{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261012-r02-counting-principle-0044", "pilot_number": "R0009-05369", "number": 5369, "family_id": "qf-density12-counting-full-product-prefix", "title": { "zh-Hant": "第 5369 題：完整直積的總數與有序前綴", "en": "Question 5369: Full Product Choices and Prefix Counts" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261012-r02-counting-principle-0044", "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-counting-principle" ], "title_zh": "完整直積的總數與有序前綴", "stem_zh": "有限選擇協議；完整直積階段的選項數依序sizes=[2, 6]，所有跨階段組合均允許。依序答 (直積總數, 各前綴直積數列表)，零階段空直積有一個空選擇序列。", "answer": { "kind": "expression", "value": "(12, [2, 12])", "explanation_zh": "完整直積與前綴數=(12, [2, 12])。逐階段乘入選項數；零階段乘積一，遇零後的前綴皆零，非概率獨立假設。" }, "difficulty": { "band": "lower_secondary", "level": 3, "skills": [ "完整直積的總數與有序前綴" ] }, "variant": { "family_id": "qf-density12-counting-full-product-prefix", "seed": 202610122044, "parameters": { "sizes": [ 2, 6 ], "operation": "density12_counting_full_product_prefix", "domain": "finite_tagged_counting_protocol", "semantic_template": "qf-density12-counting-full-product-prefix/[2,6]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-08", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261008-advance-20261012-r02-counting-principle-0044", "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "25e6fb5f040abbd3bad38e6af237a86c0f87146781c6f7416bd3a7d78dee6aa6", "family_id": "qf-density12-counting-full-product-prefix", "seed": 202610122044, "semantic_template": "qf-density12-counting-full-product-prefix/[2,6]" }, "fields": { "title": "Full Product Choices and Prefix Counts", "stem": "Ordered stage sizes=[2, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.", "answer": { "kind": "expression", "value": "(12, [2, 12])", "explanation": "Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (12, [2, 12])." }, "skills": [ "Full Product Choices and Prefix Counts" ], "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": "有限選擇協議；完整直積階段的選項數依序sizes=[2, 6]，所有跨階段組合均允許。依序答 (直積總數, 各前綴直積數列表)，零階段空直積有一個空選擇序列。", "en": "Ordered stage sizes=[2, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "declared finite tagged choices or full-product protocol", "after": "combinatorial counts,not probabilities or independence", "relation": "implies", "rule_id": "finite_counting_protocol_scope", "conditions": [], "explanation": { "zh-Hant": "加法需互斥標籤，乘法需題目明示允許完整跨階段組合；都不是概率獨立假設。", "en": "Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption." }, "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": "ordered sizes=[2, 6]", "after": "initial accumulator=1", "relation": "implies", "rule_id": "counting_prefix_initial", "conditions": [], "explanation": { "zh-Hant": "互斥類別的空和是0；完整直積的空乘積是1，代表一個空選擇序列。", "en": "The empty disjoint sum is0;the empty full product is1,representing one empty choice sequence." }, "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": "prefix count1 := mul(1, 2)", "after": "prefix count1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依明示協議把新階段乘入，或把新互斥類別加進，保留每個前綴。", "en": "Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 prefix count1，不以小數近似替代。", "en": "Recompute prefix count1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "prefix count1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "prefix through position0;new size=2", "after": "prefix total=2", "relation": "implies", "rule_id": "counting_prefix_update", "conditions": [], "explanation": { "zh-Hant": "完整直積遇0後皆為0；互斥和仍按原順序加，兩種協議不可互換。", "en": "A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "prefix count2 := mul(2, 6)", "after": "prefix count2 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依明示協議把新階段乘入，或把新互斥類別加進，保留每個前綴。", "en": "Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 prefix count2，不以小數近似替代。", "en": "Recompute prefix count2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "6" ], "output": "12", "label": "prefix count2" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "prefix through position1;new size=6", "after": "prefix total=12", "relation": "implies", "rule_id": "counting_prefix_update", "conditions": [], "explanation": { "zh-Hant": "完整直積遇0後皆為0；互斥和仍按原順序加，兩種協議不可互換。", "en": "A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols." }, "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", [ "2", "12" ] ] }, "conclusion": { "canonical_value": "(12, [2, 12])", "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-20261008-advance-20261012-r02-counting-principle-foundations", "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r02-counting-principle-foundations/questions.jsonl", "batch_sha256": "0fe31383b4044b85bc1f62d7e619a7dc1b3a078203cd4e7413dfad935f5e17e9", "original_line": 44, "original_line_sha256": "d87d789fbef169b6c176ffb3bf5bf061e1f59d30280f14f9c0c7b5dbdb591b7c", "question_compact_sha256": "25e6fb5f040abbd3bad38e6af237a86c0f87146781c6f7416bd3a7d78dee6aa6", "english_row_compact_sha256": "56566d6fc57be31ab4bef7d9cd98961d080a711c6f44fa6d5ae2118add19b209", "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r02-counting-principle-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r02-counting-principle-foundations.en.jsonl" }, "mko_bindings": { "mko-counting-principle": { "sha256": "0a0619ebbab98b96543b42abc4e66d927aeea705c65ee85b3a9badb6d3b3cf57", "english_sha256": "122425d1403554c17fc2a0e18e7b67e807bc50d412bad0250aa17710f6a34102", "original_path": "public/data/mko/counting-principle.json", "english_path": "public/data/i18n/editions/mko-counting-principle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f", "original_candidate_sha256": "1a44e9d1754c7d1c25d45fcf90830fef8eed0e63d6cb0b6a8f7d5b80467fd5fc", "original_candidate_line_sha256": "1a44e9d1754c7d1c25d45fcf90830fef8eed0e63d6cb0b6a8f7d5b80467fd5fc" }, "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-20261026-005-p10", "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56", "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cf25af6a8d24d6e060e5fa23f4ce5276b848928b3d3e75b8ab2c4a580be54fe1", "arithmetic_steps": 2 }, "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." } }