{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r01-mathematical-induction-0001","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c10aef4447762f37288f3cdf9a156a15ecc6c53501930f9695a6e3917c3c95e","family_id":"qf-density12-induction-successor-index-pair","seed":202610121001,"semantic_template":"qf-density12-induction-successor-index-pair/0"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=0. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(0, 1)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (0, 1)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0002","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832083e4b50ef066b74dafa220d3ecb7d5d015f3bbc4ed440d8ac3f1addd73cd","family_id":"qf-density12-induction-successor-index-pair","seed":202610121002,"semantic_template":"qf-density12-induction-successor-index-pair/1"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=1. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(1, 2)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (1, 2)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0003","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"598d441269cbcf2799a06c4699268abc08ecfc4bad30189bab88b43b06bba65c","family_id":"qf-density12-induction-successor-index-pair","seed":202610121003,"semantic_template":"qf-density12-induction-successor-index-pair/2"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=2. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(2, 3)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (2, 3)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0004","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bef03e0a0beb353e2e3fe0d941d53369956ddd55acccff18118f20b257b6c4c","family_id":"qf-density12-induction-successor-index-pair","seed":202610121004,"semantic_template":"qf-density12-induction-successor-index-pair/3"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=3. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(3, 4)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (3, 4)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0005","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36c9f816a0104d276144178e86ab2cb5117ce24718e6f8bc2cc224a1ee19ce20","family_id":"qf-density12-induction-successor-index-pair","seed":202610121005,"semantic_template":"qf-density12-induction-successor-index-pair/4"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=4. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(4, 5)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (4, 5)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0006","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2e15076ba7fa6fc1367595b8e7368b74a1e53290b5991f5bd47f2fdb9069543","family_id":"qf-density12-induction-successor-index-pair","seed":202610121006,"semantic_template":"qf-density12-induction-successor-index-pair/5"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=5. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(5, 6)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (5, 6)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0007","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a075284a32d0fcbca6fc25ab35c7b36336c295d7b75d70b8eea5458840a9923f","family_id":"qf-density12-induction-successor-index-pair","seed":202610121007,"semantic_template":"qf-density12-induction-successor-index-pair/6"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=6. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(6, 7)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (6, 7)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0008","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cbf9f3b337d4eee2d61acdbf13a7c87c4a9d5c48b5d874cb1a0e9b968a07ba4","family_id":"qf-density12-induction-successor-index-pair","seed":202610121008,"semantic_template":"qf-density12-induction-successor-index-pair/7"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=7. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(7, 8)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (7, 8)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0009","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810cee1832e6960c48694f4fb7c31300f51ab03f5586768c56819e64d8801004","family_id":"qf-density12-induction-successor-index-pair","seed":202610121009,"semantic_template":"qf-density12-induction-successor-index-pair/8"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=8. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(8, 9)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (8, 9)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0010","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"485d81aedf5fa5cd825a28bda645b08ca6ff081f762acf87980b7a867676b275","family_id":"qf-density12-induction-successor-index-pair","seed":202610121010,"semantic_template":"qf-density12-induction-successor-index-pair/9"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=9. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(9, 10)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (9, 10)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0011","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9ce00daf1e4218484c2adc30c88aadf61137169b83590d76166a26a880f4b55","family_id":"qf-density12-induction-successor-index-pair","seed":202610121011,"semantic_template":"qf-density12-induction-successor-index-pair/10"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=10. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(10, 11)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (10, 11)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0012","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d6cafcbfa321bda4af923bbbbcef541bae5640c2e75d512b791895b612fcafd","family_id":"qf-density12-induction-successor-index-pair","seed":202610121012,"semantic_template":"qf-density12-induction-successor-index-pair/11"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=11. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(11, 12)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (11, 12)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0013","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a40839fca1280fa543947036e790388367d516fe3dce568fd7eb577cc73f974","family_id":"qf-density12-induction-successor-index-pair","seed":202610121013,"semantic_template":"qf-density12-induction-successor-index-pair/12"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=12. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(12, 13)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (12, 13)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0014","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5db028aa0ec92aaf25f2692acea9e396e2fe75437f926477594159991e992879","family_id":"qf-density12-induction-successor-index-pair","seed":202610121014,"semantic_template":"qf-density12-induction-successor-index-pair/13"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=13. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(13, 14)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (13, 14)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0015","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e98419e5d4a6015f24d15351994ec90785edad5f04de5a12ad4d4f1d7b672000","family_id":"qf-density12-induction-successor-index-pair","seed":202610121015,"semantic_template":"qf-density12-induction-successor-index-pair/14"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=14. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(14, 15)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (14, 15)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0016","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c582ebf18bb1ddfea79a71c57e9c30773a1ed99ef9192e4e1aff6c59154e4cb0","family_id":"qf-density12-induction-successor-index-pair","seed":202610121016,"semantic_template":"qf-density12-induction-successor-index-pair/15"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=15. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(15, 16)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (15, 16)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0017","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f950f1d00431a9424613f1798d98431fed0aa557801c293c02386835af39e772","family_id":"qf-density12-induction-successor-index-pair","seed":202610121017,"semantic_template":"qf-density12-induction-successor-index-pair/16"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=16. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(16, 17)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (16, 17)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0018","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55b1c9cd5b610fc2f09962d89eeeea26ae165b82f44725133e5e5c2a68255495","family_id":"qf-density12-induction-successor-index-pair","seed":202610121018,"semantic_template":"qf-density12-induction-successor-index-pair/17"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=17. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(17, 18)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (17, 18)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0019","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"254d54e7ec3cab94c4650cba254b78fcf4d140e0f4d1bb215dc77067fa01edda","family_id":"qf-density12-induction-successor-index-pair","seed":202610121019,"semantic_template":"qf-density12-induction-successor-index-pair/18"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=18. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(18, 19)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (18, 19)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0020","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5d1a7921efa65086522c7a9434273378c22761627240ebf0aa2110b1d51f4db","family_id":"qf-density12-induction-successor-index-pair","seed":202610121020,"semantic_template":"qf-density12-induction-successor-index-pair/19"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=19. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(19, 20)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (19, 20)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0021","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f5efdbf997372425a072f351a7861d230bddcfb69cc5128d39c19c27857b12e","family_id":"qf-density12-induction-successor-index-pair","seed":202610121021,"semantic_template":"qf-density12-induction-successor-index-pair/20"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=20. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(20, 21)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (20, 21)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0022","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de01904c16790d2d71914811c2c65eb97bda187cc5bdb41872656bd691bbe602","family_id":"qf-density12-induction-successor-index-pair","seed":202610121022,"semantic_template":"qf-density12-induction-successor-index-pair/21"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=21. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(21, 22)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (21, 22)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0023","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd10c51aae86b001ea6747fdb06e614cc419e9b9d9eb0696dcf00fe74c085cf4","family_id":"qf-density12-induction-successor-index-pair","seed":202610121023,"semantic_template":"qf-density12-induction-successor-index-pair/22"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=22. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(22, 23)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (22, 23)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0024","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07206a16dee2ce62d86c4a6ff096c205775b7848e52f69118d9acd59b66e5609","family_id":"qf-density12-induction-successor-index-pair","seed":202610121024,"semantic_template":"qf-density12-induction-successor-index-pair/23"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=23. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(23, 24)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (23, 24)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0025","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7df0249a099554e35e37b1fa29847cf3e327ddb7c5ce9d9dc959fc7e549c2e60","family_id":"qf-density12-induction-successor-index-pair","seed":202610121025,"semantic_template":"qf-density12-induction-successor-index-pair/24"},"fields":{"title":"Induction-Hypothesis and Successor Indices","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=24. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.","answer":{"kind":"expression","value":"(24, 25)","explanation":"Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (24, 25)."},"skills":["Induction-Hypothesis and Successor Indices"],"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-20261008-advance-20261012-r01-mathematical-induction-0026","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f4849764c6e263239583423d8d31d95db30402f3dda167e4e3f5be3d11a6258","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121026,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[\"base_only\",-2]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-2, a(k+1)=a(k)+(99), through=0. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-2]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-2]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0027","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"222d5927a19984a602d3b7c7cbf7da08dc003f8e27d24861c11270a3086f27af","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121027,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-11,-1,2]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-11, a(k+1)=a(k)+(-1), through=2. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-11, -12, -13]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-11, -12, -13]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0028","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eec7775d4b2d707eaf661eb42169f83ae7351221b7b193f231711756601aaff","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121028,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-10,0,3]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-10, a(k+1)=a(k)+(0), through=3. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-10, -10, -10, -10]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-10, -10, -10, -10]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0029","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0992855b09f6a9cfc5ef4c67350e60b9ee5d39244ba167d9be2ba124ade38148","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121029,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-9,1,4]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-9, a(k+1)=a(k)+(1), through=4. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-9, -8, -7, -6, -5]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-9, -8, -7, -6, -5]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0030","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af8bc55405ecfafbb2fb68f072b52d6334989fcde25c1df39a9549e3560bb680","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121030,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-8,2,5]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-8, a(k+1)=a(k)+(2), through=5. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-8, -6, -4, -2, 0, 2]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-8, -6, -4, -2, 0, 2]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0031","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33c56f3be521a2b8134fb8a2ab2d26d86d3da0ffd7a54822c26e9f1becaea703","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121031,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-7,-2,6]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-7, a(k+1)=a(k)+(-2), through=6. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-7, -9, -11, -13, -15, -17, -19]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-7, -9, -11, -13, -15, -17, -19]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0032","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97e648c12c5dcb8c10b25828f49c3aa691181f72460fd6a469a06f315658e9ee","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121032,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-6,-1,7]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-6, a(k+1)=a(k)+(-1), through=7. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-6, -7, -8, -9, -10, -11, -12, -13]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-6, -7, -8, -9, -10, -11, -12, -13]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0033","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f73ac7eec4b420ad7028b22e9375d16d22384627625e8a1aa7e06fef167ecc10","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121033,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-5,0,8]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-5, a(k+1)=a(k)+(0), through=8. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-5, -5, -5, -5, -5, -5, -5, -5, -5]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-5, -5, -5, -5, -5, -5, -5, -5, -5]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0034","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e321c827d53843913d62211833a17be0021df149472a9cc49348e37f40f8d97b","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121034,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-4,1,1]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-4, a(k+1)=a(k)+(1), through=1. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-4, -3]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-4, -3]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0035","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4822a80aab33579847db2b3d55fd6ecaa8831a03e7278b88df77ac63e6921e6d","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121035,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-3,2,2]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-3, a(k+1)=a(k)+(2), through=2. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-3, -1, 1]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-3, -1, 1]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0036","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2c3a03bdf7aeb9ae2af7a83f5862d8a63abc171819d6f1285d7b583b0cc4b2b","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121036,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-2,-2,3]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-2, a(k+1)=a(k)+(-2), through=3. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-2, -4, -6, -8]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-2, -4, -6, -8]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0037","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eba17c5c818554032de68e6aa2c4fef2cc7d0bb5d406fcefdf45fe743ea248a","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121037,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[-1,-1,4]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=-1, a(k+1)=a(k)+(-1), through=4. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[-1, -2, -3, -4, -5]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [-1, -2, -3, -4, -5]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0038","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bb06c8d1c871b475a8a4e5686c7a2727154342fd63312fb64f1bd18c0d89de7","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121038,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[0,0,5]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=0, a(k+1)=a(k)+(0), through=5. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[0, 0, 0, 0, 0, 0]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [0, 0, 0, 0, 0, 0]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0039","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df59ff96a7b88d2d097f03f422788aef518ed6daa0dff354fe47ba579b15d9fd","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121039,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[1,1,6]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=1, a(k+1)=a(k)+(1), through=6. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5, 6, 7]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [1, 2, 3, 4, 5, 6, 7]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0040","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d43d7e2d5255b81ca81f0e4758566060df310d456b94de29c970d75da204ec4b","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121040,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[2,2,7]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=2, a(k+1)=a(k)+(2), through=7. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[2, 4, 6, 8, 10, 12, 14, 16]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [2, 4, 6, 8, 10, 12, 14, 16]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0041","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832ce347a4430b9c9fd8c6f9bcccd7953782a94a20b466c186093e97395c81be","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121041,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[3,-2,8]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=3, a(k+1)=a(k)+(-2), through=8. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[3, 1, -1, -3, -5, -7, -9, -11, -13]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [3, 1, -1, -3, -5, -7, -9, -11, -13]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0042","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c191c24e98c7be23eba0ffdbe0800ccad0f05e1891831c33a65e75847e1a0ed9","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121042,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[4,-1,1]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=4, a(k+1)=a(k)+(-1), through=1. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[4, 3]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [4, 3]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0043","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cf3b40f545678444c3158ac6c9ff1834bd91cb77d6e2d336e144db04cf38695","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121043,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[5,0,2]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=5, a(k+1)=a(k)+(0), through=2. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[5, 5, 5]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [5, 5, 5]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0044","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e83f653f931f06014c7d948e2c739192aface08b95e5cab3201c16e140aa75","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121044,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[6,1,3]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=6, a(k+1)=a(k)+(1), through=3. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[6, 7, 8, 9]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [6, 7, 8, 9]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0045","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fd00a0c51dfd0797a24ddd80ef5f516064e7af2d41838d56923c374a87157a1","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121045,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[7,2,4]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=7, a(k+1)=a(k)+(2), through=4. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[7, 9, 11, 13, 15]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [7, 9, 11, 13, 15]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0046","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d324f33575fd0773ed86eb07fae926206943db88972235417585c2f746fad88","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121046,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[8,-2,5]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=8, a(k+1)=a(k)+(-2), through=5. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[8, 6, 4, 2, 0, -2]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [8, 6, 4, 2, 0, -2]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0047","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e95ec014f049819c76da38f176bfc6aa2039b91615a419f95224d31c0dec714d","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121047,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[9,-1,6]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=9, a(k+1)=a(k)+(-1), through=6. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[9, 8, 7, 6, 5, 4, 3]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [9, 8, 7, 6, 5, 4, 3]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0048","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e3d45604997acfa1b1bce4ae5ee5d1c3b9cfa36d3033cf17f33942f6d333c53","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121048,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[10,0,7]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=10, a(k+1)=a(k)+(0), through=7. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[10, 10, 10, 10, 10, 10, 10, 10]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [10, 10, 10, 10, 10, 10, 10, 10]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0049","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"425c3f118b4827387d8db3c56a48618910c383c8d00c1b18686d305f6ad69abb","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121049,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[11,1,8]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=11, a(k+1)=a(k)+(1), through=8. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[11, 12, 13, 14, 15, 16, 17, 18, 19]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [11, 12, 13, 14, 15, 16, 17, 18, 19]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0050","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e64a47ed6bf2c3055f5a00c5c84afb373d5f5a7a8ed9593be51f67407aaeb1ba","family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121050,"semantic_template":"qf-density12-induction-affine-recursive-prefix/[12,2,1]"},"fields":{"title":"Inclusive Finite Recurrence Prefix","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=12, a(k+1)=a(k)+(2), through=1. List a(0) through a(through), inclusive.","answer":{"kind":"expression","value":"[12, 14]","explanation":"Start at the given initial value and add d repeatedly; through=0 lists only the initial value. Natural indices do not require nonnegative sequence values. The exact result, retaining the requested order, is [12, 14]."},"skills":["Inclusive Finite Recurrence Prefix"],"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-20261008-advance-20261012-r01-mathematical-induction-0051","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7892cfbf32777d5c89f7e874bc433f823c5985c4ab5c7bc4564faa8661bbbb7e","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121051,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"0\",\"0\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=3, increment d=2; candidate c(k)=(2)k+(3), n=0. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (0, 0)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0052","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c00897a72b3dcd52fcf66bcf5814f35f13ed69136408de9ab854c594a3e082b","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121052,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"-2\",\"0\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=3, increment d=2; candidate c(k)=(2)k+(1), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(-2, 0)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (-2, 0)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0053","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9f969e4cf0b75ef42b30356c28c859f225808e85a3f5d5f8d31efb32dc2ed5","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121053,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"0\",\"1\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=2; candidate c(k)=(3)k+(1), n=2. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(0, 1)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (0, 1)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0054","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"156badf88908c1888bde5c1f6c349d99db1d2dcf4064ea94787f3b12ce6e709b","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121054,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"4\",\"-9\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=-1; candidate c(k)=(-10)k+(5), n=3. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(4, -9)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (4, -9)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0055","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2be282072bd646d01eb5f4a578ed4498517ec1822e4a0d779c44719886249362","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121055,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"5\",\"-8\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=0; candidate c(k)=(-8)k+(3), n=4. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(5, -8)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (5, -8)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0056","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ff8ef91dfeb4b04f400ca3a11e2916d553fd017fe74f8292e312e2abfe52f6","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121056,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"6\",\"-7\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-1, increment d=1; candidate c(k)=(-6)k+(5), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(6, -7)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (6, -7)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0057","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdbc0bb2cd60d2cc11f74ff23ac841d493d146c3269d45df70dee154fac48361","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121057,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"7\",\"-6\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=-1; candidate c(k)=(-7)k+(7), n=0. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(7, -6)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (7, -6)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0058","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"000b924e212b1b8566bc5ba2bace9d58cba29c6721e54df060beff81b973379b","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121058,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"8\",\"-5\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=0; candidate c(k)=(-5)k+(9), n=1. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(8, -5)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (8, -5)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0059","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3b1a8ba5612753ebf2c59a6785ddb179efd8c5cbd1a86773e8c248a98b91658","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121059,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"9\",\"-4\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=1; candidate c(k)=(-3)k+(7), n=2. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(9, -4)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (9, -4)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0060","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c238360a6aa0ec3230cf33f8236d1f357121cae69bdde6a45f35d42d946a20e5","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121060,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"10\",\"-3\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-1, increment d=-1; candidate c(k)=(-4)k+(9), n=3. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(10, -3)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (10, -3)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0061","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d40dd5a136c75d81409bc6ccf74a1cddfe9c0f702e06dab9b5e3e15f3a284f6","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121061,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"11\",\"-2\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=0; candidate c(k)=(-2)k+(11), n=4. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(11, -2)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (11, -2)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0062","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72a70cf1054f47c45d579ed97aedd6f999f0b6693d41e2142da1176d11526fe7","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121062,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"12\",\"-1\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=1; candidate c(k)=(0)k+(13), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(12, -1)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (12, -1)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0063","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89320804788f0fbbda7febb91662e52d3742d7a63d1063be725596974428017","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121063,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"13\",\"0\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=-1; candidate c(k)=(-1)k+(11), n=0. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(13, 0)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (13, 0)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0064","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fde130141216c102eaaeb6306afa716fa7469ded17c317f4951c81146f400e4","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121064,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"14\",\"1\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-1, increment d=0; candidate c(k)=(1)k+(13), n=1. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(14, 1)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (14, 1)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0065","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10ec5f5292f6ceeaeba9997f760b5b782977e2a6a33c73d56355474bef761283","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121065,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"15\",\"2\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=1; candidate c(k)=(3)k+(15), n=2. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(15, 2)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (15, 2)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0066","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9287482560a088938ccf89989d70ec8711722f80b7528b0ade3754aea783a06f","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121066,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"16\",\"3\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=-1; candidate c(k)=(2)k+(17), n=3. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(16, 3)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (16, 3)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0067","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17ddc4f1be49c100bdba592d7c854b5d0e7d59602dd1d56388aef209a8e535ee","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121067,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"17\",\"4\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=0; candidate c(k)=(4)k+(15), n=4. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(17, 4)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (17, 4)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0068","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7c55e6257770b56d9dc5a445ca324d6f642fdcb0129a09d97c40c2b4e4beb2a","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121068,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"18\",\"5\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-1, increment d=1; candidate c(k)=(6)k+(17), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(18, 5)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (18, 5)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0069","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"783518e71427aea6b576a6bace04f4d55b18f0609ec3e308cd9accd799955382","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121069,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"19\",\"6\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=-1; candidate c(k)=(5)k+(19), n=0. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(19, 6)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (19, 6)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0070","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02feed077b262e03d1ad37fc0c617d80fa08bc00fe0ef5f7b2ad1019c11c2cb1","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121070,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"20\",\"7\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=0; candidate c(k)=(7)k+(21), n=1. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(20, 7)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (20, 7)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0071","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3caf4f7f0fc3fc0ceda72e7ba70b2713a946380971be868f85048ae1a65c8321","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121071,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"21\",\"8\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=1; candidate c(k)=(9)k+(19), n=2. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(21, 8)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (21, 8)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0072","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01c26278480572422cc82bd08a1c85bcd3dc360382f952d7be17197c50b6278e","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121072,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"22\",\"9\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-1, increment d=-1; candidate c(k)=(8)k+(21), n=3. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(22, 9)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (22, 9)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0073","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b3120f54188e68e07e10be6357ce702e50db6351e6bc515eef6319beb6e9d97","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121073,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"23\",\"10\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=0; candidate c(k)=(10)k+(23), n=4. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(23, 10)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (23, 10)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0074","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"467a8616919e08ee59b369e21249e7a881376168baeb89c06a59d06b506bd99e","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121074,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"24\",\"11\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=1, increment d=1; candidate c(k)=(12)k+(25), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(24, 11)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (24, 11)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0075","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dcee1688af6cfc4ec197edbe4e9c3c189f31370884c66c8a13e8b1b5408d6a8","family_id":"qf-density12-induction-candidate-step-defect","seed":202610121075,"semantic_template":"qf-density12-induction-candidate-step-defect/[\"25\",\"12\"]"},"fields":{"title":"Separate Base and Successor-Step Defects","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=-2, increment d=-1; candidate c(k)=(11)k+(23), n=0. Give signed integers (c(0)−s, c(n+1)−c(n)−d).","answer":{"kind":"expression","value":"(25, 12)","explanation":"Substitute at the base and adjacent indices separately; a base defect and step defect are different obligations, and finite checks are not universal induction. The exact result, retaining the requested order, is (25, 12)."},"skills":["Separate Base and Successor-Step Defects"],"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-20261008-advance-20261012-r01-mathematical-induction-0076","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9534121de75bf3f23320dd1d71bc41124999d48650ad002161d083383061088","family_id":"qf-density12-induction-range-obligations","seed":202610121076,"semantic_template":"qf-density12-induction-range-obligations/0"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(0) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(1, 0, 1)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (1, 0, 1)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0077","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d2cda00f61e422eec3964d3658c1f6c12f1e9b7984b5061aba94fb634407c6a","family_id":"qf-density12-induction-range-obligations","seed":202610121077,"semantic_template":"qf-density12-induction-range-obligations/1000"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1000) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(1001, 1000, 1001)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (1001, 1000, 1001)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0078","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b95942f01bb750fbd24474da4dd6a082068280fbfcfa1e4f314cd575b308073","family_id":"qf-density12-induction-range-obligations","seed":202610121078,"semantic_template":"qf-density12-induction-range-obligations/1"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(2, 1, 2)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (2, 1, 2)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0079","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514141db3e1818f7990b25140563f440f31d779df4a9c65cd6acd77ec121e7ba","family_id":"qf-density12-induction-range-obligations","seed":202610121079,"semantic_template":"qf-density12-induction-range-obligations/2"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(2) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(3, 2, 3)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (3, 2, 3)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0080","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7940d5f25f11c84d2aac214bdd75496db8ba95cbb54e703731abb22ea2d70a8","family_id":"qf-density12-induction-range-obligations","seed":202610121080,"semantic_template":"qf-density12-induction-range-obligations/3"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(3) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(4, 3, 4)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (4, 3, 4)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0081","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7845771c3d13e7770d940999c47ffeeee39fc216a1f5cb3e91a0a491857dc1b","family_id":"qf-density12-induction-range-obligations","seed":202610121081,"semantic_template":"qf-density12-induction-range-obligations/4"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(4) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(5, 4, 5)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (5, 4, 5)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0082","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a348a9451b63ec8a68c816d31e0cc21373b96c537e33d1d58fdd1477e120e89","family_id":"qf-density12-induction-range-obligations","seed":202610121082,"semantic_template":"qf-density12-induction-range-obligations/5"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(5) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(6, 5, 6)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (6, 5, 6)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0083","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776caa887d1447f07efae0309ec0f0253b5b4b41d6e29c33f68b213c9b4265ca","family_id":"qf-density12-induction-range-obligations","seed":202610121083,"semantic_template":"qf-density12-induction-range-obligations/6"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(6) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(7, 6, 7)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (7, 6, 7)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0084","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"400417ab82719812f2903564552bb88adbc4a9e4f7bb39d86a2ae8c45fa6ec79","family_id":"qf-density12-induction-range-obligations","seed":202610121084,"semantic_template":"qf-density12-induction-range-obligations/7"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(7) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(8, 7, 8)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (8, 7, 8)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0085","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2bc4818b51365312d721c60cfa44e96b9906bdfdcd4e19e0b7487eaafb2458","family_id":"qf-density12-induction-range-obligations","seed":202610121085,"semantic_template":"qf-density12-induction-range-obligations/8"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(8) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(9, 8, 9)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (9, 8, 9)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0086","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb28f76186bd2ad9a4fd39ace24f91be2af2ff81a8ceaf91e75d6347424a4b90","family_id":"qf-density12-induction-range-obligations","seed":202610121086,"semantic_template":"qf-density12-induction-range-obligations/9"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(9) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(10, 9, 10)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (10, 9, 10)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0087","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a475da074f5b42139675ba0163cb3c8cff8945e0ff37e868c8620d6b26690222","family_id":"qf-density12-induction-range-obligations","seed":202610121087,"semantic_template":"qf-density12-induction-range-obligations/10"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(10) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(11, 10, 11)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (11, 10, 11)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0088","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da6bf566ce3c507d946492e49d4807b716258cec2932e137e23ab9beda8a4c8b","family_id":"qf-density12-induction-range-obligations","seed":202610121088,"semantic_template":"qf-density12-induction-range-obligations/11"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(11) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(12, 11, 12)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (12, 11, 12)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0089","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97560e48364b26a93302279e38bbfb3703e0d2df990670df916e30c81db3cc3b","family_id":"qf-density12-induction-range-obligations","seed":202610121089,"semantic_template":"qf-density12-induction-range-obligations/12"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(12) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(13, 12, 13)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (13, 12, 13)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0090","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4e3af2bd92c7f10e02fe72203f1d07f72b691ea2de7337ee86671753adc4498","family_id":"qf-density12-induction-range-obligations","seed":202610121090,"semantic_template":"qf-density12-induction-range-obligations/13"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(13) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(14, 13, 14)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (14, 13, 14)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0091","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c2f12125c463da66c144c2b2f8b4b6e0c053598eb6247286a6677a2ecb3da14","family_id":"qf-density12-induction-range-obligations","seed":202610121091,"semantic_template":"qf-density12-induction-range-obligations/14"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(14) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(15, 14, 15)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (15, 14, 15)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0092","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e75ef540489b99183b3830b568c3de0f762062f2771108970ffbba69ac5a7657","family_id":"qf-density12-induction-range-obligations","seed":202610121092,"semantic_template":"qf-density12-induction-range-obligations/15"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(15) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(16, 15, 16)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (16, 15, 16)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0093","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b1c71f87089124be884dda57ff4de95100c25d21831b20d1a5372dec01ec69","family_id":"qf-density12-induction-range-obligations","seed":202610121093,"semantic_template":"qf-density12-induction-range-obligations/16"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(16) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(17, 16, 17)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (17, 16, 17)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0094","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9886f0ddeaf68a0425f6095efd63407a3fff0cb7c52473e28d3bfb909eac6cb3","family_id":"qf-density12-induction-range-obligations","seed":202610121094,"semantic_template":"qf-density12-induction-range-obligations/17"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(17) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(18, 17, 18)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (18, 17, 18)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0095","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3247a79cf84da72b19a6f3bd8bda1f43fb2e3c0664d3d11679cd572ff3e648a4","family_id":"qf-density12-induction-range-obligations","seed":202610121095,"semantic_template":"qf-density12-induction-range-obligations/18"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(18) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(19, 18, 19)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (19, 18, 19)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0096","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac43f9a3fe45e960a11ccf0e09106c7c48f1320060088830fe75179334dc405c","family_id":"qf-density12-induction-range-obligations","seed":202610121096,"semantic_template":"qf-density12-induction-range-obligations/19"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(19) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(20, 19, 20)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (20, 19, 20)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0097","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a4a06a96ef8fbe8dc953d0639fa4e59e60d296971212a41c277c2042597b3b8","family_id":"qf-density12-induction-range-obligations","seed":202610121097,"semantic_template":"qf-density12-induction-range-obligations/20"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(20) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(21, 20, 21)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (21, 20, 21)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0098","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6839c45818d9119d74beacb82ed1f5e64c7bccb3c2a74c594383b0ae671202","family_id":"qf-density12-induction-range-obligations","seed":202610121098,"semantic_template":"qf-density12-induction-range-obligations/21"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(21) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(22, 21, 22)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (22, 21, 22)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0099","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2456633772ab249217bdb880e0dd62bb2dd68fea3b0431cdac72ad1c60270c7a","family_id":"qf-density12-induction-range-obligations","seed":202610121099,"semantic_template":"qf-density12-induction-range-obligations/22"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(22) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(23, 22, 23)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (23, 22, 23)."},"skills":["Finite Coverage and Next Uncovered Index"],"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-20261008-advance-20261012-r01-mathematical-induction-0100","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f56361e729ed8a3d722a486ef97f49edf32f851e7d2ee75efa155cf3d6e6d1c","family_id":"qf-density12-induction-range-obligations","seed":202610121100,"semantic_template":"qf-density12-induction-range-obligations/23"},"fields":{"title":"Finite Coverage and Next Uncovered Index","stem":"N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(23) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).","answer":{"kind":"expression","value":"(24, 23, 24)","explanation":"Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (24, 23, 24)."},"skills":["Finite Coverage and Next Uncovered Index"],"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."}}