{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r01-finite-unital-ring-0001","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2477a51f994c75e8e89141045a4276f9913a00bcc24a0e9e44627990005dbcb","family_id":"qf-mod27-ring-qualification","seed":202610271001,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,1,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 1, 2, 4], [0, 3, 0, 3, 0, 3], [0, 4, 2, 0, 4, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0002","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3a14bc7a113c48f7f2bebf5ae0dc72387a412d81039ed1145e44d4af3acfc7c","family_id":"qf-mod27-ring-qualification","seed":202610271002,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 0, 5, 2], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 2, 0, 2, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0003","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d35ce8c11b2b4288356cc58e6fbb21d07f7e52f979a0c433a1118a3a814b35","family_id":"qf-mod27-ring-qualification","seed":202610271003,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 0, 3], [0, 3, 2, 3, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0004","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec40de336947ce89050dcb1c5008b0b3a0f8c01e9eecb8aa32c3440d80507c16","family_id":"qf-mod27-ring-qualification","seed":202610271004,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 5, 0, 2], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 2, 2, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0005","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f248d1c68ad19a55edf8e9c7aa2c5c96f2480dd51b722b25df1511cbcc828ff3","family_id":"qf-mod27-ring-qualification","seed":202610271005,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0006","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f3b4145c8138126e45502b407da0da835746f26d98a3be33334f0b421add3a9","family_id":"qf-mod27-ring-qualification","seed":202610271006,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,5,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 5, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0007","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85207c5caa8aa9b68ebac09b007195937b2716f6641cd2f358cdfa393899eeaf","family_id":"qf-mod27-ring-qualification","seed":202610271007,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 4, 3, 4], [0, 4, 0, 3, 4, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0008","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09e71d2c77dbff2f1e0cbd6885ea12bbe806e4024649657fd8a360a0c91cb521","family_id":"qf-mod27-ring-qualification","seed":202610271008,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 5, 5, 3], [0, 4, 2, 5, 1, 3], [0, 5, 0, 3, 3, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0009","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd3fa0138267c2281a51dfd4d723ea4937f140e08c90a2567edb359a845f8596","family_id":"qf-mod27-ring-qualification","seed":202610271009,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 0, 3], [0, 3, 3, 2, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0010","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae7d44ed737d410b9584fa31327468233e11ad96699a0af0a140fedb3024bfed","family_id":"qf-mod27-ring-qualification","seed":202610271010,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 5, 0, 3], [0, 4, 4, 0, 4, 0], [0, 5, 3, 3, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0011","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2556e1fff4a63928bc484fcb6d6199044836a8e368472d5e9617a9644746c6","family_id":"qf-mod27-ring-qualification","seed":202610271011,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 4, 3, 0], [0, 3, 3, 2, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0012","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52b001af80fee42b19169b03b29050796b5f729c462fe5223e9b5a7cf8c7d323","family_id":"qf-mod27-ring-qualification","seed":202610271012,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 4, 3, 0], [0, 4, 3, 3, 4, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0013","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d89577c412bebbc8ac33bc8b41b7df96d693ce23ec13e57ed2621c995a6db3ed","family_id":"qf-mod27-ring-qualification","seed":202610271013,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 3, 4, 4], [0, 4, 0, 4, 3, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0014","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47176cb0cdefc1dca7b6e6b273e2e7e2fb3c60bb8d6b949c5e51fde9d3cde13d","family_id":"qf-mod27-ring-qualification","seed":202610271014,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 5, 4], [0, 5, 0, 4, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0015","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5572c75437bd177a91f32ee34bbd375a78a553acb210d0454ed153506d7fff42","family_id":"qf-mod27-ring-qualification","seed":202610271015,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 4, 4], [0, 3, 0, 3, 0, 3], [0, 4, 4, 0, 2, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0016","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e50e3706b58656e3b0db5edeed7aa143cddfd76d5494c3cdeee3f4c7699b34","family_id":"qf-mod27-ring-qualification","seed":202610271016,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 5, 4], [0, 5, 4, 0, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0017","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7b22e4f6a408352a5069b54ed03a3a8a31dd07fd101411b501413bbd2015fbe","family_id":"qf-mod27-ring-qualification","seed":202610271017,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 4, 4, 0], [0, 3, 4, 1, 2, 5], [0, 4, 4, 2, 2, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0018","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e319a439d51719cb52e0ea02f9f110fac6f00223005518a288997e81a93414","family_id":"qf-mod27-ring-qualification","seed":202610271018,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 3, 4, 0], [0, 4, 3, 4, 3, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0019","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445edbb3b973d502cd11dda62aafaf2a3db9907da81c22aa829079aa6756285b","family_id":"qf-mod27-ring-qualification","seed":202610271019,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 3, 5, 5], [0, 4, 2, 5, 1, 3], [0, 5, 0, 5, 3, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0020","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3c56e29d4e07ef89dd397cfa77b20a25814ea477917229994a2037dc0ea1435","family_id":"qf-mod27-ring-qualification","seed":202610271020,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 4, 5], [0, 5, 0, 4, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0021","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd3f9b910e3d5a88c368b97d5ec0cafd6390336a3d275e8a19eff995bbb9fcd","family_id":"qf-mod27-ring-qualification","seed":202610271021,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,1,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 1, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, No, No, No, Yes, No, multiplication_associativity)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0022","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed94d63dca49cfdf995c667c3dfc9af01ca434aca0a5eda139a573d89879628","family_id":"qf-mod27-ring-qualification","seed":202610271022,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 4, 5], [0, 5, 4, 0, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0023","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e0a99e532e62550c88ac3fe335a7bcafae41a2ce5cd9b3acc2e4df00519334d","family_id":"qf-mod27-ring-qualification","seed":202610271023,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 5, 0, 5], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 5, 2, 0, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0024","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8aae7a27948c6c0b2846cc89e15dab63e12e59d6f5f8dbad63d5e3172801ab24","family_id":"qf-mod27-ring-qualification","seed":202610271024,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 3, 0, 5], [0, 4, 4, 0, 4, 0], [0, 5, 3, 5, 0, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 1, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0025","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8d569d157c42356a11a0f37d6c8d4e524f314c4bbcede399b7ed508aac954ff","family_id":"qf-mod27-ring-qualification","seed":202610271025,"semantic_template":"qf-mod27-ring-qualification/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"},"fields":{"title":"Qualify Both Complete Ring Tables without Assuming Their Identities","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 4, 1, 0, 1, 4], [0, 1, 2, 3, 4, 5], [0, 0, 3, 3, 0, 3], [0, 1, 4, 0, 4, 1], [0, 4, 5, 3, 1, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Give (n,zero,one,additive_abelian?,multiplication_associative?,left_distributive?,right_distributive?,distinct_identities?,qualified?,first_failure). Missing or nonunique identities and no failure are represented by null. Failure priority is additive_structure, multiplication_associativity, left_distributivity, right_distributivity, then distinct_identities.","answer":{"kind":"expression","value":"(6, 0, 2, Yes, Yes, Yes, Yes, Yes, Yes, null)","explanation":"Audit every relevant element pair or triple, retaining the distinct obligation flags even after the first failure. The qualification flag is their conjunction, not a few passing examples. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 0, 2, Yes, Yes, Yes, Yes, Yes, Yes, null)."},"skills":["Qualify Both Complete Ring Tables without Assuming Their Identities"],"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-20261027-r01-finite-unital-ring-0026","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d9326cad47e53dad452bae004ab4041078ef4a05cfed7dba9d8f5577d2884b","family_id":"qf-mod27-ring-additive-inverse","seed":202610271026,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 0, 2, 4], [0, 3, 0, 3, 0, 3], [0, 4, 2, 0, 4, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 5, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0027","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fc3f39917eab3187d3bb1103d44ae53d29141f2b93e93e22ee063239a5c750b","family_id":"qf-mod27-ring-additive-inverse","seed":202610271027,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 0, 5, 2], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 2, 0, 2, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 0, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0028","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6fbff0c1df9f16ed8f466f3be7ff6cd6be6e3588f738be0ccb56a1ce3baeeb4","family_id":"qf-mod27-ring-additive-inverse","seed":202610271028,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 0, 3], [0, 3, 2, 3, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0029","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58986de6d17da8183715b7d24a51f16eccc1cefc4112f9111bff85187f920fa7","family_id":"qf-mod27-ring-additive-inverse","seed":202610271029,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 5, 0, 2], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 2, 2, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(4, 0, 4, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 4, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0030","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6613dd26957b2849ce134cf219d85e037bb629922900f7667028bf8ef488ad8","family_id":"qf-mod27-ring-additive-inverse","seed":202610271030,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(3, 0, 2, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 2, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0031","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b92c9bd25378d0a6ceb8da46fcde2148d89eedc29ba56ef69ff767dcfc03b8ba","family_id":"qf-mod27-ring-additive-inverse","seed":202610271031,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(2, 0, 4, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 4, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0032","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2595a8c57c4d6ce580bf43028f8e92e1cfa12b7312dfdb62022cc9b3517b226","family_id":"qf-mod27-ring-additive-inverse","seed":202610271032,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 4, 3, 4], [0, 4, 0, 3, 4, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 5, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0033","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8816fa713cc7fa65e05fcbca45140bb1a2f5c62599fac01cac48157e833fe919","family_id":"qf-mod27-ring-additive-inverse","seed":202610271033,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 5, 5, 3], [0, 4, 2, 5, 1, 3], [0, 5, 0, 3, 3, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 0, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0034","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39f5c279aeed21f20963e3a10dd58dc47b6c4a09605b1fdb43c82362a314c7a9","family_id":"qf-mod27-ring-additive-inverse","seed":202610271034,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 0, 3], [0, 3, 3, 2, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0035","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddf45786c4bcad077d4489d3a7c412ed53a497042f8ff968aec06ce5a2356429","family_id":"qf-mod27-ring-additive-inverse","seed":202610271035,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 5, 0, 3], [0, 4, 4, 0, 4, 0], [0, 5, 3, 3, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(4, 0, 4, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 4, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0036","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c9e711766dc6dfd468064dcc12cebf4c4fd7dd83e62d8055b1fd5b8847e63de","family_id":"qf-mod27-ring-additive-inverse","seed":202610271036,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 3, 0], [0, 3, 3, 2, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(3, 0, 2, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 2, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0037","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74c90569146aab286e09cc6e937bba09cf123ba8edae46419045d30b0068604f","family_id":"qf-mod27-ring-additive-inverse","seed":202610271037,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 4, 3, 0], [0, 4, 3, 3, 4, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0038","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dea0c90fc37fd2cb02ce90d6476a7b37275924ab64618ce074866a03a696ae2","family_id":"qf-mod27-ring-additive-inverse","seed":202610271038,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 3, 4, 4], [0, 4, 0, 4, 3, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(1, 0, 5, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 5, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0039","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91b118d4eeff8e3bf0e3829b26f058a0b59057eb95f65b91c80f5217dcb8aeb0","family_id":"qf-mod27-ring-additive-inverse","seed":202610271039,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 5, 4], [0, 5, 0, 4, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 0, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0040","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02fa792788f5ad668651983bf0b456c58dc6e3a9eedc6ecd0c38dfbda7b49475","family_id":"qf-mod27-ring-additive-inverse","seed":202610271040,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 4, 4], [0, 3, 0, 3, 0, 3], [0, 4, 4, 0, 2, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(5, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0041","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8540862eaa7247205f0d79752472313757ce813ab14c792335f47f7d59170a82","family_id":"qf-mod27-ring-additive-inverse","seed":202610271041,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 5, 4], [0, 5, 4, 0, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(4, 0, 5, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 5, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0042","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c77c9c6a1537da6793368f11dc33ea12c13e6003e08e67046f67ce4e2e19043b","family_id":"qf-mod27-ring-additive-inverse","seed":202610271042,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 4, 4, 0], [0, 3, 4, 1, 2, 5], [0, 4, 4, 2, 2, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(3, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0043","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f6986741cdc967a3057c7986bb7aed38e71a6ceade2fe86c70263919a17462","family_id":"qf-mod27-ring-additive-inverse","seed":202610271043,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 3, 4, 0], [0, 4, 3, 4, 3, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0044","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba127007cf8adcccaba4a831ae0d50b234b76ad9d4b0f63f8577382efe81cb0f","family_id":"qf-mod27-ring-additive-inverse","seed":202610271044,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 3, 5, 5], [0, 4, 2, 5, 1, 3], [0, 5, 0, 5, 3, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(1, 0, 4, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 4, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0045","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e1347f18d1e849674a90664bc63988a030b9fe6470f646498dded1b338f7309","family_id":"qf-mod27-ring-additive-inverse","seed":202610271045,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 4, 5], [0, 5, 0, 4, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(0, 0, 0, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, 0, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0046","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86582022d1acb37e310b68f04581a6995b7fa54e85ffaf1b43456d165b46d4c5","family_id":"qf-mod27-ring-additive-inverse","seed":202610271046,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(5, 0, 2, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 0, 2, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0047","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b345799e6d1576104525936a49266c9d3136d2557db3380717d8db4f84bfdd78","family_id":"qf-mod27-ring-additive-inverse","seed":202610271047,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 4, 5], [0, 5, 4, 0, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(4, 0, 5, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 0, 5, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0048","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4390045b269406e7e370daae2198cda2d8ed5ddda17657b50732dfcd7c5561db","family_id":"qf-mod27-ring-additive-inverse","seed":202610271048,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 5, 0, 5], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 5, 2, 0, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(3, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0049","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee28a03fb1edfb2325281f027daee9334bcb6bbc3079cf47f879ecfa57d545b","family_id":"qf-mod27-ring-additive-inverse","seed":202610271049,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 3, 0, 5], [0, 4, 4, 0, 4, 0], [0, 5, 3, 5, 0, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(2, 0, 1, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, 1, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0050","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ab17d3e8d8e065f9d9935dc1c1961e262db764c3e2ec4e5ea534b5f8dac1cdb","family_id":"qf-mod27-ring-additive-inverse","seed":202610271050,"semantic_template":"qf-mod27-ring-additive-inverse/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"element\":1,\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"},"fields":{"title":"Find an Additive Inverse in an Already Qualified Ring","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 4, 1, 0, 1, 4], [0, 1, 2, 3, 4, 5], [0, 0, 3, 3, 0, 3], [0, 1, 4, 0, 4, 1], [0, 4, 5, 3, 1, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1. Give (element,zero,inverse,left_sum,right_sum).","answer":{"kind":"expression","value":"(1, 0, 4, 0, 0)","explanation":"Use the additive table and its actual zero to find the inverse, then check both orders. A multiplicative inverse is not requested or guaranteed. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, 4, 0, 0)."},"skills":["Find an Additive Inverse in an Already Qualified Ring"],"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-20261027-r01-finite-unital-ring-0051","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0219cde40bf851c96263f509f5b5c43714af60cff1aae97056b057b47ba3017","family_id":"qf-mod27-ring-power","seed":202610271051,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"element\":1,\"exponent\":1,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 0, 2, 4], [0, 3, 0, 3, 0, 3], [0, 4, 2, 0, 4, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1, exponent=1. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(1, 1, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0052","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc422f563609150f3b2f2128abc0120bf9b5a3a1421f192f86a17662adb1d597","family_id":"qf-mod27-ring-power","seed":202610271052,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"element\":0,\"exponent\":2,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 0, 5, 2], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 2, 0, 2, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0, exponent=2. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(0, 2, 1, 0)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, 1, 0)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0053","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45c4c118c81db1e148022be592932e31cf4f4f40829e00464da693fd569202c5","family_id":"qf-mod27-ring-power","seed":202610271053,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"element\":5,\"exponent\":3,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 0, 3], [0, 3, 2, 3, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5, exponent=3. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(5, 3, 1, 5)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 3, 1, 5)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0054","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9d5f32c92abf5ef3e1896efbfa4876692cb29f2e1e1764311d555480cb2a7f","family_id":"qf-mod27-ring-power","seed":202610271054,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"element\":4,\"exponent\":4,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 5, 0, 2], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 2, 2, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4, exponent=4. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(4, 4, 1, 4)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 4, 1, 4)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0055","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03c8167ac0bec304d4b02d48815bc7572dd6faed0844618eaa260d0809f98cc9","family_id":"qf-mod27-ring-power","seed":202610271055,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"element\":3,\"exponent\":5,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3, exponent=5. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(3, 5, 1, 3)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 5, 1, 3)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0056","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"661579dca877b55ce0de5546db27a30e5d737f310c32c700016aee3fc5d02105","family_id":"qf-mod27-ring-power","seed":202610271056,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"element\":2,\"exponent\":6,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2, exponent=6. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(2, 6, 1, 4)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 6, 1, 4)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0057","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6326c4e340adf57e5ca6c08957dd6edbe52477b3f1a0fec8a7d4d882baf81eff","family_id":"qf-mod27-ring-power","seed":202610271057,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"element\":1,\"exponent\":7,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 4, 3, 4], [0, 4, 0, 3, 4, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1, exponent=7. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(1, 7, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 7, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0058","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4daf72b07ce0f8d9ceadcabfe50ce4fe8b3ff3c011962756bd7f8d371b714e40","family_id":"qf-mod27-ring-power","seed":202610271058,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"element\":0,\"exponent\":8,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 5, 5, 3], [0, 4, 2, 5, 1, 3], [0, 5, 0, 3, 3, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0, exponent=8. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(0, 8, 1, 0)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 8, 1, 0)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0059","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ad24afbef015dc3e24820e10310b312132947b513da2daa351afc7c172394e9","family_id":"qf-mod27-ring-power","seed":202610271059,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"element\":5,\"exponent\":9,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 0, 3], [0, 3, 3, 2, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5, exponent=9. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(5, 9, 1, 5)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 9, 1, 5)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0060","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"370ecba3e95ac4c31fe6ddce976423bc66ddb8dbf0a73148639c6fb27cdb0c5d","family_id":"qf-mod27-ring-power","seed":202610271060,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"element\":4,\"exponent\":10,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 5, 0, 3], [0, 4, 4, 0, 4, 0], [0, 5, 3, 3, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4, exponent=10. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(4, 10, 1, 4)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 10, 1, 4)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0061","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f75e8e73590e1fe42663c0e70a6d2256eaa30c54c2f0ee4256daf83ad12064a","family_id":"qf-mod27-ring-power","seed":202610271061,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"exponent\":11,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 3, 0], [0, 3, 3, 2, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3, exponent=11. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(3, 11, 1, 3)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 11, 1, 3)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0062","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac8c3bfeec44400e4c1910dff31f520f73c6ea8ef855410717c9e352b81e961","family_id":"qf-mod27-ring-power","seed":202610271062,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"element\":2,\"exponent\":12,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 4, 3, 0], [0, 4, 3, 3, 4, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2, exponent=12. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(2, 12, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 12, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0063","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e113410408318e7488ea9a32b6a33c684a1079aa0860d323ae89f6b9de8f1f","family_id":"qf-mod27-ring-power","seed":202610271063,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"element\":1,\"exponent\":13,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 3, 4, 4], [0, 4, 0, 4, 3, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1, exponent=13. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(1, 13, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 13, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0064","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"159d54797df64cfa15488e01d8d4bb6141bc6bb03f217496533415405dcdf764","family_id":"qf-mod27-ring-power","seed":202610271064,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":0,\"exponent\":14,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 5, 4], [0, 5, 0, 4, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0, exponent=14. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(0, 14, 1, 0)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 14, 1, 0)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0065","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42db1f7b40c78c58e79c576f3c7a5cc0aaf1f5539ee0bdd5410d9094925b0379","family_id":"qf-mod27-ring-power","seed":202610271065,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"element\":5,\"exponent\":15,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 4, 4], [0, 3, 0, 3, 0, 3], [0, 4, 4, 0, 2, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5, exponent=15. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(5, 15, 1, 5)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 15, 1, 5)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0066","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4a3f7e51f063aa65bfad8b4a406fe9c408663ced28aa7ae20bf2e0c80bcc60b","family_id":"qf-mod27-ring-power","seed":202610271066,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":4,\"exponent\":16,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 5, 4], [0, 5, 4, 0, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4, exponent=16. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(4, 16, 1, 5)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 16, 1, 5)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0067","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d28115a5507c87f598d495e64568203794d313047746bf8417d27ecb00ee5620","family_id":"qf-mod27-ring-power","seed":202610271067,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"element\":3,\"exponent\":17,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 4, 4, 0], [0, 3, 4, 1, 2, 5], [0, 4, 4, 2, 2, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3, exponent=17. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(3, 17, 1, 3)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 17, 1, 3)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0068","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"185266f4793768e84140b02b641e27dbad124229ecaa8bfdcb78d2b10513b1ae","family_id":"qf-mod27-ring-power","seed":202610271068,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"element\":2,\"exponent\":18,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 3, 4, 0], [0, 4, 3, 4, 3, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2, exponent=18. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(2, 18, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 18, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0069","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7293e4772f99c5d3f8577d3b97b440fe9cd14562e30cc00887bc0ef1711725d","family_id":"qf-mod27-ring-power","seed":202610271069,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"element\":1,\"exponent\":19,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 3, 5, 5], [0, 4, 2, 5, 1, 3], [0, 5, 0, 5, 3, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1, exponent=19. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(1, 19, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 19, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0070","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97539e9d96c3b98fb72bb58634cbc2506d530539e212f64b9f96fc63795fb5fb","family_id":"qf-mod27-ring-power","seed":202610271070,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"element\":0,\"exponent\":20,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 4, 5], [0, 5, 0, 4, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=0, exponent=20. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(0, 20, 1, 0)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 20, 1, 0)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0071","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3833bc5471eea793bf76774589372d743ae9c678177c6759b955f339290fe547","family_id":"qf-mod27-ring-power","seed":202610271071,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"element\":5,\"exponent\":21,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=5, exponent=21. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(5, 21, 1, 5)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 21, 1, 5)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0072","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbdcf3eaa2a3ccffbb77de61953a50739d9468b1766d58c1b73572b2f70c069e","family_id":"qf-mod27-ring-power","seed":202610271072,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"element\":4,\"exponent\":22,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 4, 5], [0, 5, 4, 0, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=4, exponent=22. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(4, 22, 1, 4)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 22, 1, 4)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0073","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef12fd6632defb6e1bff1650d00f967bb3a8e2fb9b6b7b2737609d021aa7f699","family_id":"qf-mod27-ring-power","seed":202610271073,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"element\":3,\"exponent\":23,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 5, 0, 5], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 5, 2, 0, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=3, exponent=23. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(3, 23, 1, 3)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 23, 1, 3)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0074","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa7e671e7e4e036dcd5f870fdb3a3f10a783499b7a2b50064eea1a5a14856df","family_id":"qf-mod27-ring-power","seed":202610271074,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"element\":2,\"exponent\":24,\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 3, 0, 5], [0, 4, 4, 0, 4, 0], [0, 5, 3, 5, 0, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=2, exponent=24. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(2, 24, 1, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 24, 1, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0075","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34e709619561db0d5c853bb8248403ff7e95dba6d106bb36d54bc77451e6c546","family_id":"qf-mod27-ring-power","seed":202610271075,"semantic_template":"qf-mod27-ring-power/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"element\":1,\"exponent\":25,\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"},"fields":{"title":"A Nonnegative Ring Power Starts at Its Actual One","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 4, 1, 0, 1, 4], [0, 1, 2, 3, 4, 5], [0, 0, 3, 3, 0, 3], [0, 1, 4, 0, 4, 1], [0, 4, 5, 3, 1, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. The tables must qualify first. Selected element=1, exponent=25. Give (element,exponent,one,result).","answer":{"kind":"expression","value":"(1, 25, 2, 1)","explanation":"Start at the actual multiplicative identity and iterate the multiplication table. Exponent zero returns that identity, including when the selected element is the additive zero; this is the declared algebraic power convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 25, 2, 1)."},"skills":["A Nonnegative Ring Power Starts at Its Actual One"],"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-20261027-r01-finite-unital-ring-0076","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f856d0d479d42ea988706977b868b9bb59291427d6acb75af4037f7127fa21d8","family_id":"qf-mod27-ring-identities","seed":202610271076,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,0,2,4],[0,3,0,3,0,3],[0,4,2,0,4,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 0, 2, 4], [0, 3, 0, 3, 0, 3], [0, 4, 2, 0, 4, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0077","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4937294c2248b6bf820e7b412eb301028788e4c1870be0dc3f1539a152ec0ac1","family_id":"qf-mod27-ring-identities","seed":202610271077,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,3,5,0,4],[2,3,5,4,1,0],[3,5,4,0,2,1],[4,0,1,2,5,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,0,5,2],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,2,0,2,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 5, 0, 4], [2, 3, 5, 4, 1, 0], [3, 5, 4, 0, 2, 1], [4, 0, 1, 2, 5, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 0, 5, 2], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 2, 0, 2, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0078","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69040be1a5d35339c164072c32da3f7887b8d2dbea5c7a5d5aedf4117c002d22","family_id":"qf-mod27-ring-identities","seed":202610271078,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,4,5,3,0],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,3,5,1,0,2],[5,0,1,4,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,0,3],[0,3,2,3,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 5, 3, 0], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 3, 5, 1, 0, 2], [5, 0, 1, 4, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 0, 3], [0, 3, 2, 3, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0079","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f42ba3ed3c1e4115371fef23b61f7cdd7f82c49d2e1ae5435a3f43d0e01855","family_id":"qf-mod27-ring-identities","seed":202610271079,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,4,0,5,3],[2,4,5,1,3,0],[3,0,1,5,2,4],[4,5,3,2,0,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,5,5,0,2],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,2,2,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 4, 0, 5, 3], [2, 4, 5, 1, 3, 0], [3, 0, 1, 5, 2, 4], [4, 5, 3, 2, 0, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 5, 5, 0, 2], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 2, 2, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0080","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c4f032e6fdb87aad7ccfb88fbef64d4e96aeab0616a9bf9ee150529347391ef","family_id":"qf-mod27-ring-identities","seed":202610271080,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0081","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e623662db7b79e39f44bd3806b77acc9038640da96e0d92f6a7830bf6c563601","family_id":"qf-mod27-ring-identities","seed":202610271081,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,0,3,4],[2,5,4,1,0,3],[3,0,1,4,5,2],[4,3,0,5,2,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,4,4,2,0],[0,3,4,1,2,5],[0,4,2,2,4,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0082","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3549a21dbb9776080ce59d84742be2c743dd7f969b5445b6f4dfd5ef214d2c05","family_id":"qf-mod27-ring-identities","seed":202610271082,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,4,2,5,0],[2,4,0,5,1,3],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,0,3,1,2,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,4,3,4],[0,4,0,3,4,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 2, 5, 0], [2, 4, 0, 5, 1, 3], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 0, 3, 1, 2, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 4, 3, 4], [0, 4, 0, 3, 4, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0083","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffb20f6eb8b6718f5e16315a997510e8296d16828de222873caaa725272b9359","family_id":"qf-mod27-ring-identities","seed":202610271083,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,5,2,0,4],[2,5,0,4,3,1],[3,2,4,5,1,0],[4,0,3,1,5,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,5,5,3],[0,4,2,5,1,3],[0,5,0,3,3,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 5, 5, 3], [0, 4, 2, 5, 1, 3], [0, 5, 0, 3, 3, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0084","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ad2ac0cd40b268274454a9ce95b8ce20f1d27427c26a636ca8d003806b962e6","family_id":"qf-mod27-ring-identities","seed":202610271084,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,5,4,2,0],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,2,1,5,0,3],[5,0,4,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,0,3],[0,3,3,2,0,2],[0,4,0,0,4,4],[0,5,3,2,4,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 4, 2, 0], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 0, 4, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 0, 3], [0, 3, 3, 2, 0, 2], [0, 4, 0, 0, 4, 4], [0, 5, 3, 2, 4, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0085","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef19aa2b0a0718d822c19ac4d29875dbfae60d686a339aa4754ea75336a7258","family_id":"qf-mod27-ring-identities","seed":202610271085,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,0,4,5,2],[2,0,5,1,3,4],[3,4,1,5,2,0],[4,5,3,2,0,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,5,0,3],[0,4,4,0,4,0],[0,5,3,3,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 4, 5, 2], [2, 0, 5, 1, 3, 4], [3, 4, 1, 5, 2, 0], [4, 5, 3, 2, 0, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 5, 0, 3], [0, 4, 4, 0, 4, 0], [0, 5, 3, 3, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0086","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84d014e9d6008942040a44e81f38e09a2b25406c309a89c0054972e36c6bb7d9","family_id":"qf-mod27-ring-identities","seed":202610271086,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,4,5,0,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,0,5,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,3,3,0],[0,3,3,2,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 4, 5, 0, 2], [2, 4, 3, 0, 5, 1], [3, 5, 0, 2, 1, 4], [4, 0, 5, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 3, 3, 0], [0, 3, 3, 2, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0087","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1615a92502c18b31b8f0918a0101c8892aeddb00694b1365d32dd31cfff0c564","family_id":"qf-mod27-ring-identities","seed":202610271087,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,3,0,5,2,4],[2,0,4,1,5,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,4,3,2,1,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,4,3,0],[0,4,3,3,4,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 3, 0, 5, 2, 4], [2, 0, 4, 1, 5, 3], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 4, 3, 0], [0, 4, 3, 3, 4, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0088","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e89d17153fc38bc64755493fa1627bb889128e7f985aa22db5b7038e5d832a5","family_id":"qf-mod27-ring-identities","seed":202610271088,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,2,0],[2,3,0,1,5,4],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,0,4,2,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,0,2],[0,3,0,3,4,4],[0,4,0,4,3,3],[0,5,2,4,3,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 2, 0], [2, 3, 0, 1, 5, 4], [3, 5, 1, 4, 0, 2], [4, 2, 5, 0, 3, 1], [5, 0, 4, 2, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 0, 2], [0, 3, 0, 3, 4, 4], [0, 4, 0, 4, 3, 3], [0, 5, 2, 4, 3, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0089","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b75adefb936ab7160f77775fa6f4ea1917a9c4a76dd33c6a38d9f9b041ce55e","family_id":"qf-mod27-ring-identities","seed":202610271089,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,5,0,2,3],[2,5,0,4,3,1],[3,0,4,5,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,5,4],[0,5,0,4,4,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 0, 2, 3], [2, 5, 0, 4, 3, 1], [3, 0, 4, 5, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 5, 4], [0, 5, 0, 4, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0090","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca2e189467a8dd9125a2aaf4069c99be99679872a05a65d01ef8a2e7c3522be","family_id":"qf-mod27-ring-identities","seed":202610271090,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,5,2,3,0],[2,5,4,1,0,3],[3,2,1,0,5,4],[4,3,0,5,2,1],[5,0,3,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,4,4],[0,3,0,3,0,3],[0,4,4,0,2,2],[0,5,4,3,2,1]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 4, 4], [0, 3, 0, 3, 0, 3], [0, 4, 4, 0, 2, 2], [0, 5, 4, 3, 2, 1]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0091","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0dbfe592036928357b5f9776511439b336f35bd654dfb04f11b83b65bb2cf2","family_id":"qf-mod27-ring-identities","seed":202610271091,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,0,5,3,2],[2,0,5,4,1,3],[3,5,4,0,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,5,4],[0,5,4,0,4,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 5, 3, 2], [2, 0, 5, 4, 1, 3], [3, 5, 4, 0, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 5, 4], [0, 5, 4, 0, 4, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0092","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11e0adbc34d3422b6f1e5f2bef16ddc79f66418caff4e020dc4f8591da3cb7f2","family_id":"qf-mod27-ring-identities","seed":202610271092,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,0,5,2],[2,3,4,5,0,1],[3,0,5,2,1,4],[4,5,0,1,2,3],[5,2,1,4,3,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,4,4,0],[0,3,4,1,2,5],[0,4,4,2,2,0],[0,5,0,5,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 0, 5, 2], [2, 3, 4, 5, 0, 1], [3, 0, 5, 2, 1, 4], [4, 5, 0, 1, 2, 3], [5, 2, 1, 4, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 4, 4, 0], [0, 3, 4, 1, 2, 5], [0, 4, 4, 2, 2, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0093","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81a0453d7d1752807383af0c3f41aff5524f727deb9d022b5fc5b48e16ac54e","family_id":"qf-mod27-ring-identities","seed":202610271093,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,0,2,5,3],[2,0,3,5,1,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,4,3,5],[0,3,4,3,4,0],[0,4,3,4,3,0],[0,5,5,0,0,5]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 0, 2, 5, 3], [2, 0, 3, 5, 1, 4], [3, 2, 5, 4, 0, 1], [4, 5, 1, 0, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 4, 3, 5], [0, 3, 4, 3, 4, 0], [0, 4, 3, 4, 3, 0], [0, 5, 5, 0, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0094","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e6af62ad733d41ed8bb97aeacce59b1a8428416dab2312965366866746aaf44","family_id":"qf-mod27-ring-identities","seed":202610271094,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,3,4,0,2],[2,3,0,1,5,4],[3,4,1,5,2,0],[4,0,5,2,3,1],[5,2,4,0,1,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,2,0],[0,3,0,3,5,5],[0,4,2,5,1,3],[0,5,0,5,3,3]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 4, 0, 2], [2, 3, 0, 1, 5, 4], [3, 4, 1, 5, 2, 0], [4, 0, 5, 2, 3, 1], [5, 2, 4, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 2, 0], [0, 3, 0, 3, 5, 5], [0, 4, 2, 5, 1, 3], [0, 5, 0, 5, 3, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0095","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6412c5e77b015a1e830aa6a4eb6ee488397a4c6db66167edb82f5e62bdedb427","family_id":"qf-mod27-ring-identities","seed":202610271095,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,4,0,3,2],[2,4,0,5,1,3],[3,0,5,4,2,1],[4,3,1,2,5,0],[5,2,3,1,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,2,0,0],[0,3,2,1,5,4],[0,4,0,5,4,5],[0,5,0,4,5,4]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 0, 3, 2], [2, 4, 0, 5, 1, 3], [3, 0, 5, 4, 2, 1], [4, 3, 1, 2, 5, 0], [5, 2, 3, 1, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 2, 0, 0], [0, 3, 2, 1, 5, 4], [0, 4, 0, 5, 4, 5], [0, 5, 0, 4, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0096","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a3214905a40d543d89ffa79190c8355ae82c215e02862d80e4b43bd974858f","family_id":"qf-mod27-ring-identities","seed":202610271096,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,4,2,0,3],[2,4,5,1,3,0],[3,2,1,0,5,4],[4,0,3,5,2,1],[5,3,0,4,1,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,0,5,5],[0,3,0,3,3,0],[0,4,5,3,1,2],[0,5,5,0,2,2]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0097","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7f99b3032306e3cd25c02c99c4ad7e7543ad60c075f59c2e62ac907a0ceec9","family_id":"qf-mod27-ring-identities","seed":202610271097,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,0,4,2,3],[2,0,4,5,3,1],[3,4,5,0,1,2],[4,2,3,1,5,0],[5,3,1,2,0,4]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,3,5,4],[0,3,3,3,0,0],[0,4,5,0,4,5],[0,5,4,0,5,4]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 4, 2, 3], [2, 0, 4, 5, 3, 1], [3, 4, 5, 0, 1, 2], [4, 2, 3, 1, 5, 0], [5, 3, 1, 2, 0, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 3, 5, 4], [0, 3, 3, 3, 0, 0], [0, 4, 5, 0, 4, 5], [0, 5, 4, 0, 5, 4]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0098","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b601110fa115cd665b39f708abd9e0501f62a19d7261a6d33cab3405b90c741","family_id":"qf-mod27-ring-identities","seed":202610271098,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,3,0,2,4],[2,3,5,4,1,0],[3,0,4,2,5,1],[4,2,1,5,0,3],[5,4,0,1,3,2]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,2,5,0,5],[0,3,5,1,4,2],[0,4,0,4,4,0],[0,5,5,2,0,2]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 3, 0, 2, 4], [2, 3, 5, 4, 1, 0], [3, 0, 4, 2, 5, 1], [4, 2, 1, 5, 0, 3], [5, 4, 0, 1, 3, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 5, 0, 5], [0, 3, 5, 1, 4, 2], [0, 4, 0, 4, 4, 0], [0, 5, 5, 2, 0, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0099","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8366f8c7b5d5adb776dc26db358fa530015d855af4abeae8b6dd24a98ac85fe1","family_id":"qf-mod27-ring-identities","seed":202610271099,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,5,0,2,3,4],[2,0,3,4,5,1],[3,2,4,5,1,0],[4,3,5,1,0,2],[5,4,1,0,2,3]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,1,5,4,3],[0,3,5,3,0,5],[0,4,4,0,4,0],[0,5,3,5,0,3]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 0, 2, 3, 4], [2, 0, 3, 4, 5, 1], [3, 2, 4, 5, 1, 0], [4, 3, 5, 1, 0, 2], [5, 4, 1, 0, 2, 3]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 1, 5, 4, 3], [0, 3, 5, 3, 0, 5], [0, 4, 4, 0, 4, 0], [0, 5, 3, 5, 0, 3]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 1, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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-20261027-r01-finite-unital-ring-0100","batch_id":"batch-20261014-advance-20261027-r01-finite-unital-ring-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d24cd28ce25fef4385fb9e64abaeb79c562106e42993212165ee47c98abf989","family_id":"qf-mod27-ring-identities","seed":202610271100,"semantic_template":"qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,4,3,5,0,2],[2,3,1,4,5,0],[3,5,4,0,2,1],[4,0,5,2,1,3],[5,2,0,1,3,4]],\"multiplication\":[[0,0,0,0,0,0],[0,4,1,0,1,4],[0,1,2,3,4,5],[0,0,3,3,0,3],[0,1,4,0,4,1],[0,4,5,3,1,2]],\"n\":6}"},"fields":{"title":"Keep Additive Zero and Multiplicative One Distinct","stem":"Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 4, 3, 5, 0, 2], [2, 3, 1, 4, 5, 0], [3, 5, 4, 0, 2, 1], [4, 0, 5, 2, 1, 3], [5, 2, 0, 1, 3, 4]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 4, 1, 0, 1, 4], [0, 1, 2, 3, 4, 5], [0, 0, 3, 3, 0, 3], [0, 1, 4, 0, 4, 1], [0, 4, 5, 3, 1, 2]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).","answer":{"kind":"expression","value":"(0, 2, Yes)","explanation":"Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, Yes)."},"skills":["Keep Additive Zero and Multiplicative One Distinct"],"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."}}