant": "固定row每個實座標的有限加乘分配律，對任意u,v,alpha,beta同時成立，才給全稱線性證書。後續代值只是核本題數值，不是用一個成功例子證未知函數線性。row表示實值函數，不是primal向量本身。", "en": "Finite distributivity for a fixed row holds for arbitrary real u,v,alpha,beta and gives the universal linearity certificate.The subsequent substitution checks this instance,not linearity of an unspecified function from one passing test.The row represents a real-valued function,not the primal vector itself." }, "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": "S003", "uses": [ "G001", "S002" ], "before": "dual functional original evaluation coordinate0 product := mul(15, 2)", "after": "dual functional original evaluation coordinate0 product = 30", "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 dual functional original evaluation coordinate0 product，不以小數近似替代。", "en": "Recompute dual functional original evaluation coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "2" ], "output": "30", "label": "dual functional original evaluation coordinate0 product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "dual functional original evaluation coordinate1 product := mul(-1, 17)", "after": "dual functional original evaluation coordinate1 product = -17", "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 dual functional original evaluation coordinate1 product，不以小數近似替代。", "en": "Recompute dual functional original evaluation coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "17" ], "output": "-17", "label": "dual functional original evaluation coordinate1 product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "dual functional original evaluation complete sum := sum(30, -17)", "after": "dual functional original evaluation complete sum = 13", "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 dual functional original evaluation complete sum，不以小數近似替代。", "en": "Recompute dual functional original evaluation complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "30", "-17" ], "output": "13", "label": "dual functional original evaluation complete sum" } } ], "math_model": { "result": [ [ "15", "-1" ], [ "2", "17" ], "13", "linear_functional_not_primal_vector" ] }, "conclusion": { "canonical_value": "([15, -1], [2, 17], 13, linear_functional_not_primal_vector)", "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-r02-dual-vector-space-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/questions.jsonl", "batch_sha256": "710a303eff4a61aad547a2ca9501595d14cef5d84a5e44e51d08864c41eb9f14", "original_line": 15, "original_line_sha256": "e5ae21ee83ba2be7a9aec2e32a1c8d1445e6d596ab28901a4c2c7be9b5bd2d03", "question_compact_sha256": "6909cad67f003ddaa2d94821a112430d52b5698fb521583d8da918d71af6bc0a", "english_row_compact_sha256": "91680eacbe7f57209017c2ef9aa853dd6e5b1fb6e93cf36076be6ed83ecd4a95", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.jsonl" }, "mko_bindings": { "mko-dual-vector-space": { "sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "english_sha256": "1e23ee2fa2f740284ff7f2e81139957b6196a81326cb161bae3a56d54fe07153", "original_path": "public/data/mko/source13-20261018-dual-vector-space.json", "english_path": "public/data/i18n/editions/mko-dual-vector-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "9a0df4557b352dc8e59fc4bfd181f3dc26a40f8be666b17256a644013654c8bb", "original_candidate_line_sha256": "9a0df4557b352dc8e59fc4bfd181f3dc26a40f8be666b17256a644013654c8bb" }, "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": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1b5ac6f73c77903de5f5ca3da559ef62d2e65c0ff91d7aab64a42db47ca4e575", "arithmetic_steps": 3 }, "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." } }