{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261027-r02-norm-induced-metric-0090", "pilot_number": "R0025-13140", "number": 13140, "family_id": "qf-mod27-normmetric-induced-qualification", "title": { "zh-Hant": "第 13140 題：範數誘導度量", "en": "Question 13140: Distinguish a Universal Norm Certificate from Finite Samples" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261027-r02-norm-induced-metric-0090", "batch_id": "batch-20261014-advance-20261027-r02-norm-induced-metric-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-norm-induced-metric" ], "title_zh": "範數誘導度量", "stem_zh": "同一已認證範數以d(x,y)=norm(x-y)定義距離。操作=normmetric-induced-qualification；資料={\"norm_name\":\"l1\",\"dimension\":15,\"evidence_mode\":\"named_formula_certificate\"}。請依 [norm_name,dimension,evidence_mode,sufficient,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(l1, 15, named_formula_certificate, 是, universal_from_norm_axioms)", "explanation_zh": "Wave4根節點核對=(l1, 15, named_formula_certificate, 是, universal_from_norm_axioms)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；誘導性來自範數公理的全稱證書；不宣稱每個metric皆由norm誘導或空間完備。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "範數誘導度量" ] }, "variant": { "family_id": "qf-mod27-normmetric-induced-qualification", "seed": 202610272090, "parameters": { "norm_name": "l1", "dimension": 15, "evidence_mode": "named_formula_certificate", "domain": "Named_norm_induced_metric_on_Rn", "operation": "normmetric-induced-qualification", "semantic_template": "qf-mod27-normmetric-induced-qualification/{\"dimension\":15,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式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-20261027-r02-norm-induced-metric-0090", "batch_id": "batch-20261014-advance-20261027-r02-norm-induced-metric-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "c2b0bd89cd6a6ac6861dbcdb26d85837ad6345864a23a7824719051c7ff5151d", "family_id": "qf-mod27-normmetric-induced-qualification", "seed": 202610272090, "semantic_template": "qf-mod27-normmetric-induced-qualification/{\"dimension\":15,\"evidence_mode\":\"named_formula_certificate\",\"norm_name\":\"l1\"}" }, "fields": { "title": "Distinguish a Universal Norm Certificate from Finite Samples", "stem": "Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=15, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.", "answer": { "kind": "expression", "value": "(l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms)", "explanation": "The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms)." }, "skills": [ "Distinguish a Universal Norm Certificate from Finite Samples" ], "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": "同一已認證範數以d(x,y)=norm(x-y)定義距離。操作=normmetric-induced-qualification；資料={\"norm_name\":\"l1\",\"dimension\":15,\"evidence_mode\":\"named_formula_certificate\"}。請依 [norm_name,dimension,evidence_mode,sufficient,boundary] 欄序回答。", "en": "Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=15, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "certified named norm=l1;fixed real space R^15;d(x,y)=norm(x-y)", "after": "d>=0;d(x,y)=0 iff x-y=0 iff x=y", "relation": "implies", "rule_id": "norm_metric_positivity_and_separation", "conditions": [], "explanation": { "zh-Hant": "命名l1或linfinity的範數公理給非負與零向量分離，代入x−y後得到metric的分離責任。", "en": "The certified l1 or linfinity norm axioms give nonnegativity and separation at the zero vector.Substitute x−y to obtain metric separation." }, "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": "arbitrary x,y in R^15;y-x=-(x-y)", "after": "norm(y-x)=abs(-1)*norm(x-y)=norm(x-y)", "relation": "implies", "rule_id": "norm_metric_symmetry_absolute_homogeneity", "conditions": [], "explanation": { "zh-Hant": "用所有實純量的絕對齊次性在−1處套用，不依某一對數值相同猜對稱。", "en": "Apply absolute homogeneity at scalar−1,not an inference from one pair of equal numeric distances." }, "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": "arbitrary x,y,z in R^15;x-z=(x-y)+(y-z)", "after": "d(x,z)<=d(x,y)+d(y,z) by norm triangle inequality", "relation": "implies", "rule_id": "norm_metric_triangle_universal_decomposition", "conditions": [], "explanation": { "zh-Hant": "同一向量差的分解加上範數全稱三角不等式，對所有三點成立；不主張每個metric都可如此誘導，也不給完備性。", "en": "The vector-difference identity and universal norm triangle inequality apply to every triple.This does not assert every metric is norm-induced or establish completeness." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "norm=l1;dimension=15;selected evidence=named_formula_certificate", "after": "evidence sufficient=是;sample failure does not deny named formula metricity", "relation": "implies", "rule_id": "induced_metric_evidence_mode_not_property_negation", "conditions": [], "explanation": { "zh-Hant": "全稱範數公式證書足夠，有限距離樣本沒有覆蓋metric三公理所有點。No只判此證據不足，不說已知命名公式不是metric。", "en": "A universal named-norm certificate suffices;finite distance samples do not cover all points of the metric axioms.No judges the evidence insufficient,not the known named formula nonmetric." }, "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": [ "l1", 15, "named_formula_certificate", true, "universal_from_norm_axioms" ] }, "conclusion": { "canonical_value": "(l1, 15, named_formula_certificate, 是, universal_from_norm_axioms)", "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-20261027-r02-norm-induced-metric-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations/questions.jsonl", "batch_sha256": "d4488185ba942b622850b55a9c53a0c7d5409e86894a1ac5b64db546aef03b98", "original_line": 90, "original_line_sha256": "1b8d3505f4ed3625ea14dec7c0c7b2994630b4576317f7292012aae24efd76a4", "question_compact_sha256": "c2b0bd89cd6a6ac6861dbcdb26d85837ad6345864a23a7824719051c7ff5151d", "english_row_compact_sha256": "48a03a9bd3b86f17242446528d25ee799f0ff0a6e14ab8ae180d8895570ceb2f", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations.en.jsonl" }, "mko_bindings": { "mko-norm-induced-metric": { "sha256": "579d0833bcf98eaea0df0559e17c92ecaaa60c6760d51154ff23dcc92d83da4f", "english_sha256": "b7e34d77c8ec2850d00096ccb3e031e9c6b20a860974880e45e7fe95d8bbb5bb", "original_path": "public/data/mko/norm-induced-metric.json", "english_path": "public/data/i18n/editions/mko-norm-induced-metric.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924", "original_candidate_sha256": "dc5af36482833b6ca12806bfe4e951554d672d34c62bf4b3b7dfbda4f7f00b70", "original_candidate_line_sha256": "dc5af36482833b6ca12806bfe4e951554d672d34c62bf4b3b7dfbda4f7f00b70" }, "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-20261026-007-p26", "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": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5", "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "8e07a96bae8b118c97223ab28da418b730d94103ef2d4f2b7b1c8d9f9bf874d1", "arithmetic_steps": 0 }, "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." } }