{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0001","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99d47bb493ce049e5d514bb84e236ac9f52a233a69e0cda09825889c3c85cc89","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155001,"semantic_template":"qf-density15-sequence-window-boundary-queries/[0,[[\"untyped\"],[\"untyped\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=0; only 0≤n<L is allowed for a_n=1+(2)n. Ordered candidate indices=[-1, 0, 1]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(0, [No, No, No], [invalid-index, invalid-index, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (0, [No, No, No], [invalid-index, invalid-index, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0002","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ecce330735f91090cac85c9c96d75ac772416d9a19e67a1d81d1445dd91e109","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155002,"semantic_template":"qf-density15-sequence-window-boundary-queries/[1,[[\"typed\",\"0\"],[\"untyped\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=1; only 0≤n<L is allowed for a_n=0+(0)n. Ordered candidate indices=[0, 1, -1]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(1, [Yes, No, No], [0, invalid-index, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (1, [Yes, No, No], [0, invalid-index, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0003","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ab1535080ac4972206dac8a25c9decf5c943dfaa9a8944fd07c6a74467f752","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155003,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-10\"],[\"typed\",\"-7\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-10+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-10, -7, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-10, -7, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0004","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a030a755d651c89d06755c7980d42183d54b0634fb6da94b9cfd4193023caf2","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155004,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-9\"],[\"typed\",\"-12\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-9, -12, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-9, -12, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0005","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a70cfde319d5a2e9e11f52fdb3b5975b5a3944a9c0ca96137f999cedfa0a71","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155005,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-8\"],[\"typed\",\"-8\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-8+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-8, -8, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-8, -8, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0006","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39573e8eb4b9ba735cae41c8f9b24934662e0a911eb42d4e481affdb8b62194d","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155006,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-7\"],[\"typed\",\"-4\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-7+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-7, -4, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-7, -4, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0007","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d259a731bf95252d54c7b39320abb9051f1aeab0cca3bbf6bae550f853ca5dca","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155007,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-6\"],[\"typed\",\"-9\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-6, -9, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-6, -9, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0008","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1a4fe86c2f2757559e13e70e1841ae44194d93aaad0c6b8d3426417803bcc5","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155008,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-5\"],[\"typed\",\"-5\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-5+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-5, -5, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-5, -5, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0009","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c56b26cb481fd0770df38144693c269c638208243b5a5cd4a24c19a0e3f4a817","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155009,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-4\"],[\"typed\",\"-1\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-4+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-4, -1, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-4, -1, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0010","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f4a14cf6f682478d42ccc007b41249cf30607f41b3570e8c4a69f20ebae9946","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155010,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-3\"],[\"typed\",\"-6\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-3+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-3, -6, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-3, -6, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0011","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4528d0c050f87f6fc725ebff5902d7adc56583ebd4e8c009a04b6faafc93a03","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155011,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-2\"],[\"typed\",\"-2\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-2+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-2, -2, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-2, -2, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0012","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b74b04fc2fa42fa2055f9ddf2068cbadccf02bd006dddb5de9b7de81c0355368","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155012,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-1\"],[\"typed\",\"2\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-1+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [-1, 2, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-1, 2, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0013","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29537429a7b7afc3b226c3aa42c509d7561891045dd7550581144b6a400d5d1e","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155013,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"0\"],[\"typed\",\"-3\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=0+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [0, -3, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [0, -3, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0014","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e0fd4354896e5144e065b92c5ee684532987eedbf8086e679bf2118696fdf17","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155014,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"1\"],[\"typed\",\"1\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=1+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [1, 1, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [1, 1, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0015","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7676cf929dbb88a454df0b81c0bac7bd796d5d51adf61fc0310758b66cf6c835","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155015,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"2\"],[\"typed\",\"5\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=2+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [2, 5, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [2, 5, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0016","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9de2ffbd3cb4149affb8b4083a8d5315d082a2c3d8eeac5b636da808bba030e","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155016,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"3\"],[\"typed\",\"0\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=3+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [3, 0, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [3, 0, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0017","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61beb6be6472838e31aed739e9ad157f0e536033a877eff168a0183b98564d24","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155017,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"4\"],[\"typed\",\"4\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=4+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [4, 4, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [4, 4, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0018","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5baa7d947802dfadb6bd1386c5bf55ad7a72ec121ab5f0168e7af8be66c4189","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155018,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"5\"],[\"typed\",\"8\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=5+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [5, 8, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [5, 8, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0019","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79b95eed272630f9ad236a555016244292110e4169bbc0bc6063beb4cb75dd72","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155019,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"6\"],[\"typed\",\"3\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [6, 3, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [6, 3, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0020","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67fb192fa3533ebd87840cf0620aaf484fef75892e405561485ef138df36af2","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155020,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"7\"],[\"typed\",\"7\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=7+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [7, 7, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [7, 7, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0021","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1130205bc9ab37f251c8f2a7e3907911707246dba30b7e901224245f4d68ba9","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155021,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"8\"],[\"typed\",\"11\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=8+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [8, 11, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [8, 11, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0022","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96ff3d03bc94d31cc7105c10d0f87e8a7fe1ea91842d91fd07c185cc6aa88df1","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155022,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"9\"],[\"typed\",\"6\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [9, 6, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [9, 6, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0023","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73653656cf173bff9410f2a86a168ebbbacf3c97f09f2265f75c19adce96bc96","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155023,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"10\"],[\"typed\",\"10\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=10+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [10, 10, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [10, 10, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0024","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fbf913a07ec5a5b4c2518f99eeed7052ed0de0715c3cafa8c66b96d0940a597","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155024,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"11\"],[\"typed\",\"14\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=11+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [11, 14, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [11, 14, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0025","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f38012722f6b836a493b80704a4c4d150ff13afd03373105f80214045448ac","family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155025,"semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"12\"],[\"typed\",\"9\"],[\"untyped\"]]]"},"fields":{"title":"Finite Sequence Query Bounds before Evaluation","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=12+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).","answer":{"kind":"expression","value":"(4, [Yes, Yes, No], [12, 9, invalid-index])","explanation":"Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [12, 9, invalid-index])."},"skills":["Finite Sequence Query Bounds before Evaluation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0026","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"233e62b8efdda49aa136bc22533885ddb7aee173bb8b558b90f1ed9587fea77d","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155026,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[\"empty_window\"]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([], [], [])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([], [], [])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0027","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"119d588a660ee5a753b4fe18eda2466436a8e8b62640ed67060da33173c0c0ed","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155027,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[0],[\"0\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(7), start=0, length=1. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([0], [0], [])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([0], [0], [])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0028","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19e5814e322f4c96f1e05a503ff323e04c3a3a37bba43a31d7bc5cc0dbc99c1c","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155028,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[0,1,2],[\"-10\",\"-9\",\"-8\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-10, a_(n+1)=a_n+(1), start=0, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([0, 1, 2], [-10, -9, -8], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([0, 1, 2], [-10, -9, -8], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0029","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edafd8930741575db1ae465f04484904be8d55bbd824fbcf7ae5d0f400b76dee","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155029,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[1,2,3],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-9, a_(n+1)=a_n+(-1), start=1, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([1, 2, 3], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([1, 2, 3], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0030","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9359b7aef3cb706f910c9258ca7c6adc162c09f445494f2d95c80c542e9e69cf","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155030,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[2,3,4],[\"-8\",\"-8\",\"-8\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-8, a_(n+1)=a_n+(0), start=2, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([2, 3, 4], [-8, -8, -8], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([2, 3, 4], [-8, -8, -8], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0031","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66c38a5ed824be638824cd6c7e80830f72259ee3b122e69463fb5442f26b6df6","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155031,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[3,4,5],[\"-4\",\"-3\",\"-2\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-7, a_(n+1)=a_n+(1), start=3, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([3, 4, 5], [-4, -3, -2], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([3, 4, 5], [-4, -3, -2], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0032","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4272c4a0c940cd78926779fab5c4fc033320edb1cb3ebd03f61bede618c887a3","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155032,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[4,5,6],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-6, a_(n+1)=a_n+(-1), start=4, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([4, 5, 6], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([4, 5, 6], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0033","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c3cde7a0979c89da2fa1e97597886445f7adc4813351501453410723b6fc29","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155033,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[5,6,7],[\"-5\",\"-5\",\"-5\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-5, a_(n+1)=a_n+(0), start=5, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([5, 6, 7], [-5, -5, -5], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([5, 6, 7], [-5, -5, -5], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0034","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b631fcaee145c6a20ac2241f702fb98fa0dca64f5a59148c7e57e8a4e151f3b","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155034,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[6,7,8],[\"2\",\"3\",\"4\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-4, a_(n+1)=a_n+(1), start=6, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([6, 7, 8], [2, 3, 4], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([6, 7, 8], [2, 3, 4], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0035","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930dbb2ea2d6bdabb8e75144c7066cdadb635cbc428a390870ecc85a50e8ddb9","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155035,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[7,8,9],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-3, a_(n+1)=a_n+(-1), start=7, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([7, 8, 9], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([7, 8, 9], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0036","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a12352a109f8112f1e7da964f7dbbdea59d2bc162ba85841bc8616a9d2420820","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155036,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[8,9,10],[\"-2\",\"-2\",\"-2\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-2, a_(n+1)=a_n+(0), start=8, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([8, 9, 10], [-2, -2, -2], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([8, 9, 10], [-2, -2, -2], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0037","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9b982c390c8ddf6e07ea5fbe6fcb5f3ab44d8304555a3d6abac1d7c13f1f1bf","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155037,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[9,10,11],[\"8\",\"9\",\"10\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-1, a_(n+1)=a_n+(1), start=9, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([9, 10, 11], [8, 9, 10], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([9, 10, 11], [8, 9, 10], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0038","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ec2502833a38400b449bfdca6419bd7a8f2c5e88ebc87f42d1d30fff2f95b7f","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155038,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[10,11,12],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(-1), start=10, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([10, 11, 12], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([10, 11, 12], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0039","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee0a2f0274c7b740522326415626d163a9908ee79f770c1ccfc708f920555a9","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155039,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[11,12,13],[\"1\",\"1\",\"1\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=1, a_(n+1)=a_n+(0), start=11, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([11, 12, 13], [1, 1, 1], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([11, 12, 13], [1, 1, 1], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0040","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe896c5a06c91171448ca2037a3ed916d0cfe0f21208edfb421aaf6557e2f32d","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155040,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[12,13,14],[\"14\",\"15\",\"16\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=2, a_(n+1)=a_n+(1), start=12, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([12, 13, 14], [14, 15, 16], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([12, 13, 14], [14, 15, 16], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0041","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51520a5337a103fffdf73a610fbab4fadf359226ed94513c7b6d736522ac875f","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155041,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[13,14,15],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=3, a_(n+1)=a_n+(-1), start=13, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([13, 14, 15], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([13, 14, 15], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0042","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd5aaf8965ac3608e206093464936c882a58ad1ca9b948355032fc111493e0e","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155042,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[14,15,16],[\"4\",\"4\",\"4\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=4, a_(n+1)=a_n+(0), start=14, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([14, 15, 16], [4, 4, 4], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([14, 15, 16], [4, 4, 4], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0043","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be05d054d70559314b09324087d6abf09b53fb59e2fc964b9b3baa162ed9f843","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155043,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[15,16,17],[\"20\",\"21\",\"22\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=5, a_(n+1)=a_n+(1), start=15, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([15, 16, 17], [20, 21, 22], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([15, 16, 17], [20, 21, 22], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0044","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cf7fa803c5f9ed49f3d51abd04d8d24fe7257c19b8221eef500f2753e5250ea","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155044,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[16,17,18],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=6, a_(n+1)=a_n+(-1), start=16, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([16, 17, 18], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([16, 17, 18], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0045","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3070d8469be5e40cb99106ef819fb64be330c638f802d4ddbbb8f2517dd4dd04","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155045,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[17,18,19],[\"7\",\"7\",\"7\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=7, a_(n+1)=a_n+(0), start=17, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([17, 18, 19], [7, 7, 7], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([17, 18, 19], [7, 7, 7], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0046","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df5d07d2e615070243a4f8d5e51a6506a2e55667e255b8d164d929b85aeb9757","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155046,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[18,19,20],[\"26\",\"27\",\"28\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=8, a_(n+1)=a_n+(1), start=18, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([18, 19, 20], [26, 27, 28], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([18, 19, 20], [26, 27, 28], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0047","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f50658dabdc5af26fbd2c8c93d6f8fe5da2b6fe444aa095fe81b460b5bf578d","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155047,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[19,20,21],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=9, a_(n+1)=a_n+(-1), start=19, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([19, 20, 21], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([19, 20, 21], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0048","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a7e4bed19a58c46d350ef8f578b873705c3854b772cadf3244b390262397490","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155048,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[20,21,22],[\"10\",\"10\",\"10\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=10, a_(n+1)=a_n+(0), start=20, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([20, 21, 22], [10, 10, 10], [0, 0])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([20, 21, 22], [10, 10, 10], [0, 0])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0049","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"672fbaf6f6af6fcb031e776e1faeb659adb14350a81f573846cdfd289e680e88","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155049,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[21,22,23],[\"32\",\"33\",\"34\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=11, a_(n+1)=a_n+(1), start=21, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([21, 22, 23], [32, 33, 34], [1, 1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([21, 22, 23], [32, 33, 34], [1, 1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0050","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77d0382638e0cbf849077cff25676085c9ac12d0d7f78dbd2477e1dacde2cb2b","family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155050,"semantic_template":"qf-density15-sequence-affine-recurrence-window/[[22,23,24],[\"-10\",\"-11\",\"-12\"]]"},"fields":{"title":"Ordered Recurrence Window and Adjacent Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=12, a_(n+1)=a_n+(-1), start=22, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.","answer":{"kind":"expression","value":"([22, 23, 24], [-10, -11, -12], [-1, -1])","explanation":"Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([22, 23, 24], [-10, -11, -12], [-1, -1])."},"skills":["Ordered Recurrence Window and Adjacent Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0051","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc8771a67434e672faefb20316e0475766e66f706ed2817fea81dccbc7f5847b","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155051,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[0,0,0,0],[0,0]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[0, 0, 0, 0] and candidate nexts=[0, 0]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], Yes, Yes)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], Yes, Yes)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0052","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c343a3a17f3ed3c720bf035691bdc55bfeaafdfc5819fbc1c6cf8fc18b651228","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155052,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[1,1,2,2],[3,4]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[1, 1, 2, 2] and candidate nexts=[3, 4]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0053","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"971fdedc7fe6f7b10247bc07e95f6cad8b86833729286b3c3ab6d1fd78248b6a","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155053,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-10,0,-10,1],[2,-1]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-10, 0, -10, 1] and candidate nexts=[2, -1]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0054","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03c8dce6abe873e70f68bdd7e6d95a0691e8da062317289fdc1f8c857e57c272","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155054,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-9,0,-9,1],[3,-2]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-9, 0, -9, 1] and candidate nexts=[3, -2]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0055","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eed73b85c40cf763dece916980231150606b01af46deb0d7666d10e4988c76e","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155055,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-8,0,-8,1],[4,-3]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-8, 0, -8, 1] and candidate nexts=[4, -3]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0056","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5000fbd7a560340b228c0b692f5fcb3476c3fdd5f2679aa5afa4469ad86dce","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155056,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-7,0,-7,1],[5,-4]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-7, 0, -7, 1] and candidate nexts=[5, -4]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0057","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab167bd21955e0c779f1071714f0cfe8473173ec80b344328f601e5036287377","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155057,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-6,0,-6,1],[6,-5]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-6, 0, -6, 1] and candidate nexts=[6, -5]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0058","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abdbea3a8bff49fc2291649ab7bd23d9d61370ffdb5be09e13fcd5ce3c3a04fc","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155058,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-5,0,-5,1],[7,-6]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-5, 0, -5, 1] and candidate nexts=[7, -6]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0059","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20cf0d66817b1e1719c46fd0e366e1ec03e2e6495e668b3c7529ae0bca43af30","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155059,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-4,0,-4,1],[8,-7]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-4, 0, -4, 1] and candidate nexts=[8, -7]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0060","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e891fc82698e46e6a4c822e38762881a8ea27b61a8b736a0cc49a8be1af105a","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155060,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-3,0,-3,1],[9,-8]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-3, 0, -3, 1] and candidate nexts=[9, -8]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0061","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22795c24d3b7cd14eca790436c4b015d7f3aa4c7e76fa76ba1face37efb658a1","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155061,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-2,0,-2,1],[10,-9]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-2, 0, -2, 1] and candidate nexts=[10, -9]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0062","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1f34acefd4776971834003e3619c1a58e123b698b82b8e4a9ceec25dd0aff30","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155062,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[-1,0,-1,1],[11,-10]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-1, 0, -1, 1] and candidate nexts=[11, -10]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0063","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e17178a2d7ed8974b683e91f430c76eb28ad19021c5f385672bf0175f5492e4c","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155063,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[0,0,0,1],[12,-11]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[0, 0, 0, 1] and candidate nexts=[12, -11]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0064","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1cf77f479eff11e3cc0dfedb8ce4cd4eb5ae8e9a6e3a09afa4c6b6e5b65db55","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155064,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[1,0,1,1],[13,-12]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[1, 0, 1, 1] and candidate nexts=[13, -12]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0065","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284f7dc8a8e9f82efd0aa4fade775c51f6d5c76cd15516c139717944f704b664","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155065,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[2,0,2,1],[14,-13]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[2, 0, 2, 1] and candidate nexts=[14, -13]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0066","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1db96c981b82b17036f2df9b6eac61eea2015b6d39973ecb46a5ab99e404a670","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155066,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[3,0,3,1],[15,-14]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[3, 0, 3, 1] and candidate nexts=[15, -14]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0067","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e673839484436e76325bfd8cb7faad3de561417047dd7a747b181f06552ffe5","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155067,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[4,0,4,1],[16,-15]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[4, 0, 4, 1] and candidate nexts=[16, -15]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0068","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6fe5e6d9376da87d4c8d57f6fd1ab0d78efbf0ae19e839efaf3f25bfe4a7f86","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155068,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[5,0,5,1],[17,-16]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[5, 0, 5, 1] and candidate nexts=[17, -16]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0069","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"234b1e3a36b0167c928b8be03be5dc816aabe7a5422b0e243a1d945c784488ea","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155069,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[6,0,6,1],[18,-17]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[6, 0, 6, 1] and candidate nexts=[18, -17]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0070","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7305c965ce5ee305c5dd6c27c92cdef3737c5168d7b756b996fec024c2cab351","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155070,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[7,0,7,1],[19,-18]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[7, 0, 7, 1] and candidate nexts=[19, -18]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0071","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5d9f570d1a951bf0f090dade6b083a45ab3b66660d540760cdd67dafa79504f","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155071,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[8,0,8,1],[20,-19]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[8, 0, 8, 1] and candidate nexts=[20, -19]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0072","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9e83c5989772ffab9a63334fc13a2dd5bf59a1cd65f6c41c9f3419cf0de36bf","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155072,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[9,0,9,1],[21,-20]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[9, 0, 9, 1] and candidate nexts=[21, -20]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0073","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccf5148715508a8cb4183a60182c81b7785c58e24b754e3b4a6e22d1b444c97a","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155073,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[10,0,10,1],[22,-21]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[10, 0, 10, 1] and candidate nexts=[22, -21]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0074","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81990c8a2a055469e3b12f55e113aae549eb5f5548ad093035c9ffc563066b60","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155074,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[11,0,11,1],[23,-22]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[11, 0, 11, 1] and candidate nexts=[23, -22]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0075","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"facf912f5f3f645e89af4e06d3e568e5e0c62fa3f2695c06831684c34d34de91","family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155075,"semantic_template":"qf-density15-sequence-two-finite-extensions/[[12,0,12,1],[24,-23]]"},"fields":{"title":"Two Fifth-Term Extensions of a Given Four-Term Prefix","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[12, 0, 12, 1] and candidate nexts=[24, -23]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.","answer":{"kind":"expression","value":"([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], Yes, No)","explanation":"Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], Yes, No)."},"skills":["Two Fifth-Term Extensions of a Given Four-Term 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-20261011-advance-20261015-r05-sequence-0076","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d42a38e0fb59c50fe4d5f9e8304fdc695350f7a467e3dbccb37ec71fcffb249a","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155076,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[0,0,0,0,0]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[0, 0, 0, 0, 0]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [0, 0, 0, 0], Yes)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [0, 0, 0, 0], Yes)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0077","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f58c760ff44c9a9a47b63a2d281565c5de697e793e1569e146fd2072ea3cc9","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155077,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[2,2,2,2,2]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[2, 2, 2, 2, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([2, 4, 6, 8, 10], [0, 0, 0, 0], Yes)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([2, 4, 6, 8, 10], [0, 0, 0, 0], Yes)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0078","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6f151a0b4549dbf5c79470109cd7ef97ffe11e1d7e94d55ee359b597cf52f98","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155078,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-2,-1,0,1,2]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, -1, 0, 1, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-2, -3, -3, -2, 0], [1, 1, 1, 1], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-2, -3, -3, -2, 0], [1, 1, 1, 1], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0079","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28867f090ffe6ea7c4b1295cb66c3c337705a52541825d5dca6a2cdd9d486f62","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155079,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-9,0,-9,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-9, 0, -9, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-9, -9, -18, -19, -18], [9, -9, 8, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-9, -9, -18, -19, -18], [9, -9, 8, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0080","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5637496d0d6b43656061451560a494c2a09b1a9abccc0d34bfd78d4aa4292aca","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155080,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-8,0,-8,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-8, 0, -8, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-8, -8, -16, -17, -16], [8, -8, 7, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-8, -8, -16, -17, -16], [8, -8, 7, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0081","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e911fe962ed5565e7be49e7302212211ab6aea03fbc355231c5887299b92dd1","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155081,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-7,0,-7,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-7, 0, -7, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-7, -7, -14, -15, -14], [7, -7, 6, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-7, -7, -14, -15, -14], [7, -7, 6, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0082","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"676e2e7e5da1626eb5a3b3f845798dafd241ab115565f5865dfab71d3b087e87","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155082,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-6,0,-6,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-6, 0, -6, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-6, -6, -12, -13, -12], [6, -6, 5, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-6, -6, -12, -13, -12], [6, -6, 5, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0083","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"817d2736a71840a5d643fe2c54a5eb11ac6815dbacc58a1f3bf15f94eaf3d1fc","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155083,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-5,0,-5,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-5, 0, -5, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-5, -5, -10, -11, -10], [5, -5, 4, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-5, -5, -10, -11, -10], [5, -5, 4, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0084","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ccada82349cfe04ecc744b20041c680bdce70556523a4ef1382eb4a094b073","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155084,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-4,0,-4,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-4, 0, -4, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-4, -4, -8, -9, -8], [4, -4, 3, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-4, -4, -8, -9, -8], [4, -4, 3, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0085","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb029d5b58444014ecfb2a19e70b8ce52c4f25bcb3464faf9c727bfe7c9ccaef","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155085,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-3,0,-3,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-3, 0, -3, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-3, -3, -6, -7, -6], [3, -3, 2, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-3, -3, -6, -7, -6], [3, -3, 2, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0086","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1262f2da5912c9fd1e90be026a129c32f65fbc8d6d9f059f3997b17a1ee6e80","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155086,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-2,0,-2,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, 0, -2, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-2, -2, -4, -5, -4], [2, -2, 1, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-2, -2, -4, -5, -4], [2, -2, 1, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0087","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7f0872f77c2c0e99b4c3ab95bd09dfeb1bd33aa671fa1806f5b1dc51e53e26e","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155087,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-1,0,-1,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-1, 0, -1, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([-1, -1, -2, -3, -2], [1, -1, 0, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-1, -1, -2, -3, -2], [1, -1, 0, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0088","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8c00c232d58a298969fd23aab83e09b1d7fe0e14787739dfd13e4c63bb594f","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155088,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[0,0,0,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[0, 0, 0, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([0, 0, 0, -1, 0], [0, 0, -1, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([0, 0, 0, -1, 0], [0, 0, -1, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0089","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e00ee3bdfe80df8056f0f243f240a2a2fe84d63bff55b8412649fbacbd1fefe5","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155089,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[1,0,1,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[1, 0, 1, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([1, 1, 2, 1, 2], [-1, 1, -2, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], [-1, 1, -2, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0090","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e90b62905932804c2b18eeb6b368eee6f6bed52269a8ff7136e0bd40c25e49d","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155090,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[2,0,2,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[2, 0, 2, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([2, 2, 4, 3, 4], [-2, 2, -3, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([2, 2, 4, 3, 4], [-2, 2, -3, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0091","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5316cc7d2f3da5f6d67e10147e0f7caef0a24ccad1a0b4e526f523f276feba0","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155091,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[3,0,3,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[3, 0, 3, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([3, 3, 6, 5, 6], [-3, 3, -4, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([3, 3, 6, 5, 6], [-3, 3, -4, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0092","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce54b933e262b9adf0afa274002b2d8b6d67e36442d9ecb4f8c7f07865a73af9","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155092,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[4,0,4,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[4, 0, 4, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([4, 4, 8, 7, 8], [-4, 4, -5, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([4, 4, 8, 7, 8], [-4, 4, -5, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0093","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6541d4df162b0e76e1a97afd8595de73d222767808acca640ab247d3b04cd23a","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155093,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[5,0,5,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[5, 0, 5, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([5, 5, 10, 9, 10], [-5, 5, -6, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([5, 5, 10, 9, 10], [-5, 5, -6, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0094","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3541c471535a250a8c33e931220086da08850987d04fdcef19e16f82e19536","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155094,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[6,0,6,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[6, 0, 6, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([6, 6, 12, 11, 12], [-6, 6, -7, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([6, 6, 12, 11, 12], [-6, 6, -7, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0095","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a65d1a45137b40c886741074e492ac987b88d064f380d88d99e7324d5a717108","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155095,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[7,0,7,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[7, 0, 7, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([7, 7, 14, 13, 14], [-7, 7, -8, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([7, 7, 14, 13, 14], [-7, 7, -8, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0096","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcaec66198a967efd8cb1f7b68e126a056857ff70397758f2a52cccb1e8595c7","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155096,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[8,0,8,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[8, 0, 8, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([8, 8, 16, 15, 16], [-8, 8, -9, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([8, 8, 16, 15, 16], [-8, 8, -9, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0097","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e5ee8288e11d4c37e24050802fc38d766a024939cd1669df807e56203e73f81","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155097,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[9,0,9,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[9, 0, 9, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([9, 9, 18, 17, 18], [-9, 9, -10, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([9, 9, 18, 17, 18], [-9, 9, -10, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0098","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6957c34ed63d71ac8f08d036f9eb227818bab0df135bc2ddd1dc4922cd95f753","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155098,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[10,0,10,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[10, 0, 10, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([10, 10, 20, 19, 20], [-10, 10, -11, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([10, 10, 20, 19, 20], [-10, 10, -11, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0099","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729a1ab36d183985d0975386b217b77b21632bd4365f72d1f57c48ca72852b1a","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155099,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[11,0,11,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[11, 0, 11, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([11, 11, 22, 21, 22], [-11, 11, -12, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([11, 11, 22, 21, 22], [-11, 11, -12, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r05-sequence-0100","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459aa8b844d52e5a6747f5d9309d0a0ca5620958301905484db38617cca6c5ab","family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155100,"semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[12,0,12,-1,1]"},"fields":{"title":"Five Prefix Sums and Four Ordered Differences","stem":"N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[12, 0, 12, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).","answer":{"kind":"expression","value":"([12, 12, 24, 23, 24], [-12, 12, -13, 2], No)","explanation":"Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([12, 12, 24, 23, 24], [-12, 12, -13, 2], No)."},"skills":["Five Prefix Sums and Four Ordered Differences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}