{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0074",
  "pilot_number": "R0028-14774",
  "number": 14774,
  "family_id": "qf-mod29-poly-multiply",
  "title": {
    "zh-Hant": "第 14774 題：有限交換環上的多項式環",
    "en": "Question 14774: Finite Convolution May Lose Degree over a Ring with Zero Divisors"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0074",
    "batch_id": "batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-polynomial-ring-over-finite-commutative-ring"
    ],
    "title_zh": "有限交換環上的多項式環",
    "stem_zh": "係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[3,1,3,3],\"right\":[2,3,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)",
      "explanation_zh": "Wave4第三層核對=([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限交換環上的多項式環"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-poly-multiply",
      "seed": 202610292074,
      "parameters": {
        "n": 4,
        "addition": [
          [
            1,
            2,
            3,
            0
          ],
          [
            2,
            3,
            0,
            1
          ],
          [
            3,
            0,
            1,
            2
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "multiplication": [
          [
            2,
            1,
            0,
            3
          ],
          [
            1,
            3,
            1,
            3
          ],
          [
            0,
            1,
            2,
            3
          ],
          [
            3,
            3,
            3,
            3
          ]
        ],
        "left": [
          3,
          1,
          3,
          3
        ],
        "right": [
          2,
          3,
          3,
          3
        ],
        "domain": "Finite_support_polynomial_over_finite_commutative_ring",
        "operation": "poly-multiply",
        "semantic_template": "qf-mod29-poly-multiply/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[2,3,3,3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式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-20261029-r02-polynomial-ring-over-finite-commutative-ring-0074",
    "batch_id": "batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5d31b7ff9b7cf70860d5a02223dc3b2fd72af3bbc60f7fddef2759153de6798f",
      "family_id": "qf-mod29-poly-multiply",
      "seed": 202610292074,
      "semantic_template": "qf-mod29-poly-multiply/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[2,3,3,3]}"
    },
    "fields": {
      "title": "Finite Convolution May Lose Degree over a Ring with Zero Divisors",
      "stem": "Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).",
      "answer": {
        "kind": "expression",
        "value": "([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)",
        "explanation": "Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)."
      },
      "skills": [
        "Finite Convolution May Lose Degree over a Ring with Zero Divisors"
      ],
      "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": "係數屬同一完整有限交換環且多項式有限支撐。操作=poly-multiply；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"left\":[3,1,3,3],\"right\":[2,3,3,3]}。請依 [left,right,product,degree,boundary] 欄序回答。",
        "en": "Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same carrier Fin(4);addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]];multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]]",
      "after": "actual additive zero=3;actual multiplicative one=2;additive inverses=[2, 1, 0, 3]",
      "relation": "implies",
      "rule_id": "commutative_parent_actual_tables_and_identities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "所有係數與整環論證綁同一完整父表；單位元是實際元素，不預設標籤0或1。",
        "en": "Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1."
      },
      "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": "all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null]",
      "after": "additive commute=是;multiplicative commute=是;zero differs from one=是",
      "relation": "implies",
      "rule_id": "complete_commutative_unital_parent_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核加法Abelian群、乘法結合與交換、左右分配及0≠1；整環與多項式責任在此合格前提之後，不用有限個正樣本冒充全表資格。",
        "en": "Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes."
      },
      "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": "left polynomial:raw ascending coefficient labels=[3, 1, 3, 3];actual ring zero=3",
      "after": "normalized finite support=[3, 1];last-index degree field=1",
      "relation": "implies",
      "rule_id": "polynomial_trailing_actual_zero_protocol",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只刪最高次端的實際零係數，不刪低次或中間零；空／全零輸入表示成[zero]。此協定degree是最後索引，零多項式回0，不是一般數學的零多項式degree慣例。",
        "en": "Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "right polynomial:raw ascending coefficient labels=[2, 3, 3, 3];actual ring zero=3",
      "after": "normalized finite support=[2];last-index degree field=0",
      "relation": "implies",
      "rule_id": "polynomial_trailing_actual_zero_protocol",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只刪最高次端的實際零係數，不刪低次或中間零；空／全零輸入表示成[zero]。此協定degree是最後索引，零多項式回0，不是一般數學的零多項式degree慣例。",
        "en": "Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention."
      },
      "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": "convolution coefficient0 times coefficient0:operation=multiplication;left label=3;right label=2;original row=[3, 3, 3, 3]",
      "after": "result label=3",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "accumulate product at degree0:operation=addition;left label=3;right label=3;original row=[0, 1, 2, 3]",
      "after": "result label=3",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "convolution coefficient1 times coefficient0:operation=multiplication;left label=1;right label=2;original row=[1, 3, 1, 3]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "accumulate product at degree1:operation=addition;left label=3;right label=1;original row=[0, 1, 2, 3]",
      "after": "result label=1",
      "relation": "implies",
      "rule_id": "same_parent_operation_not_integer_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作與原次序查表；元素標籤不是整數加乘的值，不能用模數猜父環結構。",
        "en": "Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "product result:raw ascending coefficient labels=[3, 1];actual ring zero=3",
      "after": "normalized finite support=[3, 1];last-index degree field=1",
      "relation": "implies",
      "rule_id": "polynomial_trailing_actual_zero_protocol",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只刪最高次端的實際零係數，不刪低次或中間零；空／全零輸入表示成[zero]。此協定degree是最後索引，零多項式回0，不是一般數學的零多項式degree慣例。",
        "en": "Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "normalized product=[3, 1];last-index degree field=1",
      "after": "degree_additivity_not_claimed_with_zero_divisors",
      "relation": "implies",
      "rule_id": "polynomial_convolution_zero_divisor_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐有限支撐作父環卷積，再刪尾零。首項係數可因零因子相乘成零，甚至兩個非零多項式乘成零，故不無條件宣稱degree相加或域性質。",
        "en": "Convolve finite supports using parent-ring operations,then trim trailing zeros.Leading coefficients may annihilate through zero divisors,and two nonzero polynomials may multiply to zero.No unconditional degree-additivity or field property is claimed."
      },
      "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": [
      [
        3,
        1
      ],
      [
        2
      ],
      [
        3,
        1
      ],
      1,
      "degree_additivity_not_claimed_with_zero_divisors"
    ]
  },
  "conclusion": {
    "canonical_value": "([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)",
    "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-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations/questions.jsonl",
      "batch_sha256": "c6361950b4ea53bcde82fdfe52c0207bd7351e746d4b53920f02b60ee5fe3432",
      "original_line": 74,
      "original_line_sha256": "2546085d095952feea9c8cc68ba3d1818027c79b12c66e9b4e848f8a0335a804",
      "question_compact_sha256": "5d31b7ff9b7cf70860d5a02223dc3b2fd72af3bbc60f7fddef2759153de6798f",
      "english_row_compact_sha256": "458e197b2664180fb1d4c345537c28174035e4ebd9c8e9f3a8078ed2ec4a9c4e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-polynomial-ring-over-finite-commutative-ring": {
        "sha256": "738ac2a85874f0a45327552d44abd7bfe8f789812c2f6c29c8f80d592ebb8a67",
        "english_sha256": "6e6e1f5cd46d4615a07facebc94f250a9638d9c5f3145355998233dac87d03bc",
        "original_path": "public/data/mko/polynomial-ring-over-finite-commutative-ring.json",
        "english_path": "public/data/i18n/editions/mko-polynomial-ring-over-finite-commutative-ring.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "466f72389eb454e24322e59df2871ec206c351545a4a7b072d39fb5f6afc2697",
    "original_candidate_sha256": "864112bd23419cb2046c69a02492e013d13e64a6da34cb7290b1490aef7a444a",
    "original_candidate_line_sha256": "864112bd23419cb2046c69a02492e013d13e64a6da34cb7290b1490aef7a444a"
  },
  "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-20261026-007-p29",
    "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": "bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae",
      "question_first_witness_sha256": "61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "65ac161ae72a24da7c4ef00c7d70fd836fffe9911dc0c6565ac813cd53f02553",
      "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."
  }
}