{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0001","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c2d22ecded7110741da1cf120eeff50860afa3c830787d56733351cb041cf9","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215001,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[0,1],\"left_sign\":1,\"right\":[4,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=0/1, right=4/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(0, 4, 1, 1, No)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, 1, 1, No)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0002","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e5e1490a127bb54d85d62bc4900e6a81b421039376c9bbd09ec26251bc821a1","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215002,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[1,1],\"left_sign\":-1,\"right\":[5,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=1/1, right=5/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(1, 5, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0003","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2b9790cd568113bc5fbd58c0bb4df9caf484d12e9eaa2ea3e707471f716a7db","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215003,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[2,1],\"left_sign\":-1,\"right\":[6,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=2/1, right=6/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(2, 6, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0004","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"237e729d30fd860b54eb9995ac2ea6b2fb187aaa79fd8cbe848a4c75758c12af","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215004,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[3,1],\"left_sign\":-1,\"right\":[7,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=3/1, right=7/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(3, 7, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0005","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdfdb7cfcf08e8a39a9e8a5707fafe68f6638313343bcd34c3bb6531df972f1c","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215005,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[4,1],\"left_sign\":-1,\"right\":[8,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=4/1, right=8/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(4, 8, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0006","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88e86a1d2d42dfd37e3e9d845d239afdf05b69d1ca30a818f66a983b6e38dfb1","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215006,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[5,1],\"left_sign\":1,\"right\":[9,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=5/1, right=9/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(5, 9, 1, 1, No)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, 1, 1, No)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0007","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd43008893130be95836f90a73ddb9d122f0ff0a83970e72c8288aee54b652bb","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215007,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[6,1],\"left_sign\":-1,\"right\":[10,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=6/1, right=10/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(6, 10, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0008","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"176642f207c326bdb41cfe2d65516e500e0c4a3fbecfb1f1f33f97f014020a7e","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215008,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[7,1],\"left_sign\":-1,\"right\":[11,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=7/1, right=11/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(7, 11, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0009","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544a650e05138c972a17163fd26ab743ed3b11422b239168f717938042a73d1b","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215009,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[8,1],\"left_sign\":-1,\"right\":[12,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=8/1, right=12/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(8, 12, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0010","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f0cb87817e6682c320151635df16ac549cacdaea8decc10785bc0a1bcc4b9f4","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215010,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[9,1],\"left_sign\":-1,\"right\":[13,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=9/1, right=13/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(9, 13, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0011","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda09707f578cb4aeeac72faf0089ebdd38dfd058ca3346629da1143c500d480","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215011,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[10,1],\"left_sign\":1,\"right\":[14,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=10/1, right=14/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(10, 14, 1, 1, No)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, 1, 1, No)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0012","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37dbd75d0913e9070776d1d637c57df46375039c13fbb318c21e90cfe6ed472c","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215012,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[11,1],\"left_sign\":-1,\"right\":[15,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=11/1, right=15/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(11, 15, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0013","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d776aa467509212870d4513d7b07feea4a9345a0b95200f8babeb196129bf691","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215013,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[12,1],\"left_sign\":-1,\"right\":[16,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=12/1, right=16/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(12, 16, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0014","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df1691aa18c23ef36d9f2d74e14da0f6a9f07c69234459e29595228c845ed043","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215014,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[13,1],\"left_sign\":-1,\"right\":[17,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=13/1, right=17/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(13, 17, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0015","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a01a0e20b2098c9b2cefd1d95b90d01fa477ae5f1c18be557efaca330c07e10d","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215015,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[14,1],\"left_sign\":-1,\"right\":[18,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=14/1, right=18/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(14, 18, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0016","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"473e899381b6f9be1af53cca39d51bc7e00f88449acaa554cb7c5d8f9c3152bb","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215016,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[15,1],\"left_sign\":1,\"right\":[19,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(15, 19, 1, 1, No)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, 1, 1, No)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0017","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0300bdc343252b59b6606268b8fd633f6d603b4232634baaa43570cbdbaf2f5f","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215017,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[16,1],\"left_sign\":-1,\"right\":[20,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=16/1, right=20/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(16, 20, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0018","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d0e80df637689cf215b19c69da401666cdcba7070a7a9371f4551ca019efc36","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215018,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[17,1],\"left_sign\":-1,\"right\":[21,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=17/1, right=21/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(17, 21, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0019","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96606bc459f984a765b918e8ac6ebb715895b3adef766d82bf8610ee7ab8a7dc","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215019,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[18,1],\"left_sign\":-1,\"right\":[22,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=18/1, right=22/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(18, 22, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0020","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcfd8d021c1285ea641b0f763fbef6a597f0668a247fe8c85dbf64520281143d","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215020,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[19,1],\"left_sign\":-1,\"right\":[23,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=19/1, right=23/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(19, 23, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0021","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be0148068396944910b97dd9e43586115459c4978cda278c04ee8402f21bb3b6","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215021,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[20,1],\"left_sign\":1,\"right\":[24,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=20/1, right=24/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(20, 24, 1, 1, No)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, 1, 1, No)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0022","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08fe3e8831e0c2201bd70cc793e628cc9fd0375e8384c0f52da7bf864b1ea681","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215022,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[21,1],\"left_sign\":-1,\"right\":[25,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(21, 25, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0023","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8805193b0f23b5c3c65235fce6bbf9ebfe49db415c5486b69d81d410a79c5f05","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215023,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[22,1],\"left_sign\":-1,\"right\":[26,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=22/1, right=26/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(22, 26, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0024","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea298075cd0feef084f5ef87cb47f96564bce1aaca3c0a682a6a15bc3cc1c5ff","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215024,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[23,1],\"left_sign\":-1,\"right\":[27,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(23, 27, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0025","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"992064b0b3a3168317117500fb90b4500e8c54217a171dc8e0247296bd810e2e","family_id":"qf-app21-bisection-bracket-qualification","seed":202610215025,"semantic_template":"qf-app21-bisection-bracket-qualification/{\"left\":[24,1],\"left_sign\":-1,\"right\":[28,1],\"right_sign\":1}"},"fields":{"title":"Qualify Declared Sign Data without an Existence Claim","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=24/1, right=28/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).","answer":{"kind":"expression","value":"(24, 28, -1, 1, Yes)","explanation":"Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, 1, Yes)."},"skills":["Qualify Declared Sign Data without an Existence Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0026","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d641fc2cb6fe8d21f82eb9d56621e9f3f888c4f78bfec2da8b3d8ccca430c71","family_id":"qf-app21-bisection-single-update","seed":202610215026,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[0,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[4,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=0/1, right=4/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(0, 4, -1, -1, 1, 2, [2, 4, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, -1, -1, 1, 2, [2, 4, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0027","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"614d1d808e35bf4830dc32e3a5a2c848020927069fdb0fe8d0cada8164cf0baf","family_id":"qf-app21-bisection-single-update","seed":202610215027,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[1,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[5,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=1/1, right=5/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(1, 5, -1, 1, 1, 3, [1, 3, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, 1, 3, [1, 3, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0028","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa5e480be8a06f2dcaee0559c2b41e6e9d52d2364a78001fa98aa2f3f49f8f6c","family_id":"qf-app21-bisection-single-update","seed":202610215028,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[2,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[6,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=2/1, right=6/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(2, 6, -1, 0, 1, 4, [4, 4, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 0, 1, 4, [4, 4, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0029","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a223e9fc971120b464fcfab95217c4e23e26e28ec1eff9734b4428a88dde0eb3","family_id":"qf-app21-bisection-single-update","seed":202610215029,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[3,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[7,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=3/1, right=7/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(3, 7, -1, -1, 1, 5, [5, 7, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, -1, 1, 5, [5, 7, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0030","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdf79f6b4e225f50d9163ca6c68940fa4f47ca854f73c0b01d6297d2bdcf3d0a","family_id":"qf-app21-bisection-single-update","seed":202610215030,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[4,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[8,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=4/1, right=8/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(4, 8, -1, 1, 1, 6, [4, 6, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, 1, 6, [4, 6, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0031","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5087b2c851f1e8df8ec02e0b478cf44fff37652310b188974fa91bb1618e6092","family_id":"qf-app21-bisection-single-update","seed":202610215031,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[5,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[9,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=5/1, right=9/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(5, 9, -1, 0, 1, 7, [7, 7, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, -1, 0, 1, 7, [7, 7, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0032","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60bcb274c8fdfa72f34038e15e25bc27a2092fda14bcefbc37cddcec4e6eddcd","family_id":"qf-app21-bisection-single-update","seed":202610215032,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[6,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[10,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=6/1, right=10/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(6, 10, -1, -1, 1, 8, [8, 10, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, -1, 1, 8, [8, 10, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0033","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af4b8284ae1430aa40f3b932d6837d7750f457e04bcc513ed5cd5ba0c55429a","family_id":"qf-app21-bisection-single-update","seed":202610215033,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[7,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[11,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=7/1, right=11/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(7, 11, -1, 1, 1, 9, [7, 9, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, 1, 9, [7, 9, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0034","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e412c6cb78d850c0cd54a997c03ddc8e2581d93d79dcc0b142de871b9104ec03","family_id":"qf-app21-bisection-single-update","seed":202610215034,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[8,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[12,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=8/1, right=12/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(8, 12, -1, 0, 1, 10, [10, 10, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 0, 1, 10, [10, 10, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0035","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73f215b3abd1d276fc01c10a386d1cfe979097c71c26b06be020b28d5104f896","family_id":"qf-app21-bisection-single-update","seed":202610215035,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[9,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[13,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=9/1, right=13/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(9, 13, -1, -1, 1, 11, [11, 13, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, -1, 1, 11, [11, 13, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0036","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a19c922c4eaccaf7bd09ea26cf6dfe8a2712d08be37eb3b7e638f73abadfcc5","family_id":"qf-app21-bisection-single-update","seed":202610215036,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[10,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[14,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=10/1, right=14/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(10, 14, -1, 1, 1, 12, [10, 12, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, -1, 1, 1, 12, [10, 12, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0037","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"477ee298e53b216a405d2be84abd4df62de81429895fa5fa650537097a315283","family_id":"qf-app21-bisection-single-update","seed":202610215037,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[11,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[15,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=11/1, right=15/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(11, 15, -1, 0, 1, 13, [13, 13, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 0, 1, 13, [13, 13, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0038","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78a297a3f41bbf120e0ebee0aad77227fc1c1f2632f822eca7b37e8a9146bf7e","family_id":"qf-app21-bisection-single-update","seed":202610215038,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[12,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[16,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=12/1, right=16/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(12, 16, -1, -1, 1, 14, [14, 16, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, -1, 1, 14, [14, 16, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0039","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"761827a776bb79f5a2bc274d0463c601dfc71aa3b8aa6a9012435dbc61daacba","family_id":"qf-app21-bisection-single-update","seed":202610215039,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[13,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[17,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=13/1, right=17/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(13, 17, -1, 1, 1, 15, [13, 15, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, 1, 15, [13, 15, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0040","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef1d31bb7375f63b38eb087686bff59f27b6cd11665984df685f6239ae3a6258","family_id":"qf-app21-bisection-single-update","seed":202610215040,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[14,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[18,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=14/1, right=18/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(14, 18, -1, 0, 1, 16, [16, 16, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 0, 1, 16, [16, 16, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0041","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"887bdfa07665cbdbd2f1b7bbdc5de01b67f3d8c02bac0cabf33ca3d94668108a","family_id":"qf-app21-bisection-single-update","seed":202610215041,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[15,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[19,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(15, 19, -1, -1, 1, 17, [17, 19, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, -1, -1, 1, 17, [17, 19, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0042","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a361be35d1af68a72421cd00cce4bbaa2d6a5b12ac26f77259494606c077683b","family_id":"qf-app21-bisection-single-update","seed":202610215042,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[16,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[20,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=16/1, right=20/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(16, 20, -1, 1, 1, 18, [16, 18, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, 1, 18, [16, 18, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0043","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9852edffbc65028b0db1fa466ee8be6236b006b887607605780a5f900fba0a6d","family_id":"qf-app21-bisection-single-update","seed":202610215043,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[17,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[21,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=17/1, right=21/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(17, 21, -1, 0, 1, 19, [19, 19, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 0, 1, 19, [19, 19, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0044","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b73655ee62d0aa7bcf98a67499cfa5d78aa46fe08906140c39ded05393203f09","family_id":"qf-app21-bisection-single-update","seed":202610215044,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[18,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[22,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=18/1, right=22/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(18, 22, -1, -1, 1, 20, [20, 22, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, -1, 1, 20, [20, 22, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0045","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aee0082973c34e632605158c5fae16d502b9aa517f43a8c8aacdcd26d5c7a966","family_id":"qf-app21-bisection-single-update","seed":202610215045,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[19,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[23,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=19/1, right=23/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(19, 23, -1, 1, 1, 21, [19, 21, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, 1, 21, [19, 21, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0046","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6b6d4e13b69e53d59fc8e98c419e787af99d2bed3d58b035a516e4e36eebcf6","family_id":"qf-app21-bisection-single-update","seed":202610215046,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[20,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[24,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=20/1, right=24/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(20, 24, -1, 0, 1, 22, [22, 22, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, -1, 0, 1, 22, [22, 22, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0047","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23c109336581da54186932c838314ee0b792dd48aa036cd445c65ab573e6f5e1","family_id":"qf-app21-bisection-single-update","seed":202610215047,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[21,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[25,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(21, 25, -1, -1, 1, 23, [23, 25, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, -1, 1, 23, [23, 25, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0048","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bd413a9bfcc1eeff6dd0059297765183d3aaad914af2a84b6312cd1407aa157","family_id":"qf-app21-bisection-single-update","seed":202610215048,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[22,1],\"left_sign\":-1,\"mid_sign\":1,\"right\":[26,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=22/1, right=26/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(22, 26, -1, 1, 1, 24, [22, 24, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, 1, 24, [22, 24, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0049","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81401cfe1c40af972e340defe23fff873b605a35e8607db572dc81d3238c385f","family_id":"qf-app21-bisection-single-update","seed":202610215049,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[23,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[27,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(23, 27, -1, 0, 1, 25, [25, 25, 0, 0])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 0, 1, 25, [25, 25, 0, 0])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0050","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07a91bd3a34bee412760f717c145aaa002c571609974cd5f057fb61b3ea0bc1a","family_id":"qf-app21-bisection-single-update","seed":202610215050,"semantic_template":"qf-app21-bisection-single-update/{\"left\":[24,1],\"left_sign\":-1,\"mid_sign\":-1,\"right\":[28,1],\"right_sign\":1}"},"fields":{"title":"One Sign-Bracket Update with Explicit Endpoint-Zero Priority","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=24/1, right=28/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.","answer":{"kind":"expression","value":"(24, 28, -1, -1, 1, 26, [26, 28, -1, 1])","explanation":"Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, -1, 1, 26, [26, 28, -1, 1])."},"skills":["One Sign-Bracket Update with Explicit Endpoint-Zero Priority"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0051","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c60b28a512cdfe673569bfe5cf76ea4168d810b999cb0ba02d775e53f9a7112","family_id":"qf-app21-bisection-finite-trace","seed":202610215051,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[0,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[4,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=0/1, right=4/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(0, 4, -1, 1, [1], [[0, 4, 2, 1, 0, 2]], 0, 2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, -1, 1, [1], [[0, 4, 2, 1, 0, 2]], 0, 2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0052","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28877b4480636aea09183e4aa53fd883d0c383f3cd36862a4a822caed1804715","family_id":"qf-app21-bisection-finite-trace","seed":202610215052,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[1,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[5,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=1/1, right=5/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(1, 5, -1, 1, [1, 1], [[1, 5, 3, 1, 1, 3], [1, 3, 2, 1, 1, 2]], 1, 2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, [1, 1], [[1, 5, 3, 1, 1, 3], [1, 3, 2, 1, 1, 2]], 1, 2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0053","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e98f1e4254d371b272193213d4803c5badbc2347b93f346988f8bacb39f5a3a1","family_id":"qf-app21-bisection-finite-trace","seed":202610215053,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[2,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[6,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=2/1, right=6/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(2, 6, -1, 1, [1, 1, 1], [[2, 6, 4, 1, 2, 4], [2, 4, 3, 1, 2, 3], [2, 3, 5/2, 1, 2, 5/2]], 2, 5/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 1, [1, 1, 1], [[2, 6, 4, 1, 2, 4], [2, 4, 3, 1, 2, 3], [2, 3, 5/2, 1, 2, 5/2]], 2, 5/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0054","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9367bf696e506e5b3fb7d4eb7af03e284e77bf52397a1646dc4afd8c6272f194","family_id":"qf-app21-bisection-finite-trace","seed":202610215054,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[3,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[7,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=3/1, right=7/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(3, 7, -1, 1, [1, 1, 1, -1], [[3, 7, 5, 1, 3, 5], [3, 5, 4, 1, 3, 4], [3, 4, 7/2, 1, 3, 7/2], [3, 7/2, 13/4, -1, 13/4, 7/2]], 13/4, 7/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, 1, [1, 1, 1, -1], [[3, 7, 5, 1, 3, 5], [3, 5, 4, 1, 3, 4], [3, 4, 7/2, 1, 3, 7/2], [3, 7/2, 13/4, -1, 13/4, 7/2]], 13/4, 7/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0055","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a3dca376b23a8c802d5b09ec537a3cb6106a043c9e2e5c0804b40d037e57353","family_id":"qf-app21-bisection-finite-trace","seed":202610215055,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[4,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[8,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=4/1, right=8/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(4, 8, -1, 1, [1], [[4, 8, 6, 1, 4, 6]], 4, 6)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, [1], [[4, 8, 6, 1, 4, 6]], 4, 6)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0056","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"557dd5dfcb509ec17e5e5f71ff7ed3118f62bf9e4147f558eaab48ffeb2b039d","family_id":"qf-app21-bisection-finite-trace","seed":202610215056,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[5,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[9,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=5/1, right=9/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(5, 9, -1, 1, [1, 1], [[5, 9, 7, 1, 5, 7], [5, 7, 6, 1, 5, 6]], 5, 6)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, -1, 1, [1, 1], [[5, 9, 7, 1, 5, 7], [5, 7, 6, 1, 5, 6]], 5, 6)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0057","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f7a5f01f9b2a4cef11cb65e9ecf7f01d64cc344a9e64a956b7dab36b30eddd0","family_id":"qf-app21-bisection-finite-trace","seed":202610215057,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[6,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[10,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=6/1, right=10/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(6, 10, -1, 1, [1, 1, 1], [[6, 10, 8, 1, 6, 8], [6, 8, 7, 1, 6, 7], [6, 7, 13/2, 1, 6, 13/2]], 6, 13/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, 1, [1, 1, 1], [[6, 10, 8, 1, 6, 8], [6, 8, 7, 1, 6, 7], [6, 7, 13/2, 1, 6, 13/2]], 6, 13/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0058","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aada08ab63ebf0cdda35b648428e90045022edc64baaf458cff4f56b066e0df","family_id":"qf-app21-bisection-finite-trace","seed":202610215058,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[7,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[11,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=7/1, right=11/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(7, 11, -1, 1, [1, 1, 1, -1], [[7, 11, 9, 1, 7, 9], [7, 9, 8, 1, 7, 8], [7, 8, 15/2, 1, 7, 15/2], [7, 15/2, 29/4, -1, 29/4, 15/2]], 29/4, 15/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, [1, 1, 1, -1], [[7, 11, 9, 1, 7, 9], [7, 9, 8, 1, 7, 8], [7, 8, 15/2, 1, 7, 15/2], [7, 15/2, 29/4, -1, 29/4, 15/2]], 29/4, 15/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0059","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24dc26d4e79aa4188ffd676e4624032bb1630420a5f716e82640b2084eb4e340","family_id":"qf-app21-bisection-finite-trace","seed":202610215059,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[8,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[12,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=8/1, right=12/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(8, 12, -1, 1, [1], [[8, 12, 10, 1, 8, 10]], 8, 10)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 1, [1], [[8, 12, 10, 1, 8, 10]], 8, 10)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0060","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969217750201ac1b89faa5f9eb13630ac4cd6b21879a97b1be6dccc89f3beffe","family_id":"qf-app21-bisection-finite-trace","seed":202610215060,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[9,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[13,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=9/1, right=13/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(9, 13, -1, 1, [1, 1], [[9, 13, 11, 1, 9, 11], [9, 11, 10, 1, 9, 10]], 9, 10)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, 1, [1, 1], [[9, 13, 11, 1, 9, 11], [9, 11, 10, 1, 9, 10]], 9, 10)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0061","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b89e4f14a5e8ae5c2fc069e6ece36cbdc7f679ad0b2b9c09e148ee1b378f3daf","family_id":"qf-app21-bisection-finite-trace","seed":202610215061,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[10,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[14,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=10/1, right=14/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(10, 14, -1, 1, [1, 1, 1], [[10, 14, 12, 1, 10, 12], [10, 12, 11, 1, 10, 11], [10, 11, 21/2, 1, 10, 21/2]], 10, 21/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, -1, 1, [1, 1, 1], [[10, 14, 12, 1, 10, 12], [10, 12, 11, 1, 10, 11], [10, 11, 21/2, 1, 10, 21/2]], 10, 21/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0062","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcfd7e17f57f5bd3e8121e52e94c189fb6d875aff3ea9418dfa901c9d022b4ff","family_id":"qf-app21-bisection-finite-trace","seed":202610215062,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[11,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[15,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=11/1, right=15/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(11, 15, -1, 1, [1, 1, 1, -1], [[11, 15, 13, 1, 11, 13], [11, 13, 12, 1, 11, 12], [11, 12, 23/2, 1, 11, 23/2], [11, 23/2, 45/4, -1, 45/4, 23/2]], 45/4, 23/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 1, [1, 1, 1, -1], [[11, 15, 13, 1, 11, 13], [11, 13, 12, 1, 11, 12], [11, 12, 23/2, 1, 11, 23/2], [11, 23/2, 45/4, -1, 45/4, 23/2]], 45/4, 23/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0063","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4468ce4ce54bce7f6bbe463e2f7250f8278463114797743a820a30bffaa9460e","family_id":"qf-app21-bisection-finite-trace","seed":202610215063,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[12,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[16,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=12/1, right=16/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(12, 16, -1, 1, [1], [[12, 16, 14, 1, 12, 14]], 12, 14)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, 1, [1], [[12, 16, 14, 1, 12, 14]], 12, 14)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0064","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca4e7e9c80c6817f34f604e839e52b8ba508848344a53ee2d1bf66d30392e3b","family_id":"qf-app21-bisection-finite-trace","seed":202610215064,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[13,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[17,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=13/1, right=17/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(13, 17, -1, 1, [1, 1], [[13, 17, 15, 1, 13, 15], [13, 15, 14, 1, 13, 14]], 13, 14)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, [1, 1], [[13, 17, 15, 1, 13, 15], [13, 15, 14, 1, 13, 14]], 13, 14)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0065","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c2395e02273f8f6dc411f148da6a8b4e57d86078d61dab6e8e15f1b1cd281e4","family_id":"qf-app21-bisection-finite-trace","seed":202610215065,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[14,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[18,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=14/1, right=18/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(14, 18, -1, 1, [1, 1, 1], [[14, 18, 16, 1, 14, 16], [14, 16, 15, 1, 14, 15], [14, 15, 29/2, 1, 14, 29/2]], 14, 29/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 1, [1, 1, 1], [[14, 18, 16, 1, 14, 16], [14, 16, 15, 1, 14, 15], [14, 15, 29/2, 1, 14, 29/2]], 14, 29/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0066","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ebae86305f2f32eda7ecceb0eec6fdffc5f06073f9c02371cf89fa89b0250bc","family_id":"qf-app21-bisection-finite-trace","seed":202610215066,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[15,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[19,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=15/1, right=19/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(15, 19, -1, 1, [1, 1, 1, -1], [[15, 19, 17, 1, 15, 17], [15, 17, 16, 1, 15, 16], [15, 16, 31/2, 1, 15, 31/2], [15, 31/2, 61/4, -1, 61/4, 31/2]], 61/4, 31/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, -1, 1, [1, 1, 1, -1], [[15, 19, 17, 1, 15, 17], [15, 17, 16, 1, 15, 16], [15, 16, 31/2, 1, 15, 31/2], [15, 31/2, 61/4, -1, 61/4, 31/2]], 61/4, 31/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0067","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c533f37b0f39a2071d0f59786e108ffc904cc1102240081d89430b6d2c83e410","family_id":"qf-app21-bisection-finite-trace","seed":202610215067,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[16,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[20,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=16/1, right=20/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(16, 20, -1, 1, [1], [[16, 20, 18, 1, 16, 18]], 16, 18)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, [1], [[16, 20, 18, 1, 16, 18]], 16, 18)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0068","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cb6a13865ba51c73e62c320d4095267fe575a6c666e9d6dd1765dbe90694c16","family_id":"qf-app21-bisection-finite-trace","seed":202610215068,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[17,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[21,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=17/1, right=21/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(17, 21, -1, 1, [1, 1], [[17, 21, 19, 1, 17, 19], [17, 19, 18, 1, 17, 18]], 17, 18)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 1, [1, 1], [[17, 21, 19, 1, 17, 19], [17, 19, 18, 1, 17, 18]], 17, 18)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0069","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c0b8ac50494a88cf5cd21cfcb91e55cca7a446be55093d020f5ec1ba573d94f","family_id":"qf-app21-bisection-finite-trace","seed":202610215069,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[18,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[22,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=18/1, right=22/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(18, 22, -1, 1, [1, 1, 1], [[18, 22, 20, 1, 18, 20], [18, 20, 19, 1, 18, 19], [18, 19, 37/2, 1, 18, 37/2]], 18, 37/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, 1, [1, 1, 1], [[18, 22, 20, 1, 18, 20], [18, 20, 19, 1, 18, 19], [18, 19, 37/2, 1, 18, 37/2]], 18, 37/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0070","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c871f60a6edb11ad0bae5ac317e8261cf013874a9ed40ef0d523da76501bf18","family_id":"qf-app21-bisection-finite-trace","seed":202610215070,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[19,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[23,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=19/1, right=23/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(19, 23, -1, 1, [1, 1, 1, -1], [[19, 23, 21, 1, 19, 21], [19, 21, 20, 1, 19, 20], [19, 20, 39/2, 1, 19, 39/2], [19, 39/2, 77/4, -1, 77/4, 39/2]], 77/4, 39/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, [1, 1, 1, -1], [[19, 23, 21, 1, 19, 21], [19, 21, 20, 1, 19, 20], [19, 20, 39/2, 1, 19, 39/2], [19, 39/2, 77/4, -1, 77/4, 39/2]], 77/4, 39/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0071","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1df37e0bea78744a61ec50a18de0224adc02142a49fa49b197c5f681fde75f74","family_id":"qf-app21-bisection-finite-trace","seed":202610215071,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[20,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[24,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=20/1, right=24/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(20, 24, -1, 1, [1], [[20, 24, 22, 1, 20, 22]], 20, 22)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, -1, 1, [1], [[20, 24, 22, 1, 20, 22]], 20, 22)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0072","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2278385c3ac5a6c4efb51e5d449465738baf10919666dd242c6f69cd6716e93b","family_id":"qf-app21-bisection-finite-trace","seed":202610215072,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[21,1],\"left_sign\":-1,\"mid_signs\":[1,1],\"right\":[25,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=21/1, right=25/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(21, 25, -1, 1, [1, 1], [[21, 25, 23, 1, 21, 23], [21, 23, 22, 1, 21, 22]], 21, 22)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, 1, [1, 1], [[21, 25, 23, 1, 21, 23], [21, 23, 22, 1, 21, 22]], 21, 22)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0073","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28b798fbe3a812e400965957fdf438ab4dd6145bc95deb34c193ef3d9330e976","family_id":"qf-app21-bisection-finite-trace","seed":202610215073,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[22,1],\"left_sign\":-1,\"mid_signs\":[1,1,1],\"right\":[26,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=22/1, right=26/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(22, 26, -1, 1, [1, 1, 1], [[22, 26, 24, 1, 22, 24], [22, 24, 23, 1, 22, 23], [22, 23, 45/2, 1, 22, 45/2]], 22, 45/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, [1, 1, 1], [[22, 26, 24, 1, 22, 24], [22, 24, 23, 1, 22, 23], [22, 23, 45/2, 1, 22, 45/2]], 22, 45/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0074","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66178e5f0145920a84354d0a0c56ca5d2ba52cd5768cc9adc4f0e5e781a76425","family_id":"qf-app21-bisection-finite-trace","seed":202610215074,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[23,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[27,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=23/1, right=27/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0075","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb482550ba976f6e26904995c867c1bb711d5365a32867a105371a19f87bb0d9","family_id":"qf-app21-bisection-finite-trace","seed":202610215075,"semantic_template":"qf-app21-bisection-finite-trace/{\"left\":[24,1],\"left_sign\":-1,\"mid_signs\":[1],\"right\":[28,1],\"right_sign\":1}"},"fields":{"title":"A Finite Trace Stops at the First Declared Zero Midpoint","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=24/1, right=28/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.","answer":{"kind":"expression","value":"(24, 28, -1, 1, [1], [[24, 28, 26, 1, 24, 26]], 24, 26)","explanation":"Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, 1, [1], [[24, 28, 26, 1, 24, 26]], 24, 26)."},"skills":["A Finite Trace Stops at the First Declared Zero Midpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0076","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b2bacf723fa34d1d4124703f9475ea0f3fcf2c2091edb9fa7c398b413f6ad6e","family_id":"qf-app21-bisection-width-after-steps","seed":202610215076,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[-3,1],\"right\":[5,1],\"steps\":0}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-3/1, right=5/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(-3, 5, 0, 8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, 5, 0, 8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0077","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f133342c4ed1238885eb0dbf3576465cfa9a1735bdbfca5e93fc4c7014c27a7","family_id":"qf-app21-bisection-width-after-steps","seed":202610215077,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[-2,1],\"right\":[6,1],\"steps\":1}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-2/1, right=6/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(-2, 6, 1, 4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 6, 1, 4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0078","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9943adf8756a3becd4e8063f6a67937b0188f89a163b573962ea1a44ab6edfa6","family_id":"qf-app21-bisection-width-after-steps","seed":202610215078,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[-1,1],\"right\":[7,1],\"steps\":2}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-1/1, right=7/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(-1, 7, 2, 2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 7, 2, 2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0079","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf4d89e3d166dcb6df3491ed50993d22dba18b17174c0578b5ea90653c294b9d","family_id":"qf-app21-bisection-width-after-steps","seed":202610215079,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[0,1],\"right\":[8,1],\"steps\":3}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=0/1, right=8/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(0, 8, 3, 1)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 8, 3, 1)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0080","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"432cf67351c31e7c9434abacf66d7e61772b17b378d3a238220ff1ad6ba70704","family_id":"qf-app21-bisection-width-after-steps","seed":202610215080,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[1,1],\"right\":[9,1],\"steps\":4}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=1/1, right=9/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(1, 9, 4, 1/2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 9, 4, 1/2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0081","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67b0c73a205bee3f0881632944e5fc50cbed358eb3356efc8d7aa3a9e07f70b","family_id":"qf-app21-bisection-width-after-steps","seed":202610215081,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[2,1],\"right\":[10,1],\"steps\":5}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=2/1, right=10/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(2, 10, 5, 1/4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 10, 5, 1/4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0082","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c478ecb7aa5c2f24401768012f47775a1b10c9341adb58e922181aefb2d7da","family_id":"qf-app21-bisection-width-after-steps","seed":202610215082,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[3,1],\"right\":[11,1],\"steps\":6}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=3/1, right=11/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(3, 11, 6, 1/8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 11, 6, 1/8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0083","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1246a6c66cbc7ad6600770a96b6149626379aa45961d510b6839da705c2973d","family_id":"qf-app21-bisection-width-after-steps","seed":202610215083,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[4,1],\"right\":[12,1],\"steps\":0}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=4/1, right=12/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(4, 12, 0, 8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 12, 0, 8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0084","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20eee204cb9e2753c4239a853fd941e9f100c5de2194c6b97d8fc871c5afbf3c","family_id":"qf-app21-bisection-width-after-steps","seed":202610215084,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[5,1],\"right\":[13,1],\"steps\":1}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=5/1, right=13/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(5, 13, 1, 4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 13, 1, 4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0085","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1062b71636927512b3b56854d70d04f1a85f04436e00bb10b293ce2f1a5ba111","family_id":"qf-app21-bisection-width-after-steps","seed":202610215085,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[6,1],\"right\":[14,1],\"steps\":2}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=6/1, right=14/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(6, 14, 2, 2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 14, 2, 2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0086","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a2efc8f642f7b0350f26665f3aaf34981126f028c0dc33f44756c9181c39eae","family_id":"qf-app21-bisection-width-after-steps","seed":202610215086,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[7,1],\"right\":[15,1],\"steps\":3}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=7/1, right=15/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(7, 15, 3, 1)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 15, 3, 1)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0087","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f4eb9ef81d5aac8d709bebc29c865b42c644e6bf37b9e46b10df83ce54ead9","family_id":"qf-app21-bisection-width-after-steps","seed":202610215087,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[8,1],\"right\":[16,1],\"steps\":4}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=8/1, right=16/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(8, 16, 4, 1/2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 16, 4, 1/2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0088","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"677b8660436e98055780bd618974348e2f63222d9d6ea9ee597a4d8a44256aa3","family_id":"qf-app21-bisection-width-after-steps","seed":202610215088,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[9,1],\"right\":[17,1],\"steps\":5}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=9/1, right=17/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(9, 17, 5, 1/4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 17, 5, 1/4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0089","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08667e171cf5f0624406d401863efacce4c1418cf7a1cacdcc251b95d906545a","family_id":"qf-app21-bisection-width-after-steps","seed":202610215089,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[10,1],\"right\":[18,1],\"steps\":6}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=10/1, right=18/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(10, 18, 6, 1/8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 18, 6, 1/8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0090","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1736839c9067e4094e75486447b3b79e04eec66d7078c41d1e50ebab7f8b6be1","family_id":"qf-app21-bisection-width-after-steps","seed":202610215090,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[11,1],\"right\":[19,1],\"steps\":0}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=11/1, right=19/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(11, 19, 0, 8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 19, 0, 8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0091","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39bb2d9dd1b212938268a3620727e7fe644d242a03952816ff470d2c75fb5041","family_id":"qf-app21-bisection-width-after-steps","seed":202610215091,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[12,1],\"right\":[20,1],\"steps\":1}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=12/1, right=20/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(12, 20, 1, 4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 20, 1, 4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0092","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef3709bab30f95f6e83baaf081d919167182de292e37846e5dbfa2ba57d9a56f","family_id":"qf-app21-bisection-width-after-steps","seed":202610215092,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[13,1],\"right\":[21,1],\"steps\":2}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=13/1, right=21/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(13, 21, 2, 2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 21, 2, 2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0093","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9039346d74beb009aabcfecb950b8c27dde5afb0abc2fd15312ff3f0fcb4a0d4","family_id":"qf-app21-bisection-width-after-steps","seed":202610215093,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[14,1],\"right\":[22,1],\"steps\":3}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=14/1, right=22/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(14, 22, 3, 1)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 22, 3, 1)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0094","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02f34eb73ab125d2ee9276da18d8bb2a4ec868f39b0fddd688aa8bd47064e3a8","family_id":"qf-app21-bisection-width-after-steps","seed":202610215094,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[15,1],\"right\":[23,1],\"steps\":4}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=15/1, right=23/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(15, 23, 4, 1/2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 23, 4, 1/2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0095","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97e0948a5e681ce0443f284c79a5ba9beaef06688c5f8a76a178ff34f6e1b2bb","family_id":"qf-app21-bisection-width-after-steps","seed":202610215095,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[16,1],\"right\":[24,1],\"steps\":5}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=16/1, right=24/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(16, 24, 5, 1/4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 24, 5, 1/4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0096","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475835e2599c5be3e80a7b8b90107f15c731c935e2ec87241878b7e17146e2b1","family_id":"qf-app21-bisection-width-after-steps","seed":202610215096,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[17,1],\"right\":[25,1],\"steps\":6}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=17/1, right=25/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(17, 25, 6, 1/8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 25, 6, 1/8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0097","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97364a4d71515d191f69e1b78bf30ff35dd22be6308067fa86bedfe2497170f6","family_id":"qf-app21-bisection-width-after-steps","seed":202610215097,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[18,1],\"right\":[26,1],\"steps\":0}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=18/1, right=26/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(18, 26, 0, 8)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 26, 0, 8)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0098","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbea5f121e63f2a7c5c2dc8043a9912c27a88d9566eb5f442a6406d0029bd473","family_id":"qf-app21-bisection-width-after-steps","seed":202610215098,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[19,1],\"right\":[27,1],\"steps\":1}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=19/1, right=27/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(19, 27, 1, 4)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 27, 1, 4)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0099","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54cfb6e033e4684e18dd77be386c5390ae22689ff753d1441baf569cf6cef394","family_id":"qf-app21-bisection-width-after-steps","seed":202610215099,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[20,1],\"right\":[28,1],\"steps\":2}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=20/1, right=28/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(20, 28, 2, 2)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 28, 2, 2)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r05-bisection-method-0100","batch_id":"batch-20261013-advance-20261021-r05-bisection-method-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d3858b70e81eb8973a43c5ca808a28039272458dde233ad2d74442f26f4cbb4","family_id":"qf-app21-bisection-width-after-steps","seed":202610215100,"semantic_template":"qf-app21-bisection-width-after-steps/{\"left\":[21,1],\"right\":[29,1],\"steps\":3}"},"fields":{"title":"Geometric Width after a Declared Number of Halvings","stem":"Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=21/1, right=29/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.","answer":{"kind":"expression","value":"(21, 29, 3, 1)","explanation":"Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 29, 3, 1)."},"skills":["Geometric Width after a Declared Number of Halvings"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}