{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-r02-polynomial-0096", "pilot_number": "R0067-34221", "number": 34221, "family_id": "qf-polynomial-addition", "title": { "zh-Hant": "第 34221 題：多項式係數逐項相加", "en": "Question 34221: Coefficientwise Polynomial Addition" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-r02-polynomial-0096", "batch_id": "batch-20261003-r02-polynomial-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-polynomial" ], "title_zh": "多項式係數逐項相加", "stem_zh": "本題 Q[x] 的整數係數向量由低次至高次排列。求 [8, 1] 與 [-8, 1] 的多項式和，以去尾端零的係數向量回答。", "answer": { "kind": "expression", "value": "[0, 2]", "explanation_zh": "每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。" }, "difficulty": { "band": "upper_secondary", "level": 3, "skills": [ "多項式係數逐項相加" ] }, "variant": { "family_id": "qf-polynomial-addition", "seed": 202610032096, "parameters": { "left": [ 8, 1 ], "right": [ -8, 1 ], "operation": "add", "semantic_template": "qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\",\"1\"]}" } }, "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-r02-polynomial-0096", "batch_id": "batch-20261003-r02-polynomial-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "38d2168b3429852b17657a5468e53858955cbe6a5e7cd8516da176eb2ce8f518", "family_id": "qf-polynomial-addition", "seed": 202610032096, "semantic_template": "qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\",\"1\"]}" }, "fields": { "title": "Coefficientwise Polynomial Addition", "stem": "In Q[x], coefficient vectors are ordered from lowest to highest power. Add [8, 1] and [-8, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].", "answer": { "kind": "expression", "value": "[0, 2]", "explanation": "Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]." }, "skills": [ "Coefficientwise Polynomial Addition" ], "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": "本題 Q[x] 的整數係數向量由低次至高次排列。求 [8, 1] 與 [-8, 1] 的多項式和，以去尾端零的係數向量回答。", "en": "In Q[x], coefficient vectors are ordered from lowest to highest power. Add [8, 1] and [-8, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0]." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-polynomial-addition, [left, operation, right]]", "after": "derived = [add, no-invented-domain]", "relation": "implies", "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank", "conditions": [], "explanation": { "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。", "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [left ORIGINAL, [8, 1]]", "after": "derived = [[8, 1], [8, 1]]", "relation": "implies", "rule_id": "Polynomial_RAW_ALL_SAFE_coefficients_trim_ONLY_trailing_zero", "conditions": [], "explanation": { "zh-Hant": "Q[x]的係數按低次到高次，raw長0..32、每格先驗SAFE，包括尾端0；只刪trailing zeros，不刪內部0。零多項式內部空向量，顯示[0]，不可改原source向量。", "en": "In Q[x],coefficients run from low to high degree.Type EVERY RAW SAFE cell of a0..32-entry list,including trailing zeros.Trim ONLY trailing zeros,not internal zeros.The zero polynomial has empty internal support but displays[0];keep the ORIGINAL source vector unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [right ORIGINAL, [-8, 1]]", "after": "derived = [[-8, 1], [-8, 1]]", "relation": "implies", "rule_id": "Polynomial_RAW_ALL_SAFE_coefficients_trim_ONLY_trailing_zero", "conditions": [], "explanation": { "zh-Hant": "Q[x]的係數按低次到高次，raw長0..32、每格先驗SAFE，包括尾端0；只刪trailing zeros，不刪內部0。零多項式內部空向量，顯示[0]，不可改原source向量。", "en": "In Q[x],coefficients run from low to high degree.Type EVERY RAW SAFE cell of a0..32-entry list,including trailing zeros.Trim ONLY trailing zeros,not internal zeros.The zero polynomial has empty internal support but displays[0];keep the ORIGINAL source vector unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Polynomial ORIGINAL matching-power coefficient0 := sum(8, -8)", "after": "Polynomial ORIGINAL matching-power coefficient0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL matching-power coefficient0，不以小數近似替代。", "en": "Recompute Polynomial ORIGINAL matching-power coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "8", "-8" ], "output": "0", "label": "Polynomial ORIGINAL matching-power coefficient0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Polynomial ORIGINAL matching-power coefficient1 := sum(1, 1)", "after": "Polynomial ORIGINAL matching-power coefficient1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL matching-power coefficient1，不以小數近似替代。", "en": "Recompute Polynomial ORIGINAL matching-power coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "Polynomial ORIGINAL matching-power coefficient1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[8, 1], [-8, 1]]", "after": "derived = [[0, 2], [0, 2]]", "relation": "implies", "rule_id": "Polynomial_ADD_EVERY_matching_power_THEN_trim", "conditions": [], "explanation": { "zh-Hant": "逐相同冪相加，缺係數才補0，合併後再刪尾0；全抵消要顯示[0]。由全部係數得到多項式和，不以單點值代替恆等式；衍生和可超RAW SAFE。", "en": "Add EVERY matching power,using0 only outside support,THEN trim trailing zeros.Full cancellation displays[0].ALL coefficients determine the polynomial sum,not one sampled value.Derived coefficient sums may exceed RAW SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "[0, 2]" }, "conclusion": { "canonical_value": "[0, 2]", "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-r02-polynomial-foundations", "batch_path": "public/data/questions/batches/batch-20261003-r02-polynomial-foundations/questions.jsonl", "batch_sha256": "2294545062c111a50b54219e05944a8e143ded85eb92974a7ac019620d363675", "original_line": 96, "original_line_sha256": "a897cdc8ef09d56263db9972584850a4d7f71a1e03d8ce0b5da5a90fc734994c", "question_compact_sha256": "38d2168b3429852b17657a5468e53858955cbe6a5e7cd8516da176eb2ce8f518", "english_row_compact_sha256": "8bd3e821669d89e89299505d6986f270326034b6a9ca9337bab89e6f3e92f16e", "english_manifest_path": "public/data/i18n/editions/batch-20261003-r02-polynomial-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261003-r02-polynomial-foundations.en.jsonl" }, "mko_bindings": { "mko-polynomial": { "sha256": "0a0d76b12363d10d9ecbe8b45d7bebb1136637db1c41e6308481c675c8db9e6c", "english_sha256": "4c5e7d9539a5612d401321e99d2e0f4667c558a65abb5f412bf6cb3518f9fbc8", "original_path": "public/data/mko/polynomial.json", "english_path": "public/data/i18n/editions/mko-polynomial.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de", "original_candidate_sha256": "b92437bcb6cf612c08a8f9b2b186f92e0d9dad21ebbf9fdeb79e1b8b4641922f", "original_candidate_line_sha256": "b92437bcb6cf612c08a8f9b2b186f92e0d9dad21ebbf9fdeb79e1b8b4641922f" }, "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-20261026-012-p68", "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c", "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57" }, "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": "576c0b1b5de8da1564d56c44caa30bc6350c63493406fce85b415fe416788405", "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." } }