{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0001","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1963e04cdd901b2ce1b663fef0c036e046fe5f0ca25db5283e5b11595ccf530","family_id":"qf-density15-composition-compatible-path-table","seed":202610154001,"semantic_template":"qf-density15-composition-compatible-path-table/[0,0,0,[],[]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(0)=∅→Fin(0)→Fin(0); complete f=[], g=[]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([0, 0, 0], [], [])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([0, 0, 0], [], [])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0002","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db2aed9381a69c5f1b23263b09d0163e6f69fe7b41022e9191e60608c1060383","family_id":"qf-density15-composition-compatible-path-table","seed":202610154002,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,0,0],[3,3,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0003","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f8434f5c4e86540049401e37cd4000c4348fc592b7fa199be77622f4eb128d7","family_id":"qf-density15-composition-compatible-path-table","seed":202610154003,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,0,0],[1,3,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0004","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f2acb02c6d7fa52542f00da9a80dfd926438fc74dd6436e8542093b5bb35c0","family_id":"qf-density15-composition-compatible-path-table","seed":202610154004,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,0,0],[0,3,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0005","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7f5cff3269f7d9c583b779c8499f3e57bfd8d9f775498cfe6dd07e30bdcb196","family_id":"qf-density15-composition-compatible-path-table","seed":202610154005,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,1,0],[3,1,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0006","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc18fdba936c5b5a413ca7fa1dafbf80aa7a100c0a15f714dd267033a07f1d7c","family_id":"qf-density15-composition-compatible-path-table","seed":202610154006,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,1,0],[1,1,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0007","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb97ddede4d45bb7d9aafab4db680b1121da6d6f59a6fde2fe9b54e2914c75dc","family_id":"qf-density15-composition-compatible-path-table","seed":202610154007,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,1,0],[0,1,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0008","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0b1fd9915472c702642d05ffd9878e7a2b34dbc4b94dfa85edff74cb62f5325","family_id":"qf-density15-composition-compatible-path-table","seed":202610154008,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,2,0],[3,0,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0009","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e804dcc04ccac5936fc6d5e4dfbaa2e77e1ac8e3893a413cbd70f38faf360cb3","family_id":"qf-density15-composition-compatible-path-table","seed":202610154009,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,2,0],[1,0,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0010","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12fa3a5e4aa7663e1c3c29b5347a58eddd2fc93d4909466339e30b3b6298eec1","family_id":"qf-density15-composition-compatible-path-table","seed":202610154010,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,2,0],[0,0,3]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0011","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6be7277389e1ae9b995ed735f5ba67b7068db734e74da0eced105ae1158ca7bd","family_id":"qf-density15-composition-compatible-path-table","seed":202610154011,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,0,1],[3,3,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0012","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb2f2b6d52799da60ef9624664c857481bdea79017b06994ed92a5ba54df7c48","family_id":"qf-density15-composition-compatible-path-table","seed":202610154012,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,0,1],[1,3,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0013","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe40799da5478a31f749cdca02950f60c3ccdc62aebe8c3a05632d7a58946e7","family_id":"qf-density15-composition-compatible-path-table","seed":202610154013,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,0,1],[0,3,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0014","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c001b90cd01e7df4ff8e5ec2bce4fa995efe51294d9db145e866cb53060eb7a","family_id":"qf-density15-composition-compatible-path-table","seed":202610154014,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,1,1],[3,1,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0015","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ded8877c938fa0adec5c2c9b919f10b7f370048160feba72e482fd73e2cad3b","family_id":"qf-density15-composition-compatible-path-table","seed":202610154015,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,1,1],[1,1,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0016","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d84c8ac59b4058a780e1fe443836d60b6aa49f7a9cf1b653096ca37ab29e43b","family_id":"qf-density15-composition-compatible-path-table","seed":202610154016,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,1,1],[0,1,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0017","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e01bb6751017d4ee59de4992ca1147875c57df779bd03d8c4caf2df8fb27b628","family_id":"qf-density15-composition-compatible-path-table","seed":202610154017,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,2,1],[3,0,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0018","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"488ef117a123c657182f9f0653a31db84108691f743a392fabb27eeeda9e2e30","family_id":"qf-density15-composition-compatible-path-table","seed":202610154018,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,2,1],[1,0,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0019","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43b143d120c8b757e9f8133f5c65103e6af7cbf9f5727df22e37d1756e6fe548","family_id":"qf-density15-composition-compatible-path-table","seed":202610154019,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,2,1],[0,0,1]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0020","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da107f6c1e3b0e2d3917f627f6516e170ee94c334ffd741f0f084e35de8f7e2d","family_id":"qf-density15-composition-compatible-path-table","seed":202610154020,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,0,2],[3,3,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0021","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95af0617b9f75eebded18511d7e5ff6715ed09141465785034421ec198ab66d","family_id":"qf-density15-composition-compatible-path-table","seed":202610154021,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,0,2],[1,3,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0022","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dcc1019fb940ae7f3ebd10f3dd90dd2447224172a0c0faedef831254e901605","family_id":"qf-density15-composition-compatible-path-table","seed":202610154022,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,0,2],[0,3,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0023","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15b9f54fcc1a96e623bbbac8ef0c9307bf5a75bc2d2581f0cce3c0ab129d3db1","family_id":"qf-density15-composition-compatible-path-table","seed":202610154023,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[0,1,2],[3,1,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0024","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1424e2f9f9a82af58dd67a58315675c540390e612667b280e55e6b1dfc994252","family_id":"qf-density15-composition-compatible-path-table","seed":202610154024,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[1,1,2],[1,1,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0025","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc4cc644429ad9b94935ef4fb4d23da614340e1e9f4064d55b29e3ac0349d77c","family_id":"qf-density15-composition-compatible-path-table","seed":202610154025,"semantic_template":"qf-density15-composition-compatible-path-table/[3,3,4,[2,1,2],[0,1,0]]"},"fields":{"title":"Complete Compatible Composition Paths and Range","stem":"First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.","answer":{"kind":"expression","value":"([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1])","explanation":"Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1])."},"skills":["Complete Compatible Composition Paths and Range"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0026","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8be3d05e88e0f3fbb21a0b22c94a821a70e315d36eba17e83c6bbe22a1f83cd7","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154026,"semantic_template":"qf-density15-composition-restricted-table-domain/[0,0,0,[],[]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(0)=∅, middle=Fin(0)=∅, target=Fin(0)=∅. Complete f=[]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([0, 0, 0], [], [], [], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([0, 0, 0], [], [], [], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0027","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba3662f4f75022175e87d3ff3101c01ab93cea00e24734ea9df5b887a39da37","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154027,"semantic_template":"qf-density15-composition-restricted-table-domain/[3,3,3,[0,2],[[0,2],[2,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 1, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([3, 3, 3], [0, 2], [0, 2], [2, 1], [1])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 3], [0, 2], [0, 2], [2, 1], [1])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0028","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee57bb21014845cf3035a2f0bc0605f7bc770dcc9a98da34710250659cd0e2f","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154028,"semantic_template":"qf-density15-composition-restricted-table-domain/[3,3,0,[],[]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(0)=∅. Complete f=[0, 1, 2]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([3, 3, 0], [], [], [], [0, 1, 2])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 0], [], [], [], [0, 1, 2])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0029","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8b815d974f6dc86c5d463059077a827afe8c27844e57eff65ccec48e067d73f","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154029,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,2],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0030","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfb9219d501bc199d44196b67ab8caf7e8612bdcbe09ed1b24563703fdfd6c2d","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154030,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,2],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0031","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79ebc6375ea27983d97267ce31b2bfb3ed039a5eb3cb337f3c03f1ce4ae071f8","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154031,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,1],[2,2],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0032","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"900a1dbc4ed06f06155e6c1a8e5ec542d23eb478909fa46f45b0f1fa3f1935f9","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154032,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,1],[2,2],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0033","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7b2375adee5272613c94934a1ecc337053a38afb8fa44ad671a30f5ac6fd95","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154033,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,1],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0034","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1f0e32142da330ec72637ba2f439a8ce306f389f4f1b4a0b83dad568623ca90","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154034,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,1],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0035","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c3323d47736ba0c933e3650005aa9e25e18a252b1fedbf174e037053120df7","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154035,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,1],[2,1],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0036","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64c8462190992657c34a2ca1fa930472b4b817b60683055b3efd5de9ea4534c1","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154036,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,1],[2,1],[3,2],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0037","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d93cbcb7af4e311ee692209f2e3906e653cb9e6019d9c08b484de54bf958082","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154037,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,2],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0038","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2bdd4eb0d2947941f8ffaca13c4d12f5b62a33b9445d8eb8f787178f862f17f","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154038,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,2],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0039","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bda05b4876e392112008d6f1c4484282fa1adc3d7867ac9b6779bfa919beb63d","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154039,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,1],[2,2],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0040","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e4c36eb8dec760527310268e588bd83b78dfb10d5551bde654e4b73394d1ba1","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154040,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,1],[2,2],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0041","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1860fd198538dfe0b0483718d52afb5095ebfcbfeff85867ccab4d3d88b17a","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154041,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,1],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0042","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33057a1eacdabdea9263b60be82a024ff359bc1dcaa33d7e999b2d484b88b256","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154042,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,1],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0043","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d995f97617868ae66ef6ddcbd44fe5a4ff1c035722f21be8b20185fd8a335066","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154043,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,1],[2,1],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0044","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75b7631dd89b0bc35ac3d1dc090eed1cd9b2d710f38d560eb6c67cdd6b8209df","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154044,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,1],[2,1],[3,1],[4,2]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0045","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0e9aeacb86176c3807153d1f678ada047bae14f3fc3ca55c85e1548b0794bd3","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154045,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,2],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0046","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfdc21855857b8748cbdaca1c1fbd7b10e3911ff8866ccf60dcd24c60ba26453","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154046,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,2],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0047","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e80dd3ef1fd27a6c87214bf00de13469c1267f215c3e398f14493afd630a243","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154047,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,1],[2,2],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0048","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea8cf3d264ed4c0937865af16cf9918f9b6b072f0fa68f7c8fcb9730df8b20a7","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154048,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,1],[2,2],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0049","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d43e0247b78eaf5e663cd2463324184b38df74491c6c91be61ae0ac2b349d1b9","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154049,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,2],[1,2],[2,1],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0050","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"458297a70b337ed302444b275a8a0f889eb2fdbf15462813fb1e91bf183fb865","family_id":"qf-density15-composition-restricted-table-domain","seed":202610154050,"semantic_template":"qf-density15-composition-restricted-table-domain/[5,3,3,[0,2],[[0,1],[1,2],[2,1],[3,2],[4,1]]]"},"fields":{"title":"Restricted Middle Domain with Original Pairing Preserved","stem":"First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.","answer":{"kind":"expression","value":"([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], [])","explanation":"Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], [])."},"skills":["Restricted Middle Domain with Original Pairing Preserved"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0051","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"058d232b697ee0fe7aa43d96d4922f59912d6761443da5d464fc7d8dc452d5cd","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154051,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"1\",\"0\"],[\"1\",\"0\"],[\"0/1\",\"1/1\",\"-1/1\"],[\"0/1\",\"1/1\",\"-1/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(1)x+(0), g(x)=(1)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0052","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1dd1c950030e4cdacab2879a1d9e7d0bd86bfa6c8abd5c8b5e91d886062f74c","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154052,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"2\",\"2\"],[\"2\",\"1\"],[\"2/1\",\"4/1\",\"0/1\"],[\"1/1\",\"3/1\",\"-1/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(1)x+(1), g(x)=(2)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0053","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"753c26cea5a28e3ab852efb00d6aede64667ad8b8ebf177a563f7af8b4c6a8e4","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154053,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"0\",\"2\"],[\"0\",\"1\"],[\"2/1\",\"2/1\",\"2/1\"],[\"1/1\",\"1/1\",\"1/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(0)x+(1), g(x)=(0)x+(2); ordered inputs=[0/1, 1/1, 2/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0054","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfa0bf1dff106ae875cc331186d69744b405e38442024b76ea0e52d13375d3ef","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154054,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-4\",\"3\"],[\"-4\",\"1\"],[\"3/1\",\"1/1\",\"7/1\"],[\"1/1\",\"-1/1\",\"5/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(2)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0055","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7d81193d6502c6ffb09a74caf46823a95de85a7cfc255ce55c91327ad820d13","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154055,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-6\",\"1\"],[\"-6\",\"3\"],[\"1/1\",\"-2/1\",\"7/1\"],[\"3/1\",\"0/1\",\"9/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(3)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0056","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"650760ba2fd4ba7744f8b560ab3cf1dd9870fe6de93403869fd1593a0c61bad4","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154056,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-8\",\"-1\"],[\"-8\",\"5\"],[\"-1/1\",\"-5/1\",\"7/1\"],[\"5/1\",\"1/1\",\"13/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(4)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0057","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b63d885b9ab77adf165f5cfa0a7a2cd983c06a80d80598ce179406640e74f12f","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154057,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-10\",\"3\"],[\"-10\",\"4\"],[\"3/1\",\"-2/1\",\"13/1\"],[\"4/1\",\"-1/1\",\"14/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(5)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0058","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98dcdf084850d7d9f3816e6f35939a762206aad8df020eb1fb2d2d619ca6decc","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154058,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-12\",\"1\"],[\"-12\",\"6\"],[\"1/1\",\"-5/1\",\"13/1\"],[\"6/1\",\"0/1\",\"18/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(6)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0059","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4faf8fcdf988700b407bd510c525a4cc7b7535142e8b71d2903c3eccdb28198f","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154059,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-14\",\"-1\"],[\"-14\",\"8\"],[\"-1/1\",\"-8/1\",\"13/1\"],[\"8/1\",\"1/1\",\"22/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(7)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0060","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65d1c5d31b046085be25733d6f6ac3b7411604359da3635ea7d6addac0f4dd2","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154060,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-16\",\"3\"],[\"-16\",\"7\"],[\"3/1\",\"-5/1\",\"19/1\"],[\"7/1\",\"-1/1\",\"23/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(8)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0061","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63066055bf852086da93f2708b9d5616cb61a7f4e2bd520743ffcf20ecbc077d","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154061,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-18\",\"1\"],[\"-18\",\"9\"],[\"1/1\",\"-8/1\",\"19/1\"],[\"9/1\",\"0/1\",\"27/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(9)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0062","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"047f716c1f74f86d64be10ae2954d3b6aa915d600078c9ea14ad7e5b6a922861","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154062,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-20\",\"-1\"],[\"-20\",\"11\"],[\"-1/1\",\"-11/1\",\"19/1\"],[\"11/1\",\"1/1\",\"31/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(10)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0063","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acfea2359cee0b7c9b82714dede58f85c4b8c35aed8c99695cc7980ad0bf1b31","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154063,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-22\",\"3\"],[\"-22\",\"10\"],[\"3/1\",\"-8/1\",\"25/1\"],[\"10/1\",\"-1/1\",\"32/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(11)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0064","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63173695117cb18174174a35c0c07c6c8a18736319cc3723b19c43f9873e5ad0","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154064,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-24\",\"1\"],[\"-24\",\"12\"],[\"1/1\",\"-11/1\",\"25/1\"],[\"12/1\",\"0/1\",\"36/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(12)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0065","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea81ec2029a853aef9b6d3e651168fe7e0e0923adc27d407632c934d77d9e2ee","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154065,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-26\",\"-1\"],[\"-26\",\"14\"],[\"-1/1\",\"-14/1\",\"25/1\"],[\"14/1\",\"1/1\",\"40/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(13)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0066","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e3d5e1e8535141b48729479a0f0391dbe95bd637527d61389a62b998127a45","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154066,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-28\",\"3\"],[\"-28\",\"13\"],[\"3/1\",\"-11/1\",\"31/1\"],[\"13/1\",\"-1/1\",\"41/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(14)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0067","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f3e5bce3cb110217866e334dbc6808651db007114e59c816e5fdbf98ebbd96","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154067,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-30\",\"1\"],[\"-30\",\"15\"],[\"1/1\",\"-14/1\",\"31/1\"],[\"15/1\",\"0/1\",\"45/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(15)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0068","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b40980f264799638eb3446a06102bd2ac54127ad10ecdccaf90a6619a18730a","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154068,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-32\",\"-1\"],[\"-32\",\"17\"],[\"-1/1\",\"-17/1\",\"31/1\"],[\"17/1\",\"1/1\",\"49/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(16)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0069","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6cca11253515673dcfc671e9ff18d609f20a3c036f7c699847c29f456ee9f1d","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154069,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-34\",\"3\"],[\"-34\",\"16\"],[\"3/1\",\"-14/1\",\"37/1\"],[\"16/1\",\"-1/1\",\"50/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(17)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0070","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23bd0012fd85c873517a5c5d9449bbe78c686cf20c6b02dea820949a7bdcaefe","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154070,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-36\",\"1\"],[\"-36\",\"18\"],[\"1/1\",\"-17/1\",\"37/1\"],[\"18/1\",\"0/1\",\"54/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(18)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0071","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ce3c6fac8617ab79b08b7ee5c575fee9fa1f1477d926d783e96b1a10df756f","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154071,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-38\",\"-1\"],[\"-38\",\"20\"],[\"-1/1\",\"-20/1\",\"37/1\"],[\"20/1\",\"1/1\",\"58/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(19)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0072","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ac3cf7797d1da457d5ec3ecbd39c2796a732ddde92cce8a9ab64bbce2888fad","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154072,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-40\",\"3\"],[\"-40\",\"19\"],[\"3/1\",\"-17/1\",\"43/1\"],[\"19/1\",\"-1/1\",\"59/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(20)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0073","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8057f4ac049f6135ce5c388e01907e88a8ce97c063fbad80a337d65751ba04","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154073,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-42\",\"1\"],[\"-42\",\"21\"],[\"1/1\",\"-20/1\",\"43/1\"],[\"21/1\",\"0/1\",\"63/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(21)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0074","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b6b269ef370114e90c629cd05bd9678d24ee92b3d8bcf3b2ab9762f44ce37a","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154074,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-44\",\"-1\"],[\"-44\",\"23\"],[\"-1/1\",\"-23/1\",\"43/1\"],[\"23/1\",\"1/1\",\"67/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(22)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0075","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7fedb23f935b28f56837374f0dd5a3c97892282ca61ab7efdb91d609257d382","family_id":"qf-density15-composition-noncommuting-affine","seed":202610154075,"semantic_template":"qf-density15-composition-noncommuting-affine/[[\"-46\",\"3\"],[\"-46\",\"22\"],[\"3/1\",\"-20/1\",\"49/1\"],[\"22/1\",\"-1/1\",\"68/1\"]]"},"fields":{"title":"Both Affine Composition Orders with Slope-First Coefficients","stem":"Total Q functions f(x)=(23)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).","answer":{"kind":"expression","value":"([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19)","explanation":"Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19)."},"skills":["Both Affine Composition Orders with Slope-First Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0076","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e286b54dd31039dbc0eb6da0bdc244ad4d13d6bbbd56f783b18c82693cdd9a4","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154076,"semantic_template":"qf-density15-composition-three-map-parentheses/[0,0,0,0,[],[],[]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(0)=∅→Fin(0)→Fin(0)→Fin(0); complete f=[], g=[], h=[]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([0, 0, 0, 0], [], [], [], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([0, 0, 0, 0], [], [], [], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0077","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776e89aa7cf289a1e8c67481e9e8131a4e3b07325753887df967bcb6e154caaf","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154077,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,0,0],[2,2,2],[0,0,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0078","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40fa9c00e07ee83e82ccaa97433e073856afaef98af1969df5403a6d21c6abe8","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154078,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,0,0],[0,2,2],[1,0,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0079","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7803389cd2d2b770853f3f3babfb190ebc70c6767dd0457e902fb9207b5b7f5","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154079,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,0,0],[1,2,2],[3,0,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0080","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93d5befe99ed512a4cf75864a639cf9aecaa1a4061dfd3e01eae2a7a2940ff2","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154080,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,1,0],[2,0,2],[0,1,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0081","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20d253ba1e3957fd634bed2bc9a35997297b4f1066dd46162ae085af7defeb6","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154081,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,1,0],[0,0,2],[1,1,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0082","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a46c0963b53d95f529eae2b195fe7f6029092d6e87d07bdbd523ac88e927929","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154082,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,1,0],[1,0,2],[3,1,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0083","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c829fb39327592137d2f4673a8b6e99223f5916b74dc412c7a341ab14cd8330","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154083,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,2,0],[2,1,2],[0,3,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0084","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d886fbe553c1eff051fae64c7c17fbcd0b33e3e09855f6a8e6e0ac870c0cb0a7","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154084,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,2,0],[0,1,2],[1,3,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0085","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a1096f50cb87b3118151612a697891f30b6ecf49a3f4802dce0e2f8d302966","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154085,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,2,0],[1,1,2],[3,3,0]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0086","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b07ecc73ad5226cb61878a8a53b1a042a761e00d5530335d6f62038065379a3","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154086,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,0,1],[2,2,0],[0,0,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0087","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeaf04923222285b18fbea9e88b4d19608de7e562638495078d4f9b51a147075","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154087,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,0,1],[0,2,0],[1,0,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0088","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"102de5691a5978594e7d5971203937ef3117c44e2c1e952e70660926b43a4afd","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154088,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,0,1],[1,2,0],[3,0,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0089","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4549f571be98ee0a8a68e737edf07aa6ee21a98ce4b59dcdd6201bcccd9be867","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154089,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,1,1],[2,0,0],[0,1,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0090","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f8dbb735053ff9189dcb86c138ef9eb8f822f730196826a2f166e0193920147","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154090,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,1,1],[0,0,0],[1,1,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0091","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a26377b93510531da309beafdd0f93c796ca88d73fa6fb54a31aa0b969ce57a2","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154091,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,1,1],[1,0,0],[3,1,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0092","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f22deae9e0707edcfdb774708d718c16fef8eb22b533047b36c17b45c5f8ea80","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154092,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,2,1],[2,1,0],[0,3,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0093","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b749f632d2e0e0ff85af9da5e1754d9dac21d6ac9c11afa1108efe77662daade","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154093,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,2,1],[0,1,0],[1,3,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0094","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c0aacc89c4b175099905af61e9f0fe502f9a1be6b10fc3e4618858b4768a831","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154094,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,2,1],[1,1,0],[3,3,1]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0095","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1115b2e0b38efefe4f5c64a42b0204ddd4e3d71d788c266ad6bb581fd114495a","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154095,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,0,2],[2,2,1],[0,0,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0096","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb65b3262a7e6a6782c964777294419f7646546b921027e5764913289629e18","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154096,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,0,2],[0,2,1],[1,0,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0097","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"924ac4bde4ff2bcb3dbe7844237c27c3bffcd7a28171948131ae6c9e52baf3d6","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154097,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,0,2],[1,2,1],[3,0,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0098","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69230e5ced9540f78bacb60bc099e47faae090e5ef66389695e86ab52203eba8","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154098,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[0,1,2],[2,0,1],[0,1,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0099","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4486bab6c22ebcf002c9468bf61874e7fb52b2c2e0027a49acca91d175358fb5","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154099,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[1,1,2],[0,0,1],[1,1,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r04-composition-0100","batch_id":"batch-20261011-advance-20261015-r04-composition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e430fb092627901294e3bfdef99a9e7e1783cb0f482a70c32857d79a89bf861f","family_id":"qf-density15-composition-three-map-parentheses","seed":202610154100,"semantic_template":"qf-density15-composition-three-map-parentheses/[3,3,3,4,[2,1,2],[1,0,1],[3,1,3]]"},"fields":{"title":"Two Parenthesizations of Three Complete Typed Maps","stem":"First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.","answer":{"kind":"expression","value":"([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)","explanation":"Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)."},"skills":["Two Parenthesizations of Three Complete Typed Maps"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}