{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0001","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d487b2de53c01fda046ffee56a41da2444677e1fd4785ba9f56baaab1d28e26","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105001,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [1, 2], -2, [0, 1], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], -2, [0, 1], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0002","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21f365c69ec1aab41578c55a916e6964df16b53b76ef6f251876c402115c9433","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105002,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [2, 4, 6], -1, [1, 2, 3], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], -1, [1, 2, 3], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0003","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d36294cd8494c684f11bb802e544f4bfc05b25a9b7a357a56836b6a4819db969","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105003,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 0, [2, 3, 4, 5], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 0, [2, 3, 4, 5], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0004","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69baa8ec5b320d6db97d5d1dd619eb0bf82fd24a49d7bd52b689b5db565be520","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105004,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [2, 4], 1, [3, 4], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], 1, [3, 4], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0005","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b31a0d8dde98ede256db839ed76247d82eb97540e17e5bff4ae393c0e9808ba","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105005,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [1, 2, 3], 2, [4, 5, 6], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], 2, [4, 5, 6], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0006","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9870a15f3fa22f44ef4a91c61de08b4efa1b9de6adad7ac9212bd98086c23ac7","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105006,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], -2, [5, 6, 7, 8], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], -2, [5, 6, 7, 8], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0007","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"299f633cd5a07063abe607df988a6e1389aa69677f31bbf45b8b8e2bcb3b6c22","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105007,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [1, 2], -1, [6, 7], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], -1, [6, 7], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0008","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c76071c02a3e23f85fd998a7f6b1cd0d88e196ed6ea0ad5bd9d81e5b91d6a3c","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105008,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [2, 4, 6], 0, [7, 8, 9], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 0, [7, 8, 9], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0009","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d972fb1dd3b7ea839478555ed3dc14bc50be7797ebe705900315e2fde32ab190","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105009,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 1, [8, 9, 10, 11], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 1, [8, 9, 10, 11], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0010","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10900a90de00428865f7bc11306f5cd30ea85c6010fda181438b7c023421879c","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105010,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [2, 4], 2, [9, 10], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], 2, [9, 10], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0011","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd7c087a1ecd00862a3361625b533d4c3b3c207496bf7a608f9d5cf5643fd1b5","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105011,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [1, 2, 3], -2, [10, 11, 12], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], -2, [10, 11, 12], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0012","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2976a340ecfd21ede1b27732e992e3faeb92416c9083e638a71ec2a3af54297","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105012,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], -1, [11, 12, 13, 14], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], -1, [11, 12, 13, 14], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0013","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39b5549775aa9d486c7c58153a894952fa596b36c71e5763a08c7e0b042f7681","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105013,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [1, 2], 0, [12, 13], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 0, [12, 13], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0014","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efd9464cde091c1b6545fc51e4a5bf04fb78f9cee834441770a2491450143878","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105014,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [2, 4, 6], 1, [13, 14, 15], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 1, [13, 14, 15], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0015","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd76f2d91523c558a864eda4b0a0711f49fdd11bf3610d5919448c1ef2b27b7d","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105015,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], 2, [14, 15, 16, 17], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], 2, [14, 15, 16, 17], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0016","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93d11be902bdc5f3e74ef77c583eab9fe7c24b968c26a262b993a364005cd0ec","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105016,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [2, 4], -2, [15, 16], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], -2, [15, 16], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0017","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b330a469d8fecfbc1f1ac112e9057a9c7aab91f31c143bcee58c69dac3a426","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105017,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [1, 2, 3], -1, [16, 17, 18], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], -1, [16, 17, 18], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0018","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2733198e89792bf65679eff985e862e0111f8a09648b59b1162a65fa36c69765","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105018,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], 0, [17, 18, 19, 20], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0019","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f3288477f5e6936fbe356a456e8ab94ae63db24e20958da5edc60d2607bd72","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105019,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [1, 2], 1, [18, 19], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 1, [18, 19], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0020","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7937469bf47cd4eeaf34c97835d48055d79f0ea825463df2366b5c53fc38c3e6","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105020,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [2, 4, 6], 2, [19, 20, 21], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [2, 4, 6], 2, [19, 20, 21], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0021","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ca1449a7626a3fc59efd9ed54d57910a474db05624f71a97ddd92e97a5d17aa","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105021,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], -2, [20, 21, 22, 23], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], -2, [20, 21, 22, 23], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0022","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb05b684a020d19114291a6d2dfda9c348942e22387df6fbe0b7072c92a3434a","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105022,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [2, 4], -1, [21, 22], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [2, 4], -1, [21, 22], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0023","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7abc3ebbd3fe6b1588dfb7b725db53e2d307b9cbb0116d1731d6f737de5b0342","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105023,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":3}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(3, [1, 2, 3], 0, [22, 23, 24], Yes, Yes, rule_applies_for_n_ge_3)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (3, [1, 2, 3], 0, [22, 23, 24], Yes, Yes, rule_applies_for_n_ge_3)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0024","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684a053e31792f3002ceb9e4123234e4b44c77aafd6d17cc256a2a88cbf1f89f","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105024,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":4}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(4, [2, 4, 6, 8], 1, [23, 24, 25, 26], Yes, Yes, rule_applies_for_n_ge_4)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (4, [2, 4, 6, 8], 1, [23, 24, 25, 26], Yes, Yes, rule_applies_for_n_ge_4)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0025","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"963b9381d1b79a82290dce41b5a83c3c3d7c3b5347055bf732d6edfd0f4640ae","family_id":"qf-mod41-recurrence-specification-qualification","seed":202611105025,"semantic_template":"qf-mod41-recurrence-specification-qualification/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":2}"},"fields":{"title":"An Order-r Total Rule Needs All r Initial Values","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).","answer":{"kind":"expression","value":"(2, [1, 2], 2, [24, 25], Yes, Yes, rule_applies_for_n_ge_2)","explanation":"Check coefficient/initial arity and the explicit scope n≥r. A total single-valued rule and complete initial state determine forward values; this does not prove a closed form, convergence or asymptotics. The exact result, retaining the requested order, is (2, [1, 2], 2, [24, 25], Yes, Yes, rule_applies_for_n_ge_2)."},"skills":["An Order-r Total Rule Needs All r Initial Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0026","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a2d086af7cd018579c990707f60b93b078b580758a8edca5f64574ed9634405","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105026,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":2,\"target_n\":5}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [0, 1], 5, [0, 1, 0, -1, -4, -11], -11, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0027","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79ad3841d203d4de9e7e473902b140ab9960829b819d49c12bff275bbad2f98f","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105027,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":3,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [1, 2, 3], 7, [1, 2, 3, 27, 177, 1175, 7811, 51919], 51919, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0028","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18069fe624dfcbcb720926a4a2da92a5478fda9554acfe0300337d7936f10b8f","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105028,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":4,\"target_n\":9}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [2, 3, 4, 5], 9, [2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071], 93071, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0029","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d0a85698658b7da0d69ed0362b5a25e700a0de21f5c14fd63804cedaf6151d8","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105029,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":2,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [3, 4], 8, [3, 4, 23, 101, 451, 2007, 8931, 39739, 176819], 176819, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0030","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa9db6a66d124e095d4ece19e4fc27589660315f616efdcb13df3b466a6baa8d","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105030,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":3,\"target_n\":6}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [4, 5, 6], 6, [4, 5, 6, 34, 121, 439, 1595], 1595, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0031","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5756b04a74554e3d48de4738d13ee12ed719db0e37ac22e4e5c37a552736782f","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105031,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":4,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [5, 6, 7, 8], 8, [5, 6, 7, 8, 138, 1190, 10392, 90842, 794122], 794122, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0032","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da51c71a743f1da4fc64d4b2eff0beb1c1f46662edbc4c5f21897a1cabb1f01d","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105032,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":2,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [6, 7], 7, [6, 7, 19, 44, 106, 255, 615, 1484], 1484, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0033","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3de18507f3242febb2a203162b4740459a6fe3fdb7fa1bdcebc1551cbcfa30a","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105033,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":3,\"target_n\":9}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [7, 8, 9], 9, [7, 8, 9, 100, 652, 4330, 28788, 191352, 1271924, 8454528], 8454528, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0034","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4520e491d604354d8c237fec351aadbb498c9302b414665489ba2ef2abd20c1c","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105034,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":4,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [8, 9, 10, 11], 7, [8, 9, 10, 11, 101, 467, 2204, 10431], 10431, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0035","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514d9430d3a58ebbf20228444b5c60455aa0dc4426f98372ed512475fd8379a6","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105035,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":2,\"target_n\":6}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [9, 10], 6, [9, 10, 60, 262, 1170, 5206, 23166], 23166, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0036","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e2101cc0676aad4030703f394a09abc3e8973e888e967100fcc24f77b0d490b","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105036,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":3,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [10, 11, 12], 8, [10, 11, 12, 66, 231, 835, 3031, 10992, 39871], 39871, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0037","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91708f665f1079c4877fbb64c4cf5c45e22e8c39659b6ed709bfcd5b091cc86b","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105037,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":4,\"target_n\":10}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [11, 12, 13, 14], 10, [11, 12, 13, 14, 259, 2231, 19483, 170313, 1488843, 13015015, 113773395], 113773395, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0038","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35fd56a75951d90674f13291a70f62013635cb270b4bdc527fb8bfe92e87537c","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105038,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":2,\"target_n\":5}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [12, 13], 5, [12, 13, 38, 89, 216, 521], 521, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0039","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84e41b8c047579900034f6643965d0ae5559e7b15830cc2529bcd77ee982d5ff","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105039,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":3,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [13, 14, 15], 7, [13, 14, 15, 173, 1127, 7485, 49765, 330785], 330785, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0040","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbcbf9357048262d7ae568b84a95ca8d23d67a3e7ad3796e3890257e88d1dae4","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105040,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":4,\"target_n\":9}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [14, 15, 16, 17], 9, [14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243], 374243, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0041","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070071d0f4745c6e04425447445f981c1cc25630a384ab70d62ed696d3585851","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105041,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":2,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [15, 16], 8, [15, 16, 92, 398, 1774, 7890, 35106, 156202, 695018], 695018, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0042","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f690ecff267ec47c7ee8aa7b00783a713fe74690bab2c0461ac712662c06d84","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105042,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":3,\"target_n\":6}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [16, 17, 18], 6, [16, 17, 18, 103, 361, 1306, 4742], 4742, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0043","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef9a93ce6ef90f9790c35c749572ad7f25da37c882ee1c30b7e68c8da4acdc7","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105043,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [17, 18, 19, 20], 8, [17, 18, 19, 20, 380, 3272, 28574, 249784, 2183564], 2183564, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0044","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60a3dc8292e151ebfb4f1b1cc2dfaab16970ef1e0a24ee3945daae666d60f4c3","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105044,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":2,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [18, 19], 7, [18, 19, 57, 134, 326, 787, 1901, 4590], 4590, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0045","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b88c576dd3832cf75803bcc569cfc078238a52e53193c9576671bf42b614ac12","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105045,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":3,\"target_n\":9}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [19, 20, 21], 9, [19, 20, 21, 246, 1602, 10640, 70742, 470218, 3125558, 20775706], 20775706, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0046","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70b928604efb1c09a0fd36bddb0c721bad9d0e059bc5e3830a0436f0f26dfbcc","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105046,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":4,\"target_n\":7}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0047","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"220174c8141d8a73a9eb21637f04faecc948924eb4ad15b74bf76ae007fce3e7","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105047,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":2,\"target_n\":6}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [21, 22], 6, [21, 22, 129, 559, 2493, 11089, 49341], 49341, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0048","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"533f0052f7a0af4ff002348864e522fef0e2c6fa12dc2e11678a1b39e70ca31c","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105048,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":3,\"target_n\":8}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (3, [22, 23, 24], 8, [22, 23, 24, 140, 491, 1777, 6453, 23404, 84895], 84895, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0049","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06f7de834357239f365046bde176e1632494f470a11b3e927393a1ddfa684728","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105049,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":4,\"target_n\":10}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (4, [23, 24, 25, 26], 10, [23, 24, 25, 26, 501, 4313, 37665, 329255, 2878285, 25161097, 219950837], 219950837, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0050","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f09b815cc8887123cd202df4e0ee92d6425aeb618f0e95e4d405cac40701522","family_id":"qf-mod41-recurrence-forward-generation","seed":202611105050,"semantic_template":"qf-mod41-recurrence-forward-generation/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":2,\"target_n\":5}"},"fields":{"title":"Shift the Full State and Append Each Exact New Scalar","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_values, target_n, prefix, target_value, certificate).","answer":{"kind":"expression","value":"(2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only)","explanation":"Generate the complete prefix through the target in declared coefficient order and retain the target value. Do not reverse the coefficient/state alignment or infer analytic behavior from finitely many terms. The exact result, retaining the requested order, is (2, [24, 25], 5, [24, 25, 76, 179, 436, 1053], 1053, exact_forward_application_only)."},"skills":["Shift the Full State and Append Each Exact New Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0051","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90500da11dc6e492228f0fcb2633b083d3dba271557c277a7460f12742b9c881","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105051,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":-2,\"initial_values\":[0,1],\"order\":2,\"target_n\":5}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-2 and initial_values=[0, 1], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 5, -11, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, -11, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0052","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b648392694ea79b2dfc8ed591577f3cbcccbd8f3572c87268031054c2a67c43","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105052,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":-1,\"initial_values\":[1,2,3],\"order\":3,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=-1 and initial_values=[1, 2, 3], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 7, 51919, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 7, 51919, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0053","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afcb90f19ba32d7ee7b1fdba443deab88c910535a86499eceea9fd6e23bd2744","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105053,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":0,\"initial_values\":[2,3,4,5],\"order\":4,\"target_n\":9}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=0 and initial_values=[2, 3, 4, 5], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 9, 93071, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 9, 93071, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0054","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"470bea95ab1363e18298beca911bba6b607f064f97abf4a96879dba76cc14622","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105054,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":1,\"initial_values\":[3,4],\"order\":2,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=1 and initial_values=[3, 4], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 8, 176819, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 8, 176819, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0055","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7943bd47eb994cfea4ea04be66363490049374c0c9d370b1b631addb1378c8ef","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105055,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":2,\"initial_values\":[4,5,6],\"order\":3,\"target_n\":6}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=2 and initial_values=[4, 5, 6], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 6, 1595, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 6, 1595, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0056","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee1e4b30aef0ebf29413febc39967f2c1062a89b0b616282debb5aca7fadfe8b","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105056,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":-2,\"initial_values\":[5,6,7,8],\"order\":4,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-2 and initial_values=[5, 6, 7, 8], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 8, 794122, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 8, 794122, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0057","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e080b5114b270127f4bdb825a3a33d912a2da768b35795582f9d98b9c6a5318c","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105057,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":-1,\"initial_values\":[6,7],\"order\":2,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=-1 and initial_values=[6, 7], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 7, 1484, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 7, 1484, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0058","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9528d8e15ddf6dc85b1b29ffb2d36be793d7141a8149a21bdfa629ebe1fca29","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105058,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":0,\"initial_values\":[7,8,9],\"order\":3,\"target_n\":9}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=0 and initial_values=[7, 8, 9], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 9, 8454528, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 9, 8454528, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0059","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ffcac41d088b18848ece60e5d587d27d6ae470bc3d110bb881378330996c02","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105059,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":1,\"initial_values\":[8,9,10,11],\"order\":4,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=1 and initial_values=[8, 9, 10, 11], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 7, 10431, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 7, 10431, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0060","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7fa001d4c5711ae9dadbca432167a40e5f70b4448f20ba98a5f4713ed39d99e","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105060,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":2,\"initial_values\":[9,10],\"order\":2,\"target_n\":6}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=2 and initial_values=[9, 10], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 6, 23166, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 6, 23166, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0061","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53cfc19bf2937dfb60673c3c25a2055f83baf4c81cf94c7e1270723393e694f4","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105061,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":-2,\"initial_values\":[10,11,12],\"order\":3,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 8, 39871, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 8, 39871, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0062","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bb8a69a0ec9cd4e457a28cf822110a7f0b726abf2e917df7eaabd8ad395020b","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105062,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":-1,\"initial_values\":[11,12,13,14],\"order\":4,\"target_n\":10}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=-1 and initial_values=[11, 12, 13, 14], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 10, 113773395, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 10, 113773395, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0063","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e8c019b751162915b6452e94ea70c2170c3b8378f2469f94a958764261f2e4","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105063,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":0,\"initial_values\":[12,13],\"order\":2,\"target_n\":5}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0064","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"369112a8b4d70fccb7ce138c9915424a47257b6ca74a4b265be6fb36034181c1","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105064,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":1,\"initial_values\":[13,14,15],\"order\":3,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=1 and initial_values=[13, 14, 15], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 7, 330785, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 7, 330785, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0065","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc4b131fb556e5fb0f90934f539554b8ca1fa45bf989dfb79956217e82404f9","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105065,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":2,\"initial_values\":[14,15,16,17],\"order\":4,\"target_n\":9}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0066","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4925010eed5f934e13969b562cb21b397e13f0e5f351658bb63bc70d30936c5","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105066,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":-2,\"initial_values\":[15,16],\"order\":2,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-2 and initial_values=[15, 16], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 8, 695018, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 8, 695018, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0067","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0913adb7d3c17721d3d3c149c4fdbd113002d90e02281e26c818825bd43c4ca3","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105067,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":-1,\"initial_values\":[16,17,18],\"order\":3,\"target_n\":6}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-1 and initial_values=[16, 17, 18], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 6, 4742, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 6, 4742, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0068","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0190620c6e2d4c8e4b4d780916c58b820be666e4b4fe11e9d08e65ac592821","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105068,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":0,\"initial_values\":[17,18,19,20],\"order\":4,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=0 and initial_values=[17, 18, 19, 20], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 8, 2183564, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 8, 2183564, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0069","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcc5dd5dbb75e44fe0c9f6bb298815b307716ba49756fb8ad92c0a0fb39df062","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105069,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":1,\"initial_values\":[18,19],\"order\":2,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=1 and initial_values=[18, 19], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 7, 4590, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 7, 4590, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0070","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b57fe4789490ab68518fe6f731b7e5f26c9421fbc0f611f074ab4b578fab8081","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105070,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6],\"constant\":2,\"initial_values\":[19,20,21],\"order\":3,\"target_n\":9}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[2, 4, 6] in oldest-to-newest order, constant=2 and initial_values=[19, 20, 21], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 9, 20775706, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 9, 20775706, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0071","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a489d6efbfe73b4d570fe27c1201ff46bc3202b20e30ac3252cc027ecf90e15f","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105071,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3,4],\"constant\":-2,\"initial_values\":[20,21,22,23],\"order\":4,\"target_n\":7}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=7. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 7, 22413, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 7, 22413, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0072","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"214d1f3dfb7d964205f86fb5618c15d461992e9aeba4a07421d8f285ad0011b2","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105072,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4],\"constant\":-1,\"initial_values\":[21,22],\"order\":2,\"target_n\":6}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[2, 4] in oldest-to-newest order, constant=-1 and initial_values=[21, 22], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=6. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 6, 49341, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 6, 49341, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0073","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"281d19da4c284bf3acf1a1c4d962ab459d7c114adaa401090c85c3588ca9ef59","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105073,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2,3],\"constant\":0,\"initial_values\":[22,23,24],\"order\":3,\"target_n\":8}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=0 and initial_values=[22, 23, 24], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=8. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(3, 3, 8, 84895, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (3, 3, 8, 84895, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0074","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c76ffa2b14d556531ebcd65d544a587206f9030535507aa0eea542c0c404f1c7","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105074,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[2,4,6,8],\"constant\":1,\"initial_values\":[23,24,25,26],\"order\":4,\"target_n\":10}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=4, coefficients=[2, 4, 6, 8] in oldest-to-newest order, constant=1 and initial_values=[23, 24, 25, 26], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=10. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(4, 4, 10, 219950837, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (4, 4, 10, 219950837, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0075","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90bb69dbc8da51fba071e7b18323ec1994e21a60691fe5454bf51f681b8016d4","family_id":"qf-mod41-recurrence-initial-value-uniqueness","seed":202611105075,"semantic_template":"qf-mod41-recurrence-initial-value-uniqueness/{\"coefficients\":[1,2],\"constant\":2,\"initial_values\":[24,25],\"order\":2,\"target_n\":5}"},"fields":{"title":"A Complete Initial State Gives a Unique Forward Prefix","stem":"Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=2 and initial_values=[24, 25], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).","answer":{"kind":"expression","value":"(2, 2, 5, 1053, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)","explanation":"Use the total single-valued update and every initial value to determine the target. The uniqueness certificate concerns this prefix under this exact specification, not a closed form or an unspecified multivalued recurrence. The exact result, retaining the requested order, is (2, 2, 5, 1053, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)."},"skills":["A Complete Initial State Gives a Unique Forward 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-20261014-advance-20261110-r05-recurrence-relation-0076","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa9408ae91ba822399664ba7a439369ceabb0cecdd95f7af59f3155913b87588","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105076,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,2],\"constant\":0,\"first_initial\":0,\"second_left\":1,\"second_right\":2,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 2] oldest-to-newest, constant=0, known a_0=0, alternative a_1=1 or 2, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (0, 1, 2, [0, 1, 2, 5, 12, 29, 70], [0, 2, 4, 10, 24, 58, 140], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0077","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34989b4dc1295b8069403d6e447da00c72fce99404a0bde161e2ece27f07ebae","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105077,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,3],\"constant\":1,\"first_initial\":1,\"second_left\":2,\"second_right\":3,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 3] oldest-to-newest, constant=1, known a_0=1, alternative a_1=2 or 3, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (1, 2, 3, [1, 2, 9, 32, 115, 410, 1461, 5204], [1, 3, 12, 43, 154, 549, 1956, 6967], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0078","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a068b2d7c46767d97ffde1446a7ebbef356c1d5917b9b76dcc68c09ad8bc104","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105078,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,4],\"constant\":2,\"first_initial\":2,\"second_left\":3,\"second_right\":4,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 4] oldest-to-newest, constant=2, known a_0=2, alternative a_1=3 or 4, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0079","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c299d14d57075224b22d994904b13d5c139722d7913cff6580b5fc33cf6165d2","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105079,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,5],\"constant\":3,\"first_initial\":3,\"second_left\":4,\"second_right\":5,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 5] oldest-to-newest, constant=3, known a_0=3, alternative a_1=4 or 5, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (3, 4, 5, [3, 4, 26, 137, 714, 3710, 19267], [3, 5, 31, 163, 849, 4411, 22907], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0080","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd750f8995ad84d50d0505763f76f26d1d0669b0f3808a79886503824564b354","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105080,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,6],\"constant\":0,\"first_initial\":4,\"second_left\":5,\"second_right\":6,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 6] oldest-to-newest, constant=0, known a_0=4, alternative a_1=5 or 6, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (4, 5, 6, [4, 5, 38, 238, 1504, 9500, 60008, 379048], [4, 6, 44, 276, 1744, 11016, 69584, 439536], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0081","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a68115652ac83267753ff38dfdd9771600257c4bc91b62b480abfb3701cfe04","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105081,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,2],\"constant\":1,\"first_initial\":5,\"second_left\":6,\"second_right\":7,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 2] oldest-to-newest, constant=1, known a_0=5, alternative a_1=6 or 7, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (5, 6, 7, [5, 6, 28, 75, 235, 696, 2098, 6285, 18865], [5, 7, 30, 82, 255, 757, 2280, 6832, 20505], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0082","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28ab9d65c4f83c426356c1533ad5625cba2712837b71b957ba493da71b875d1","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105082,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,3],\"constant\":2,\"first_initial\":6,\"second_left\":7,\"second_right\":8,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 3] oldest-to-newest, constant=2, known a_0=6, alternative a_1=7 or 8, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (6, 7, 8, [6, 7, 29, 96, 319, 1055, 3486], [6, 8, 32, 106, 352, 1164, 3846], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0083","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e26de653e26252ffd8dc2dde988c66b5406ed39b521fb0f3bfde295a2f2eb048","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105083,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,4],\"constant\":3,\"first_initial\":7,\"second_left\":8,\"second_right\":9,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 4] oldest-to-newest, constant=3, known a_0=7, alternative a_1=8 or 9, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (7, 8, 9, [7, 8, 49, 215, 961, 4277, 19033, 84689], [7, 9, 53, 233, 1041, 4633, 20617, 91737], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0084","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eaccb5a345ecce0b9559133667f7ad0374a722e698280659217095259b4091a","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105084,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,5],\"constant\":0,\"first_initial\":8,\"second_left\":9,\"second_right\":10,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 5] oldest-to-newest, constant=0, known a_0=8, alternative a_1=9 or 10, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (8, 9, 10, [8, 9, 69, 372, 2067, 11451, 63456, 351633, 1948533], [8, 10, 74, 400, 2222, 12310, 68216, 378010, 2094698], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0085","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3914108ef9431fd4a14560a95c1384a98fa3a945f15805cbef2dc852a48e2993","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105085,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,6],\"constant\":1,\"first_initial\":9,\"second_left\":10,\"second_right\":11,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 6] oldest-to-newest, constant=1, known a_0=9, alternative a_1=10 or 11, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (9, 10, 11, [9, 10, 70, 431, 2657, 16374, 100902], [9, 11, 76, 468, 2885, 17779, 109560], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0086","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f34d5ec3338460d84b6af338bcef17fe60bc7fd06a454777a99fe439ac3be0ee","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105086,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,2],\"constant\":2,\"first_initial\":10,\"second_left\":11,\"second_right\":12,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 2] oldest-to-newest, constant=2, known a_0=10, alternative a_1=11 or 12, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (10, 11, 12, [10, 11, 44, 112, 314, 854, 2338, 6386], [10, 12, 46, 118, 330, 898, 2458, 6714], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0087","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039d193b5ba093a65875133ff28299dad2c3fe41519d6ce1b4c8aaa2fafbbbc1","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105087,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,3],\"constant\":3,\"first_initial\":11,\"second_left\":12,\"second_right\":13,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 3] oldest-to-newest, constant=3, known a_0=11, alternative a_1=12 or 13, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0088","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1f9046036b039bb36652542369e659b22de3a74704800fba51172683a0e998f","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105088,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,4],\"constant\":0,\"first_initial\":12,\"second_left\":13,\"second_right\":14,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 4] oldest-to-newest, constant=0, known a_0=12, alternative a_1=13 or 14, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (12, 13, 14, [12, 13, 64, 269, 1140, 4829, 20456], [12, 14, 68, 286, 1212, 5134, 21748], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0089","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6347cad8727c4cc80f2c7bc1d248c1a911588de771497d7cfbef2349286c6f8","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105089,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,5],\"constant\":1,\"first_initial\":13,\"second_left\":14,\"second_right\":15,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 5] oldest-to-newest, constant=1, known a_0=13, alternative a_1=14 or 15, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (13, 14, 15, [13, 14, 97, 514, 2765, 14854, 79801, 428714], [13, 15, 102, 541, 2910, 15633, 83986, 451197], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0090","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21fe494760c036c3cd1aa550b45331a5b3c11d08e3a6ca99c41305f5becf8592","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105090,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,6],\"constant\":2,\"first_initial\":14,\"second_left\":15,\"second_right\":16,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 6] oldest-to-newest, constant=2, known a_0=14, alternative a_1=15 or 16, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (14, 15, 16, [14, 15, 134, 851, 5510, 35615, 230222, 1488179, 9619742], [14, 16, 140, 890, 5762, 37244, 240752, 1556246, 10059734], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0091","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"246b4cc477e11371610c5db9577abd7b8852ce205fa7ff9c05c059f2e7c0ad93","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105091,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,2],\"constant\":3,\"first_initial\":15,\"second_left\":16,\"second_right\":17,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 2] oldest-to-newest, constant=3, known a_0=15, alternative a_1=16 or 17, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (15, 16, 17, [15, 16, 50, 119, 291, 704, 1702], [15, 17, 52, 124, 303, 733, 1772], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0092","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"573a7b1869bf421bd23a454b42a756485dcce970da261232b807a38b34ab23fc","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105092,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,3],\"constant\":0,\"first_initial\":16,\"second_left\":17,\"second_right\":18,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 3] oldest-to-newest, constant=0, known a_0=16, alternative a_1=17 or 18, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (16, 17, 18, [16, 17, 83, 283, 1015, 3611, 12863, 45811], [16, 18, 86, 294, 1054, 3750, 13358, 47574], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0093","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13e76128a2191bc781e34f91c4814c7deb47db43241bf6287afd826d83f1713e","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105093,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,4],\"constant\":1,\"first_initial\":17,\"second_left\":18,\"second_right\":19,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 4] oldest-to-newest, constant=1, known a_0=17, alternative a_1=18 or 19, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (17, 18, 19, [17, 18, 124, 551, 2577, 11962, 55580, 258207, 1199569], [17, 19, 128, 570, 2665, 12371, 57480, 267034, 1240577], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0094","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43b7c7eed04fc7258ffe46ca9a2e0fb46090952e0963cf2d49fb5d44162139da","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105094,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,5],\"constant\":2,\"first_initial\":18,\"second_left\":19,\"second_right\":20,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 5] oldest-to-newest, constant=2, known a_0=18, alternative a_1=19 or 20, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (18, 19, 20, [18, 19, 115, 596, 3097, 16083, 83514], [18, 20, 120, 622, 3232, 16784, 87154], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0095","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a743fef9258615b118b85dc3198cb0f2268aabe65e08ea3d005dd46c363a4ca3","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105095,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,6],\"constant\":3,\"first_initial\":19,\"second_left\":20,\"second_right\":21,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 6] oldest-to-newest, constant=3, known a_0=19, alternative a_1=20 or 21, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (19, 20, 21, [19, 20, 161, 1009, 6379, 40295, 254531, 1607779], [19, 21, 167, 1047, 6619, 41811, 264107, 1668267], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0096","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e9db650e1a15b50bec9ebdf3f2ba67541f1e7522026ed3bcc1b24811b7a11d","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105096,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,2],\"constant\":0,\"first_initial\":20,\"second_left\":21,\"second_right\":22,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 2] oldest-to-newest, constant=0, known a_0=20, alternative a_1=21 or 22, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (20, 21, 22, [20, 21, 102, 267, 840, 2481, 7482, 22407, 67260], [20, 22, 104, 274, 860, 2542, 7664, 22954, 68900], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0097","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9313f776f858d4f3987cff0b6de4a5526eb08437f83fde49592bdc0165a96bd9","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105097,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,3],\"constant\":1,\"first_initial\":21,\"second_left\":22,\"second_right\":23,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 3] oldest-to-newest, constant=1, known a_0=21, alternative a_1=22 or 23, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (21, 22, 23, [21, 22, 88, 287, 950, 3138, 10365], [21, 23, 91, 297, 983, 3247, 10725], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0098","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36834c9b841195135a4839c2aa1613dc06f2df0f3b99e2e748b57455fa53d129","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105098,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[2,4],\"constant\":2,\"first_initial\":22,\"second_left\":23,\"second_right\":24,\"target_n\":7}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[2, 4] oldest-to-newest, constant=2, known a_0=22, alternative a_1=23 or 24, target_n=7. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (22, 23, 24, [22, 23, 138, 600, 2678, 11914, 53014, 235886], [22, 24, 142, 618, 2758, 12270, 54598, 242934], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0099","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ac001bd9bfec956139f20db6677bafadb712e864cc8b6834da965d5530f9e96","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105099,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[3,5],\"constant\":3,\"first_initial\":23,\"second_left\":24,\"second_right\":25,\"target_n\":8}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[3, 5] oldest-to-newest, constant=3, known a_0=23, alternative a_1=24 or 25, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (23, 24, 25, [23, 24, 192, 1035, 5754, 31878, 176655, 978912, 5424528], [23, 25, 197, 1063, 5909, 32737, 181415, 1005289, 5570693], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r05-recurrence-relation-0100","batch_id":"batch-20261014-advance-20261110-r05-recurrence-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0ae86b7ab50bb4ac25ce75a5860900336f95375de83ce443c3790e012374551","family_id":"qf-mod41-recurrence-missing-initial-boundary","seed":202611105100,"semantic_template":"qf-mod41-recurrence-missing-initial-boundary/{\"coefficients\":[1,6],\"constant\":0,\"first_initial\":24,\"second_left\":25,\"second_right\":26,\"target_n\":6}"},"fields":{"title":"Different Missing-Initial Completions Give Different Whole Prefixes","stem":"Use order2 coefficients=[1, 6] oldest-to-newest, constant=0, known a_0=24, alternative a_1=25 or 26, target_n=6. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary).","answer":{"kind":"expression","value":"(24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)","explanation":"Generate both prefixes including their initial values. They differ as prefixes because a_1 differs, even when the target values later coincide. The fixed No withholds uniqueness from only one initial value; it does not negate the separate prefixes_differ flag. The exact result, retaining the requested order, is (24, 25, 26, [24, 25, 174, 1069, 6588, 40597, 250170], [24, 26, 180, 1106, 6816, 42002, 258828], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique)."},"skills":["Different Missing-Initial Completions Give Different Whole Prefixes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}