{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0001","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bbdddba28ebb5c2a91118020ceb2cd8d1e25e7474ac38060f469a4c8130089d","family_id":"qf-mod29-poly-normalize","seed":202610292001,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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. Original coefficients=[1, 3, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 3, 0, 0], 0, [1, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0], 0, [1, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0002","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"085563d2dfdaa54a669d9237098b28bb26f1eb72c5de051a152380a831260081","family_id":"qf-mod29-poly-normalize","seed":202610292002,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"coefficients\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] 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. Original coefficients=[3, 0, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 0, 0, 0], 0, [3], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 0, 0], 0, [3], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0003","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b567e431aaac0094f9ddea7dab2c4faf6341add7606abacc7c33c28a8dd40728","family_id":"qf-mod29-poly-normalize","seed":202610292003,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"coefficients\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 2, 0, 0], 0, [3, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0004","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6419a625493a345e55bacdd39230c39ae8c8f8b592562b74a6d80e3162de4ee8","family_id":"qf-mod29-poly-normalize","seed":202610292004,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"coefficients\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] 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. Original coefficients=[0, 3, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([0, 3, 0, 0], 0, [0, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 0, 0], 0, [0, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0005","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"986ff9d4228a5cd1ee0a80a08d4517cf9654db3bb42f946233e75ec2be82bdb6","family_id":"qf-mod29-poly-normalize","seed":202610292005,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"coefficients\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 2, 0, 0], 0, [3, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0006","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8380ab31d5dc9aeb8575154f2036e88f1cfcae975d92e65aa83303b9229c2b7","family_id":"qf-mod29-poly-normalize","seed":202610292006,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 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. Original coefficients=[2, 0, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 0, 0, 0], 0, [2], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 0, 0], 0, [2], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0007","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67b8dc830ec3cca4338e58165fc74c6b6aa3deee7ebf16bed08cb963ca637653","family_id":"qf-mod29-poly-normalize","seed":202610292007,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"coefficients\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] 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. Original coefficients=[3, 0, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 0, 1, 1], 1, [3, 0], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 1, 1], 1, [3, 0], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0008","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc1704d13299eda0945e50801ef0ed309d8684b5fe395e663a1e8d7301efe40","family_id":"qf-mod29-poly-normalize","seed":202610292008,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"coefficients\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] 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. Original coefficients=[1, 3, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 3, 1, 1], 1, [1, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 1, 1], 1, [1, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0009","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f056bb06fab1a1c893766655db97bc4b19f2645c190a0a4ddbd9c848e4c12cf3","family_id":"qf-mod29-poly-normalize","seed":202610292009,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"coefficients\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 3, 1, 1], 1, [2, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0010","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df24d386879eda75b9bd4ac5b750303dd89e2ff70d1c22930999dc96073ab2b7","family_id":"qf-mod29-poly-normalize","seed":202610292010,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"coefficients\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] 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. Original coefficients=[3, 1, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 1, 1, 1], 1, [3], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 1, 1], 1, [3], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0011","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"146731f5b734f2b2e3e322e648212166255724dfeb0587e0ff48169e2799c734","family_id":"qf-mod29-poly-normalize","seed":202610292011,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"coefficients\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 3, 1, 1], 1, [2, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0012","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"609e01adbb739cb9616ecb05f9977708c01f8720ce9a6af98ea57decea45bda5","family_id":"qf-mod29-poly-normalize","seed":202610292012,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"coefficients\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 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. Original coefficients=[1, 2, 1, 1]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 2, 1, 1], 1, [1, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 1, 1], 1, [1, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0013","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62408b6a52ff978a012bf8640304d98420882260896c64cef78d046459c5cc9b","family_id":"qf-mod29-poly-normalize","seed":202610292013,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"coefficients\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] 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. Original coefficients=[0, 3, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([0, 3, 2, 2], 2, [0, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 2, 2], 2, [0, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0014","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21e904925dce607e3a78dfca1e65d700f501c0a5c8ed8aaf5c70e71e1c9efa1","family_id":"qf-mod29-poly-normalize","seed":202610292014,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"coefficients\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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. Original coefficients=[3, 2, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 2, 2, 2], 2, [3], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 2, 2], 2, [3], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0015","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1a3acb876c445fd81a453dd609ee7f26be3b46077522f9edd52da80ace1b269","family_id":"qf-mod29-poly-normalize","seed":202610292015,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"coefficients\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] 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. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 1, 2, 2], 2, [3, 1], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0016","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3db16908e6f69d1484047b902458cb9ff8a9cb39e61f0d6b80b51c581f47b2d6","family_id":"qf-mod29-poly-normalize","seed":202610292016,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"coefficients\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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. Original coefficients=[2, 3, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 3, 2, 2], 2, [2, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 2, 2], 2, [2, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0017","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aac944d64ac9d094c428484a6bbbf0dc1aaf8718becac9643f5d1a1b846355e","family_id":"qf-mod29-poly-normalize","seed":202610292017,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"coefficients\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 1, 2, 2], 2, [3, 1], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0018","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd730c6544de6790b9a73cf52fa7f53770beee5864e1baca063fde4c12d05ad3","family_id":"qf-mod29-poly-normalize","seed":202610292018,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"coefficients\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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. Original coefficients=[1, 2, 2, 2]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 2, 2, 2], 2, [1], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 2, 2], 2, [1], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0019","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b95274e19532ae3351c2cdce6a4751dd39694f3adaf81f04e9a977558b7127f2","family_id":"qf-mod29-poly-normalize","seed":202610292019,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"coefficients\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","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=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 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. Original coefficients=[2, 0, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 0, 3, 3], 3, [2, 0], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 3, 3], 3, [2, 0], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0020","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"968840be0e22125cf4d44cec3cd25d873e2a2a3fe0f4f4220cc5ff6cb52e38b8","family_id":"qf-mod29-poly-normalize","seed":202610292020,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"coefficients\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 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. Original coefficients=[3, 2, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 2, 3, 3], 3, [3, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 3, 3], 3, [3, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0021","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1142f1453b84924dfe3fd3557ada609dbb2f193f98359e49668be8674d48c054","family_id":"qf-mod29-poly-normalize","seed":202610292021,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"coefficients\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 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. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 2, 3, 3], 3, [1, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0022","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0248fa98773af69f427e9fb64ddd70d58a73d245fdf08327eab4c3d8dfc2d6c8","family_id":"qf-mod29-poly-normalize","seed":202610292022,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"coefficients\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 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. Original coefficients=[2, 3, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([2, 3, 3, 3], 3, [2], 0)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 3, 3], 3, [2], 0)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0023","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536a72b593e88ce50022e8d0432a417fdd6e6c10f196a5f7ad9bb8adb1030c45","family_id":"qf-mod29-poly-normalize","seed":202610292023,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"coefficients\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 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. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 2, 3, 3], 3, [1, 2], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0024","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b6a5563304d0506a8f64102cc577f7bbe5b692a30502796c1075be0928ce2f7","family_id":"qf-mod29-poly-normalize","seed":202610292024,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"coefficients\":[3,1,3,3],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","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. Original coefficients=[3, 1, 3, 3]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([3, 1, 3, 3], 3, [3, 1], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 3, 3], 3, [3, 1], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0025","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64b58fbca28658491d2e4e480f76f14978d42667b8bd364b3c04d2bee66abdfb","family_id":"qf-mod29-poly-normalize","seed":202610292025,"semantic_template":"qf-mod29-poly-normalize/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"coefficients\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Normalize a Finite-Support Coefficient Array","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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. Original coefficients=[1, 3, 0, 0, 0]. Give (raw,zero,normalized,degree).","answer":{"kind":"expression","value":"([1, 3, 0, 0, 0], 0, [1, 3], 1)","explanation":"Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0, 0], 0, [1, 3], 1)."},"skills":["Normalize a Finite-Support Coefficient Array"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0026","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81853695582271a07d811459ad862582c2ed5635f357379ae5afb32118e4764","family_id":"qf-mod29-poly-add","seed":202610292026,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 3], [2, 1], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0027","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22f7656f185fec1708966b1da7157f62485350aed008d956a8ce0a6b37a63806","family_id":"qf-mod29-poly-add","seed":202610292027,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4,\"right\":[2,3,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] 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, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3], [2, 3], [1, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [2, 3], [1, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0028","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15363488f02544842354bea523c86c91f0460461c38920d2af6f2ea802bd1ac4","family_id":"qf-mod29-poly-add","seed":202610292028,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,3,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 2], [0, 3], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0, 3], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0029","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fab52e1cd219d997cd708ccd03c239472a86a4f4e7a0bc36916e3d74d75032d","family_id":"qf-mod29-poly-add","seed":202610292029,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4,\"right\":[2,0,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] 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=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([0, 3], [2], [2, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [2], [2, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0030","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bf40d04049d4ac74341d617be45c0f232b9524cf1bcca970f2809e2424115f5","family_id":"qf-mod29-poly-add","seed":202610292030,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,3,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 2], [1, 3], [2], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [1, 3], [2], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0031","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b7b7495c7423380ffbf010ba291f0de8db228452f0e81be4779e35624bb479c","family_id":"qf-mod29-poly-add","seed":202610292031,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 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=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2], [1, 2], [3, 2], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [1, 2], [3, 2], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0032","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1a7c807ce0fe4845bf857b196feaa03e7952753b24710ff7e0a383580617707","family_id":"qf-mod29-poly-add","seed":202610292032,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4,\"right\":[1,3,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] 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, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 0], [1, 3], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 0], [1, 3], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0033","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d94025b9c32d08935f595c9f40946d7aa605657ae09a1ac73aed3969f1b227a","family_id":"qf-mod29-poly-add","seed":202610292033,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4,\"right\":[0,1,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] 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=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 3], [0], [0, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [0], [0, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0034","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37515380fa7160ca638f7425494fe17bdd27c5d214b23783ea56f7e7be76d1dd","family_id":"qf-mod29-poly-add","seed":202610292034,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,2,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2, 3], [0, 2], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [0, 2], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0035","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc346d6e952acf4647d0c1fdc2dbc1b20825c188ae83fc5f69e77e5faef51f1","family_id":"qf-mod29-poly-add","seed":202610292035,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4,\"right\":[0,3,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] 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, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3], [0, 3], [2, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [0, 3], [2, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0036","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5240da3a44a42241e1bf19e8ff301c73374ddf49f9000960053054f2d211d97a","family_id":"qf-mod29-poly-add","seed":202610292036,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2, 3], [1, 2], [2], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1, 2], [2], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0037","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d5124c3aeac52937a54e7eaffb40199f43b353071f0509fd7198191913e24d9","family_id":"qf-mod29-poly-add","seed":202610292037,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4,\"right\":[3,1,1,1]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 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=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 2], [3], [3, 2], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3], [3, 2], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0038","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83792cca126081ea4735054a54898e29941245d1786b0d02fc0f72de42c35948","family_id":"qf-mod29-poly-add","seed":202610292038,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4,\"right\":[1,0,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] 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=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([0, 3], [1, 0], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [1, 0], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0039","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d85863c06e983c3f5d138b79666e56e25028971c2e3fed26709255ee802ccf94","family_id":"qf-mod29-poly-add","seed":202610292039,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,3,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3], [1, 3], [0, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [1, 3], [0, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0040","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1a1728a211abd651426657f48eb0ff79b176001581529db343bd4a9d79ac9e9","family_id":"qf-mod29-poly-add","seed":202610292040,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4,\"right\":[2,3,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] 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, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 1], [2, 3], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2, 3], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0041","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"377b36da42da978a3631c150381e60721d93b4e4d626d4c8c8d4fba70302a38f","family_id":"qf-mod29-poly-add","seed":202610292041,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,2,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2, 3], [1], [1, 3], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1], [1, 3], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0042","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"495c152b426d27e426227107b82eac088e05d88d81fbc09763d5b639a0446ab7","family_id":"qf-mod29-poly-add","seed":202610292042,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,3,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 1], [0, 3], [1], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [0, 3], [1], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0043","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f1c56455f9fd13083d14704d94b4d58ac02e217f3060e35c5979594cb9ebd02","family_id":"qf-mod29-poly-add","seed":202610292043,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,1,2,2]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1], [0, 1], [3, 1], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1], [0, 1], [3, 1], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0044","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe601da1f1bac5cd65e29bd54cfc3873a145693177d6db4fa9b8b59dee3940d","family_id":"qf-mod29-poly-add","seed":202610292044,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4,\"right\":[3,2,3,3]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","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=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 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=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2, 0], [3, 2], [2], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 0], [3, 2], [2], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0045","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eb683a51a42fa40c179a234467894f3e128223e3b7309ed9f28b801d19978ce","family_id":"qf-mod29-poly-add","seed":202610292045,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,3,3,3]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 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, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([3, 2], [0], [0, 2], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0], [0, 2], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0046","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be951e08d6078f99a0e62e638facaa1b43b65b9e66cb5e1d138ed4f0f5683761","family_id":"qf-mod29-poly-add","seed":202610292046,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4,\"right\":[0,1,3,3]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 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=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 2], [0, 1], [2], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [0, 1], [2], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0047","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2492f2f087ba47f78c4bb6b54857dbb846fb8008bcd16a83db314a914dd1421a","family_id":"qf-mod29-poly-add","seed":202610292047,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,2,3,3]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 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=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([2], [0, 2], [1, 2], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [0, 2], [1, 2], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0048","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e47d6a5a3a38777cc466fb0961d795b4d7b7415380f351d95687124937dc38","family_id":"qf-mod29-poly-add","seed":202610292048,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[3,1,3,3]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 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=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 2], [3, 1], [1], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3, 1], [1], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0049","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee631fbd069863bd067d0e66909e84dda39240fa4ab22929622738cdad8c7580","family_id":"qf-mod29-poly-add","seed":202610292049,"semantic_template":"qf-mod29-poly-add/{\"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":"Add Polynomial Coefficients with the Parent Ring Table","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,sum,degree).","answer":{"kind":"expression","value":"([3, 1], [2], [2, 1], 1)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2], [2, 1], 1)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0050","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68742dd2f254c819a923fddd96d3f267e51d747f99478b41fd490531a639ebf8","family_id":"qf-mod29-poly-add","seed":202610292050,"semantic_template":"qf-mod29-poly-add/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0,0]}"},"fields":{"title":"Add Polynomial Coefficients with the Parent Ring Table","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).","answer":{"kind":"expression","value":"([1, 3], [2, 1], [3], 0)","explanation":"Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0)."},"skills":["Add Polynomial Coefficients with the Parent Ring Table"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0051","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"004379dcad22d52c6fbca318ecbe0dc2543770fae85014c4171b8c91b00c9039","family_id":"qf-mod29-poly-multiply","seed":202610292051,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0]}"},"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=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 3], [2, 1], [2, 3, 3], 2, 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 ([1, 3], [2, 1], [2, 3, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0052","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820596f5c9431214099ce8c305893e07417147419f464179b5a9286835795a75","family_id":"qf-mod29-poly-multiply","seed":202610292052,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4,\"right\":[2,3,0,0]}"},"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=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] 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, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3], [2, 3], [3], 0, 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], [2, 3], [3], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0053","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"515e59b91fc21516aa51f30a26c703fee14b2dbb6c3555b48ba3a1d79d1305ab","family_id":"qf-mod29-poly-multiply","seed":202610292053,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,3,0,0]}"},"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=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 2], [0, 3], [0, 2, 3], 2, 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, 2], [0, 3], [0, 2, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0054","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0d27e67d261678811d4c33d7fcda9f237a2d744accfb6d22b0b8117bdcfbdf1","family_id":"qf-mod29-poly-multiply","seed":202610292054,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"left\":[0,3,0,0],\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4,\"right\":[2,0,0,0]}"},"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=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] 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=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([0, 3], [2], [0, 3], 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 ([0, 3], [2], [0, 3], 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3ab9da82bec328a8afb1b5a49c314ca6c27cc973a19594ecf4165a423eb36a8","family_id":"qf-mod29-poly-multiply","seed":202610292055,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"left\":[3,2,0,0],\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,3,0,0]}"},"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=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 2], [1, 3], [1, 2, 2], 2, 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, 2], [1, 3], [1, 2, 2], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0056","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"307d8707da9ba1503b6253af4f11c209ba942543bcec82b2fd5f4470f058822a","family_id":"qf-mod29-poly-multiply","seed":202610292056,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[2,0,0,0],\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,0,0]}"},"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=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 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=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2], [1, 2], [2], 0, 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 ([2], [1, 2], [2], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0057","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9a009759f1ace15b932e7eab35d0147f88d9e397df663f314f9b70409844e8f","family_id":"qf-mod29-poly-multiply","seed":202610292057,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[3,0,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4,\"right\":[1,3,1,1]}"},"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=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] 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, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 0], [1, 3], [1, 0, 3], 2, 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, 0], [1, 3], [1, 0, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0058","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceab58b5f483ec6d53e25de5581f881cb41d6a84690d58cce3326fa0b965d7d7","family_id":"qf-mod29-poly-multiply","seed":202610292058,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[1,3,1,1],\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4,\"right\":[0,1,1,1]}"},"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=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] 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=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 3], [0], [1, 3], 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 ([1, 3], [0], [1, 3], 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0059","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c27f1a32fa92732cd38a95287ab0b309fd3f5158b7edfee62a08995baa060db","family_id":"qf-mod29-poly-multiply","seed":202610292059,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4,\"right\":[0,2,1,1]}"},"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, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2, 3], [0, 2], [0, 3, 3], 2, 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 ([2, 3], [0, 2], [0, 3, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0060","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bbb1a5da6d4f62b3187939ce6edc4cf144362a9b4144d08edf30f4e3557c41b","family_id":"qf-mod29-poly-multiply","seed":202610292060,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"left\":[3,1,1,1],\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4,\"right\":[0,3,1,1]}"},"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=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] 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, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3], [0, 3], [3], 0, 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], [0, 3], [3], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0061","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08f5e44a3d40a65c1e494fa7eb04e1720c872ef0380a1cadb9921a959c7c919c","family_id":"qf-mod29-poly-multiply","seed":202610292061,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"left\":[2,3,1,1],\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4,\"right\":[1,2,1,1]}"},"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, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2, 3], [1, 2], [1, 3, 2], 2, 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 ([2, 3], [1, 2], [1, 3, 2], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0062","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc9fb730b78371856b5280825d0b2789c7a03501c23f200819a7db97cf715ca","family_id":"qf-mod29-poly-multiply","seed":202610292062,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"left\":[1,2,1,1],\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4,\"right\":[3,1,1,1]}"},"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=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 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=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 2], [3], [1, 2], 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 ([1, 2], [3], [1, 2], 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0063","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0e22bf3c989b0355c722b818fb524c51331d7d7802be60b2a6eb1f961b3a543","family_id":"qf-mod29-poly-multiply","seed":202610292063,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[0,3,2,2],\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4,\"right\":[1,0,2,2]}"},"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, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] 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=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([0, 3], [1, 0], [1, 3, 3], 2, 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 ([0, 3], [1, 0], [1, 3, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0064","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"130a1f8e70abce923be1e77d2a5ae4f682f2b70e89ac0dbce60a3d6df2cf50c9","family_id":"qf-mod29-poly-multiply","seed":202610292064,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[3,2,2,2],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,3,2,2]}"},"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=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3], [1, 3], [3], 0, 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, 3], [3], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0065","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5372245dc4d64276e464574dd880dffc82783ec53df22913d0d60b9872df4567","family_id":"qf-mod29-poly-multiply","seed":202610292065,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4,\"right\":[2,3,2,2]}"},"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=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] 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, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 1], [2, 3], [2, 1, 3], 2, 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], [2, 1, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0066","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e64abfad77883e7919973553ab9891923aa3ff96e9c339390f7c091f36e112c3","family_id":"qf-mod29-poly-multiply","seed":202610292066,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"left\":[2,3,2,2],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4,\"right\":[1,2,2,2]}"},"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=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2, 3], [1], [2, 3], 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 ([2, 3], [1], [2, 3], 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0067","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81ebab290ad67c051d3653b6e643f7cf79dbee50c82cb58df802716e55b8f74","family_id":"qf-mod29-poly-multiply","seed":202610292067,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"left\":[3,1,2,2],\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,3,2,2]}"},"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=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 1], [0, 3], [0, 1, 1], 2, 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], [0, 3], [0, 1, 1], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0068","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb463a0de523b32098cfa8d0b262d25a113f1b689c7008dda43d39db9b34ff83","family_id":"qf-mod29-poly-multiply","seed":202610292068,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"left\":[1,2,2,2],\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4,\"right\":[0,1,2,2]}"},"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, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1], [0, 1], [1], 0, 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 ([1], [0, 1], [1], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0069","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f59229ff7d84aa9ae0a2233ca5c552ca1ebd336fd2779f54e01b78fd418dfce8","family_id":"qf-mod29-poly-multiply","seed":202610292069,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"left\":[2,0,3,3],\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4,\"right\":[3,2,3,3]}"},"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=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 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=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2, 0], [3, 2], [3, 0, 2], 2, 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 ([2, 0], [3, 2], [3, 0, 2], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0070","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc6ba63214eeb19ae9bf50fac6c00431167b91bf57078a7b351a8972aa04e1b7","family_id":"qf-mod29-poly-multiply","seed":202610292070,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[3,2,3,3],\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,3,3,3]}"},"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=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 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, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([3, 2], [0], [3, 2], 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, 2], [0], [3, 2], 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0071","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b89ce2b174304c5f894f73dc575eaeba576fd6e463eccaff6623f129ba449109","family_id":"qf-mod29-poly-multiply","seed":202610292071,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4,\"right\":[0,1,3,3]}"},"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=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 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=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 2], [0, 1], [0, 2, 2], 2, 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 ([1, 2], [0, 1], [0, 2, 2], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0072","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dafd504e25f1756112801b16487bae5fcc95e8869ecdeb77bbf8bfc191e227f","family_id":"qf-mod29-poly-multiply","seed":202610292072,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"left\":[2,3,3,3],\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4,\"right\":[0,2,3,3]}"},"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=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 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=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([2], [0, 2], [2], 0, 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 ([2], [0, 2], [2], 0, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0073","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e941b5fafb3e1662ba518d39813313b7c116d4861ac3f6991cfd8026cb4c6e","family_id":"qf-mod29-poly-multiply","seed":202610292073,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"left\":[1,2,3,3],\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4,\"right\":[3,1,3,3]}"},"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=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 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=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 2], [3, 1], [3, 2, 1], 2, 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 ([1, 2], [3, 1], [3, 2, 1], 2, 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."}}
{"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0075","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa05fb61b3ae94b033700cc8f11cdca19e47e6668c2239c5a3cfa523bf8527c4","family_id":"qf-mod29-poly-multiply","seed":202610292075,"semantic_template":"qf-mod29-poly-multiply/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"left\":[1,3,0,0,0],\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4,\"right\":[2,1,0,0,0]}"},"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=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).","answer":{"kind":"expression","value":"([1, 3], [2, 1], [2, 3, 3], 2, 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 ([1, 3], [2, 1], [2, 3, 3], 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0076","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83a576e36dc024d0a5bbd2b985a041ca0d081412e11a8798d76285abf3cfaa69","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292076,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":1,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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. Nonnegative exponent=1. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0077","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ae53c040cdc9ecaefcae655aaf5ddfa5e478d6eea4506abbf4da2bb08a30b8d","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292077,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"degree\":2,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,1,3],[0,3,3,0]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] 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. Nonnegative exponent=2. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0078","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"836bf65f46b6e7c29d50e95754b6cc706858eec8c3e87c32c891ce19582c973d","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292078,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"degree\":3,\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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. Nonnegative exponent=3. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0079","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1629a9424fd1a0094cb49407fe57256540a1f3296eaeff609567197ba6871d5f","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292079,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,3,0,2],[2,0,3,1],[3,2,1,0]],\"degree\":4,\"multiplication\":[[0,0,0,0],[0,2,1,3],[0,1,2,3],[0,3,3,0]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] 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. Nonnegative exponent=4. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0080","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e514377c0baa4f8a4622ab9a2d70a17705048c06a70005e3d9b290810a179a5","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292080,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],\"degree\":5,\"multiplication\":[[0,0,0,0],[0,0,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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. Nonnegative exponent=5. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0081","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7ebaa996a1a10a1fe011cb0f3518497055934045771df3503e611c7890f25ca","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292081,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":6,\"multiplication\":[[0,0,0,0],[0,3,2,1],[0,2,0,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 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. Nonnegative exponent=6. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0082","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44a468c942e48855a969f5b7db9d2e729ff9c3c270d6cbe135d8ed92cf91d0d3","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292082,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"degree\":7,\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,1,2],[3,1,2,0]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] 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. Nonnegative exponent=7. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0083","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edb27ca314800bfcf5ec3d46dcece614d520f8feb8064c0c25faef36070843c3","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292083,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"degree\":8,\"multiplication\":[[0,1,2,3],[1,1,1,1],[2,1,0,3],[3,1,3,1]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] 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. Nonnegative exponent=8. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0084","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47fbff3847c988e6af09628c05c8de3202e7948c4a02facfdbd7294a8b568f40","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292084,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"degree\":9,\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,2,3],[0,1,3,2]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] 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. Nonnegative exponent=9. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0085","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8596ce5d9e7d4f83bc792b9be7eaa7b5b5f377a1109b71121ee077f6a09a5a2b","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292085,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],\"degree\":10,\"multiplication\":[[2,1,0,3],[1,1,1,1],[0,1,2,3],[3,1,3,1]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] 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. Nonnegative exponent=10. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0086","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d6d8f1fd388184b1c5c055b0c08a2e62d659929b4c982f6cd8095696427353","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292086,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,0,3,2],[0,1,2,3],[3,2,0,1],[2,3,1,0]],\"degree\":11,\"multiplication\":[[1,1,0,0],[1,1,1,1],[0,1,3,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 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. Nonnegative exponent=11. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0087","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12081d30a95eeb50766d690c6274d8e776293879c8c4c260225c5e5266d25b6","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292087,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,0,3,1],[0,1,2,3],[3,2,1,0],[1,3,0,2]],\"degree\":12,\"multiplication\":[[3,1,2,0],[1,1,1,1],[2,1,1,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 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. Nonnegative exponent=12. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0088","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b13e14b07cdcbacaf6cb76e10509aea2bddec61549854c27bd63da62d5a348e5","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292088,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"degree\":13,\"multiplication\":[[0,1,2,3],[1,2,2,1],[2,2,2,2],[3,1,2,0]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] 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. Nonnegative exponent=13. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0089","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c701f92ad8236dd580d9979249bb7155a43e437d77622c9792ed4b1d3f5dd39a","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292089,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"degree\":14,\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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. Nonnegative exponent=14. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0090","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e573027fa5f1b576d65b052fa8e0913a06c644ce245bcb426e903a755e5079","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292090,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"degree\":15,\"multiplication\":[[2,0,2,0],[0,1,2,3],[2,2,2,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] 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. Nonnegative exponent=15. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0091","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7526157f1c4b20c80c5766f5cec0ca2b5064e3d361f4d047c7c7cfa8c5575d2d","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292091,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,0,1],[2,3,1,0],[0,1,2,3],[1,0,3,2]],\"degree\":16,\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,2,2],[3,3,2,2]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] 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. Nonnegative exponent=16. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0092","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2ef90f85546414ea94345f3b4a5365f8e4610a0aaefa8354a297f863f0fd391","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292092,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],\"degree\":17,\"multiplication\":[[2,0,2,0],[0,3,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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. Nonnegative exponent=17. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0093","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f95e000a201db70a70f7ba9d7406d189c95b82152b84bcdcaab94d2079198374","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292093,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,3,0,2],[3,2,1,0],[0,1,2,3],[2,0,3,1]],\"degree\":18,\"multiplication\":[[3,1,2,0],[1,2,2,1],[2,2,2,2],[0,1,2,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 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. Nonnegative exponent=18. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0094","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2591d74bd0ebad91773ad6744251527bce195e0c489fd2ceff27d2684cd05852","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292094,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":19,\"multiplication\":[[0,1,2,3],[1,3,1,3],[2,1,0,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","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=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 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. Nonnegative exponent=19. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0095","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab54635bda0d6cd1717b7e7ff85a7f7ae4ee0836b90ae836251148d4d14a9083","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292095,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"degree\":20,\"multiplication\":[[0,1,2,3],[1,0,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 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. Nonnegative exponent=20. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0096","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a62835b22733d32864db085235e3596d34aa8cc7308f540f94f2644749b209c","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292096,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"degree\":21,\"multiplication\":[[3,0,0,3],[0,1,2,3],[0,2,1,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 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. Nonnegative exponent=21. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0097","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"594605b0b971f69719d422862780110a8dead4b6f350502a16bb337d33e4e8a1","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292097,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[2,3,1,0],[3,2,0,1],[1,0,3,2],[0,1,2,3]],\"degree\":22,\"multiplication\":[[1,0,2,3],[0,1,2,3],[2,2,3,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 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. Nonnegative exponent=22. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0098","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adfe303efc20d1a10cea86ccd781e2f917918a76b908815c94e7778841ea67b4","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292098,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[3,2,1,0],[2,0,3,1],[1,3,0,2],[0,1,2,3]],\"degree\":23,\"multiplication\":[[3,0,0,3],[0,2,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 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. Nonnegative exponent=23. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac932b9e6e2c07039b0934ab93fea907bdf7cbff9ee5c2d68807fbf6c30de108","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292099,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":24,\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","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. Nonnegative exponent=24. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0100","batch_id":"batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d06fe4d56f72885ace3c5139a1c181149c5939f1dcf775aae599e921807ba5","family_id":"qf-mod29-poly-infinite-carrier-witness","seed":202610292100,"semantic_template":"qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"degree\":25,\"multiplication\":[[0,0,0,0],[0,1,2,3],[0,2,0,2],[0,3,2,1]],\"n\":4}"},"fields":{"title":"Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial","stem":"Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] 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. Nonnegative exponent=25. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).","answer":{"kind":"expression","value":"(25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one)","explanation":"Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one)."},"skills":["Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial"],"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."}}