{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r05-catalan-numbers-0087", "pilot_number": "R0044-22737", "number": 22737, "family_id": "qf-mod42-catalan-negative-index-boundary", "title": { "zh-Hant": "第 22737 題：Catalan數", "en": "Question 22737: Negative Indices Are Outside This Catalan Domain" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r05-catalan-numbers-0087", "batch_id": "batch-20261014-advance-20261111-r05-catalan-numbers-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-catalan-numbers" ], "title_zh": "Catalan數", "stem_zh": "C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-12, 否, null, Catalan_domain_is_natural_numbers_including_zero)", "explanation_zh": "Wave6第五層核對=(-12, 否, null, Catalan_domain_is_natural_numbers_including_zero)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "Catalan數" ] }, "variant": { "family_id": "qf-mod42-catalan-negative-index-boundary", "seed": 202611115087, "parameters": { "n": -12, "domain": "Exact_Catalan_sequence", "operation": "catalan-negative-index-boundary", "semantic_template": "qf-mod42-catalan-negative-index-boundary/{\"n\":-12}" } }, "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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r05-catalan-numbers-0087", "batch_id": "batch-20261014-advance-20261111-r05-catalan-numbers-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8a320d87b9406f6f6cfff6f65394018ef74e29ea5b440a12bd08b2cf9d01020e", "family_id": "qf-mod42-catalan-negative-index-boundary", "seed": 202611115087, "semantic_template": "qf-mod42-catalan-negative-index-boundary/{\"n\":-12}" }, "fields": { "title": "Negative Indices Are Outside This Catalan Domain", "stem": "Use negative integer n=-12. Give (n, in_domain, value_or_null, boundary).", "answer": { "kind": "expression", "value": "(-12, No, null, Catalan_domain_is_natural_numbers_including_zero)", "explanation": "Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-12, No, null, Catalan_domain_is_natural_numbers_including_zero)." }, "skills": [ "Negative Indices Are Outside This Catalan 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": "C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-negative-index-boundary；資料={\"n\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use negative integer n=-12. Give (n, in_domain, value_or_null, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=catalan-negative-index-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[n, in_domain, value_or_null, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。", "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness." }, "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 negative n=-12", "after": "in_domain=否;value_or_null=null;this declared sequence startsC0=1,not a negative-index extension", "relation": "implies", "rule_id": "catalan_negative_outside_zero_based_natural_domain", "conditions": [], "explanation": { "zh-Hant": "本序列從自然數0起，負索引保null；不擅自採用其他延伸，也不把C0改成C1。", "en": "This sequence starts at natural index0.Negative indices remain null without silently adopting an extension or shifting C0 to C1." }, "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": [ -12, false, null, "Catalan_domain_is_natural_numbers_including_zero" ] }, "conclusion": { "canonical_value": "(-12, 否, null, Catalan_domain_is_natural_numbers_including_zero)", "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-20261111-r05-catalan-numbers-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/questions.jsonl", "batch_sha256": "ceaecd210e147af2eb1f9c63d2057f658f4e7d1ec07cc95c12ee88d2eb3cb856", "original_line": 87, "original_line_sha256": "add592b61c96c0ee79eb445d04d7dfb344ac55ce62b5c62ee0f3d38e41ffee81", "question_compact_sha256": "8a320d87b9406f6f6cfff6f65394018ef74e29ea5b440a12bd08b2cf9d01020e", "english_row_compact_sha256": "67235f34fb0839fd35751dd74e63adb5d7c565e1554e7bb74d25b0369c77585f", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r05-catalan-numbers-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r05-catalan-numbers-foundations.en.jsonl" }, "mko_bindings": { "mko-catalan-numbers": { "sha256": "2d847df82bf2c534d59aedb78304a0f59d94a87bf4454e0eee1886df489e21b5", "english_sha256": "328d37ca081c47ad3e92abe05ddd5028dfeb23321136b79cc8b3d101c89e814d", "original_path": "public/data/mko/catalan-numbers.json", "english_path": "public/data/i18n/editions/mko-catalan-numbers.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4db6f09f48d29a15f89787c4aac3ae0abe7981d9f906e3fbecc88b60dd686db5", "original_candidate_sha256": "d5f914ce3f8362bee291e2537227f12cff13991c429c4ba0546437ac8a95e757", "original_candidate_line_sha256": "d5f914ce3f8362bee291e2537227f12cff13991c429c4ba0546437ac8a95e757" }, "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-p45", "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": "89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81", "question_first_witness_sha256": "b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7" }, "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": "f58aeeebf608dd04b75fbac8919847fbdc7ea1dbd312704c81e90b8eaf2c7453", "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." } }