{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-r01-power-root-0015", "pilot_number": "R0055-28215", "number": 28215, "family_id": "qf-natural-power-exact", "title": { "zh-Hant": "第 28215 題：自然數冪的精確值", "en": "Question 28215: Exact Natural-Number Power" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-r01-power-root-0015", "batch_id": "batch-20261003-r01-power-root-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-power-root" ], "title_zh": "自然數冪的精確值", "stem_zh": "在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (2)^2 的精確值。", "answer": { "kind": "numeric", "value": "4", "explanation_zh": "(2)^2=4。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "自然數冪的精確值" ] }, "variant": { "family_id": "qf-natural-power-exact", "seed": 202610031015, "parameters": { "a": 2, "n": 2, "domain": "reals", "zero_power": "empty_product", "operation": "natural_power", "semantic_template": "qf-natural-power-exact/{\"a\":2,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_source_snapshot_check", "producer_substantive_template_deduplication" ], "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-03", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261003-r01-power-root-0015", "batch_id": "batch-20261003-r01-power-root-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "05e299e45b03fdd26cddbf8a4544afbf1d8bfcddf5dcba04a4aa55e4d5ecd04a", "family_id": "qf-natural-power-exact", "seed": 202610031015, "semantic_template": "qf-natural-power-exact/{\"a\":2,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}" }, "fields": { "title": "Exact Natural-Number Power", "stem": "For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (2)^2 exactly.", "answer": { "kind": "numeric", "value": "4", "explanation": "(2)^2=4. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits." }, "skills": [ "Exact Natural-Number Power" ], "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": "在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (2)^2 的精確值。", "en": "For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (2)^2 exactly." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-natural-power-exact, natural_power]", "after": "derived = [[exact_integer_power]]", "relation": "implies", "rule_id": "source_answer_roles_or_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。", "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。", "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Legacy power ORIGINAL multiply1 := mul(1, 2)", "after": "Legacy power ORIGINAL multiply1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。", "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Legacy power ORIGINAL multiply1，不以小數近似替代。", "en": "Recompute Legacy power ORIGINAL multiply1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Legacy power ORIGINAL multiply1" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Legacy power ORIGINAL multiply2 := mul(2, 2)", "after": "Legacy power ORIGINAL multiply2 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。", "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Legacy power ORIGINAL multiply2，不以小數近似替代。", "en": "Recompute Legacy power ORIGINAL multiply2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Legacy power ORIGINAL multiply2" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [2, 2, empty_product]", "after": "derived = [4, 正]", "relation": "implies", "rule_id": "Legacy_power_ALL_n_products_EMPTYproduct_n0_and_sign_parity", "conditions": [], "explanation": { "zh-Hant": "n0的空乘積固定1（含此代數設定的0^0=1），不是極限規則。n>0時zero base得0；非零base的符號由原base與n奇偶決定。全部BigInt結果精確，不把衍生大整數截到SAFE或1e8。", "en": "The n0 empty product is1,including0^0=1 in THIS algebraic setting,not a limit rule.For n>0 a zero base gives0;otherwise sign follows base sign and exponent parity.All BigInt results remain exact without derived SAFE/1e8 clipping." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。", "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct." } }, "arithmetic": null } ], "math_model": { "result": "4" }, "conclusion": { "canonical_value": "4", "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-20261003-r01-power-root-foundations", "batch_path": "public/data/questions/batches/batch-20261003-r01-power-root-foundations/questions.jsonl", "batch_sha256": "bfa4980fbbea1d78253ffeb98cae98821ea3170cde1f1511acda1bb97d29de73", "original_line": 15, "original_line_sha256": "eff4100fd139e31d5a318b779b1fb48ec2c462181a98f16a487a6fc1a28eede0", "question_compact_sha256": "05e299e45b03fdd26cddbf8a4544afbf1d8bfcddf5dcba04a4aa55e4d5ecd04a", "english_row_compact_sha256": "f92799560e438d0f321ae51e344e335a99b834527a64ddff49fb0bcee10da22f", "english_manifest_path": "public/data/i18n/editions/batch-20261003-r01-power-root-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261003-r01-power-root-foundations.en.jsonl" }, "mko_bindings": { "mko-power-root": { "sha256": "29380d066f7612e50db0cbd16c05855d0464ed8f50858e7f348e62e899557c19", "english_sha256": "52d750ba77161d6a82d15e4c2d59f1983c52a8affd327d39f3f1b6c3756c664f", "original_path": "public/data/mko/power-root.json", "english_path": "public/data/i18n/editions/mko-power-root.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686", "original_candidate_sha256": "21f4ac8f9f9cab7f1c7e27676eaa8ddd99fa3233dc17e80666a2ba32e1601205", "original_candidate_line_sha256": "21f4ac8f9f9cab7f1c7e27676eaa8ddd99fa3233dc17e80666a2ba32e1601205" }, "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-20261026-010-p56", "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": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683", "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e" }, "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": "5d755f3bceaf7114ca9e721aaf13f0bbcafcb6a2028627ab8675feb306fe0424", "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." } }