{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r06-mobius-function-0098", "pilot_number": "R0042-21773", "number": 21773, "family_id": "qf-mod40-mobius-domain-boundary", "title": { "zh-Hant": "第 21773 題：Möbius函數", "en": "Question 21773: Zero and Negative Integers Are Outside This Mobius Domain" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r06-mobius-function-0098", "batch_id": "batch-20261014-advance-20261109-r06-mobius-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-mobius-function" ], "title_zh": "Möbius函數", "stem_zh": "positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-22, 否, null, Mobius_domain_positive_integers_only)", "explanation_zh": "Wave6第三層核對=(-22, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Möbius函數" ] }, "variant": { "family_id": "qf-mod40-mobius-domain-boundary", "seed": 202611096098, "parameters": { "n": -22, "domain": "Exact_positive_integer_Mobius_function", "operation": "mobius-domain-boundary", "semantic_template": "qf-mod40-mobius-domain-boundary/{\"n\":-22}" } }, "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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r06-mobius-function-0098", "batch_id": "batch-20261014-advance-20261109-r06-mobius-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8efddeb334e2d80fdebe8540c019259f72ad2ad258ee6c84f9f6871b2068d35e", "family_id": "qf-mod40-mobius-domain-boundary", "seed": 202611096098, "semantic_template": "qf-mod40-mobius-domain-boundary/{\"n\":-22}" }, "fields": { "title": "Zero and Negative Integers Are Outside This Mobius Domain", "stem": "Use nonpositive bounded integer n=-22. Give (n, in_positive_domain, mu_or_null, boundary).", "answer": { "kind": "expression", "value": "(-22, No, null, Mobius_domain_positive_integers_only)", "explanation": "Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-22, No, null, Mobius_domain_positive_integers_only)." }, "skills": [ "Zero and Negative Integers Are Outside This Mobius Domain" ], "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": "positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use nonpositive bounded integer n=-22. Give (n, in_positive_domain, mu_or_null, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=mobius-domain-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[n, in_positive_domain, mu_or_null, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記保留，本頁直接列來源有序欄位，不需JavaScript。tag不是額外證據，固定No保留其原聲稱範圍。", "en": "Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope." }, "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": "ORIGINAL bounded integer n=-22<=0", "after": "in_positive_domain=否;mu_or_null=null;do not apply positive-integer squarefree parity convention here", "relation": "implies", "rule_id": "mobius_nonpositive_outside_declared_domain", "conditions": [], "explanation": { "zh-Hant": "此函數宣告定義於正整數；零與負數在本模型外，null不是0的mu值，也不否認其他延伸慣例可以另定義。", "en": "This function is declared on positive integers.Zero and negatives are outside this model;null is not a value of0,nor a denial that other extensions can be defined." }, "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": [ -22, false, null, "Mobius_domain_positive_integers_only" ] }, "conclusion": { "canonical_value": "(-22, 否, null, Mobius_domain_positive_integers_only)", "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-20261109-r06-mobius-function-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r06-mobius-function-foundations/questions.jsonl", "batch_sha256": "7ed895d51f7c25fef70946caed5452254a4a2e02ad05f2e1c4b765f1ccdba585", "original_line": 98, "original_line_sha256": "b89197d8420ad9f8abec256089af0bffa80cc30e16766b2fb9d35fe71b8aef6a", "question_compact_sha256": "8efddeb334e2d80fdebe8540c019259f72ad2ad258ee6c84f9f6871b2068d35e", "english_row_compact_sha256": "87aba1c1b10a414fb1507bb56367b5a79686e1949676acfe48c5d89fc755adb3", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r06-mobius-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r06-mobius-function-foundations.en.jsonl" }, "mko_bindings": { "mko-mobius-function": { "sha256": "8991dece49cf8a59e05519f80e840deb92afafba5fef912edc1a03ad30438f84", "english_sha256": "c602666c47bdf330fa90517c2baa6b3bb4811deb05ff4c991c801c6ba0793623", "original_path": "public/data/mko/mobius-function.json", "english_path": "public/data/i18n/editions/mko-mobius-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a84fbe07a4e46e8e47a8ad331f35fd3f81b821aef6cb27e7852de854aaa96b9a", "original_candidate_sha256": "5be399794cc70c5bbbba05e9294dba551e2cdea38df4d5ee0edc2a5ab25f47f8", "original_candidate_line_sha256": "5be399794cc70c5bbbba05e9294dba551e2cdea38df4d5ee0edc2a5ab25f47f8" }, "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-20261026-009-p43", "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": "dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a", "question_first_witness_sha256": "9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "901c468aa3d03614ab9a0a5a8f5f83749ce8f7dcb2d9b18b6f18d961792b9695", "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." } }