{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0001","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04d7aebbfb2cdd93fb682634f51643a4f8bc6cd780bb0fbe1ea8c594864e3a20","family_id":"qf-arithmetic-domain-boundary","seed":20260914,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":2,\"b\":3,\"number_system\":\"N\",\"operator\":\"+\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 2+3 be classified?","answer":{"kind":"numeric","value":5,"explanation":"2+3=5, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0002","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"537c3f4db692644881a559b2d3d39ccdb4bbf72d150b403e6ac721ebdc05a4fd","family_id":"qf-arithmetic-domain-boundary","seed":20260915,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":0,\"b\":8,\"number_system\":\"N\",\"operator\":\"+\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 0+8 be classified?","answer":{"kind":"numeric","value":8,"explanation":"0+8=8, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0003","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad0c6bd9b17472a7a29e27936e5997cce8f7df0cf90e7017ee75a1b12c33839","family_id":"qf-arithmetic-domain-boundary","seed":20260916,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":11,\"b\":7,\"number_system\":\"N\",\"operator\":\"+\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 11+7 be classified?","answer":{"kind":"numeric","value":18,"explanation":"11+7=18, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0004","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e0af4bae7c29bc2d78f2811b54e2bf279da8177d2bb999842b9230e972134b","family_id":"qf-arithmetic-domain-boundary","seed":20260917,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":24,\"b\":16,\"number_system\":\"N\",\"operator\":\"+\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 24+16 be classified?","answer":{"kind":"numeric","value":40,"explanation":"24+16=40, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0005","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23def39746dd85deb83f0334c6dc21394c45eaf377830d0d42c9388e51a03798","family_id":"qf-arithmetic-domain-boundary","seed":20260918,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":100,\"b\":9,\"number_system\":\"N\",\"operator\":\"+\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 100+9 be classified?","answer":{"kind":"numeric","value":109,"explanation":"100+9=109, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0006","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8228c59a5380473dadb0836957824e3fca2a2a18b8869f83e0a92c677e74590","family_id":"qf-arithmetic-domain-boundary","seed":20260919,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":2,\"b\":4,\"number_system\":\"N\",\"operator\":\"*\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 2*4 be classified?","answer":{"kind":"numeric","value":8,"explanation":"2*4=8, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0007","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"473b2e922151c9b46b47cd0518a9ee1ce3f688cdeaca212ebaaa977ddaac9a79","family_id":"qf-arithmetic-domain-boundary","seed":20260920,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":0,\"b\":15,\"number_system\":\"N\",\"operator\":\"*\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 0*15 be classified?","answer":{"kind":"numeric","value":0,"explanation":"0*15=0, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0008","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d79f25503857d7d874b49144892807e21ef4c8fbef5819bf478e2254c46c63d","family_id":"qf-arithmetic-domain-boundary","seed":20260921,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":3,\"b\":7,\"number_system\":\"N\",\"operator\":\"*\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 3*7 be classified?","answer":{"kind":"numeric","value":21,"explanation":"3*7=21, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0009","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2158d06e8f2c8e263d2f013041eb116c8856d6d98e48ea188722eda13e6946d","family_id":"qf-arithmetic-domain-boundary","seed":20260922,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":8,\"b\":9,\"number_system\":\"N\",\"operator\":\"*\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 8*9 be classified?","answer":{"kind":"numeric","value":72,"explanation":"8*9=72, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0010","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b78d17ffe97d53f67c0c04b36d5c6d4010a08982730fd8bb67e208aa1ad30059","family_id":"qf-arithmetic-domain-boundary","seed":20260923,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":12,\"b\":5,\"number_system\":\"N\",\"operator\":\"*\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 12*5 be classified?","answer":{"kind":"numeric","value":60,"explanation":"12*5=60, which remains in the explicitly stated natural-number system N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0011","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa183137a572b8ab6ccd171570a5e7382bbaaa5ad502152cfb684bdffb3a7ba3","family_id":"qf-arithmetic-domain-boundary","seed":20260924,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":3,\"b\":5,\"number_system\":\"N\",\"operator\":\"-\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 3-5 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"3-5 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0012","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40e1878dba880c4839df2ffa5a387f98b59996bdd30cb7d2ec7a812b32326a38","family_id":"qf-arithmetic-domain-boundary","seed":20260925,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":0,\"b\":1,\"number_system\":\"N\",\"operator\":\"-\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 0-1 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"0-1 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0013","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"051b5d145d87b71261cf401844a1ba2f33080b4790067a5264586a652c7a2375","family_id":"qf-arithmetic-domain-boundary","seed":20260926,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":7,\"b\":10,\"number_system\":\"N\",\"operator\":\"-\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 7-10 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"7-10 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0014","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ac19c55474df7183ffdc3a9d54a3e126791242bf693bf92be5ff14705926903","family_id":"qf-arithmetic-domain-boundary","seed":20260927,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":14,\"b\":20,\"number_system\":\"N\",\"operator\":\"-\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 14-20 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"14-20 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0015","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016099d93b8f7a44d35f95ad05d63dcc39bf3b594d70d178630627487926f052","family_id":"qf-arithmetic-domain-boundary","seed":20260928,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":2,\"b\":9,\"number_system\":\"N\",\"operator\":\"-\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 2-9 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"2-9 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0016","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b112f57a744c4443ae300194425e9bced03d637850fa144b88ddf68ab3efb0a2","family_id":"qf-arithmetic-domain-boundary","seed":20260929,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":3,\"b\":2,\"number_system\":\"N\",\"operator\":\"/\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 3/2 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"3/2 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0017","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01755508933361adc7e5e8e583614f227bc5755f42765d3cd1cf372fcabc1c2d","family_id":"qf-arithmetic-domain-boundary","seed":20260930,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":5,\"b\":4,\"number_system\":\"N\",\"operator\":\"/\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 5/4 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"5/4 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0018","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e3a0d668d0c133e576cb8ea6fe502741450ff165dc24a807756cd80fe0dae8","family_id":"qf-arithmetic-domain-boundary","seed":20260931,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":7,\"b\":3,\"number_system\":\"N\",\"operator\":\"/\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 7/3 be classified?","answer":{"kind":"choice","value":"Not closed in N","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"7/3 is not a natural number in this instance, so the operation is not closed in N."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0019","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d488892ced7d0258d2bbc7c2a7e35aca45867a9465922ca138d4f1d05448d3ac","family_id":"qf-arithmetic-domain-boundary","seed":20260932,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":9,\"b\":0,\"number_system\":\"N\",\"operator\":\"/\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 9/0 be classified?","answer":{"kind":"choice","value":"Zero divisor; no ordinary numerical result","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"The divisor in 9/0 is zero, so this question does not treat it as an ordinary numerical division result."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-arithmetic-domain-boundary-0020","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d51e4c2c102fe914f20f76abc2a756d7a52c0aa247d67f561aaf22fa09e0b4cc","family_id":"qf-arithmetic-domain-boundary","seed":20260933,"semantic_template":"qf-arithmetic-domain-boundary/{\"a\":0,\"b\":0,\"number_system\":\"N\",\"operator\":\"/\"}"},"fields":{"title":"Arithmetic Number-System Boundary","stem":"In the number system N, including 0, how should the result of 0/0 be classified?","answer":{"kind":"choice","value":"Zero divisor; no ordinary numerical result","choices":["Not closed in N","Zero divisor; no ordinary numerical result"],"explanation":"The divisor in 0/0 is zero, so this question does not treat it as an ordinary numerical division result."},"skills":["number-system boundaries","arithmetic operations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0001","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c522b5dd40489876fe357b126dd1e23e9ba1584b39c14be2a85ba2aa90d55076","family_id":"qf-variable-expression-evaluation","seed":20260934,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":0,\"constant\":1,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=0. Evaluate the algebraic expression x+1; this is an expression, not an equation.","answer":{"kind":"numeric","value":1,"explanation":"Substitution gives 0+1=1; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0002","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa4820b654ba6d33ab281983a1b06f4b8fd2a5d0e6f28443a220a111f08b53a8","family_id":"qf-variable-expression-evaluation","seed":20260935,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":3,\"constant\":2,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=3. Evaluate the algebraic expression x+2; this is an expression, not an equation.","answer":{"kind":"numeric","value":5,"explanation":"Substitution gives 3+2=5; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0003","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb24cc834e7f4d83597897ed66e23ba94c88357d7100296a110940f4f8183c8","family_id":"qf-variable-expression-evaluation","seed":20260936,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":6,\"constant\":3,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=6. Evaluate the algebraic expression x+3; this is an expression, not an equation.","answer":{"kind":"numeric","value":9,"explanation":"Substitution gives 6+3=9; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0004","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a7a61f28e43070e827e80064729c13410cd33d68f83324cf42b1edacd07fd68","family_id":"qf-variable-expression-evaluation","seed":20260937,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":9,\"constant\":4,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=9. Evaluate the algebraic expression x+4; this is an expression, not an equation.","answer":{"kind":"numeric","value":13,"explanation":"Substitution gives 9+4=13; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0005","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85b4bfab9b2f8f2485e9eff6293f0cc31bc349b0ccb64d78a93448ce15008847","family_id":"qf-variable-expression-evaluation","seed":20260938,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":12,\"constant\":5,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=12. Evaluate the algebraic expression x+5; this is an expression, not an equation.","answer":{"kind":"numeric","value":17,"explanation":"Substitution gives 12+5=17; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0006","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63d2b1aa282c8606565ccec4af9e80867cad5cfd8e85061ef3bd06a1d3fab375","family_id":"qf-variable-expression-evaluation","seed":20260939,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":15,\"constant\":1,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=15. Evaluate the algebraic expression x+1; this is an expression, not an equation.","answer":{"kind":"numeric","value":16,"explanation":"Substitution gives 15+1=16; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0007","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5829a777f9fa38f736fbd8126b1a647a5c3f59b9486657ea043230324ffc4715","family_id":"qf-variable-expression-evaluation","seed":20260940,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":18,\"constant\":2,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=18. Evaluate the algebraic expression x+2; this is an expression, not an equation.","answer":{"kind":"numeric","value":20,"explanation":"Substitution gives 18+2=20; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0008","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eedac3aed54399adf62a34aa1f2ad8d52efd43ef27f52851128c65f77e0d3ae5","family_id":"qf-variable-expression-evaluation","seed":20260941,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":21,\"constant\":3,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=21. Evaluate the algebraic expression x+3; this is an expression, not an equation.","answer":{"kind":"numeric","value":24,"explanation":"Substitution gives 21+3=24; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0009","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1e4a7a04d993f1583a1d53f091dc0a16f7c22e5bd98296477599ac416902809","family_id":"qf-variable-expression-evaluation","seed":20260942,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":24,\"constant\":4,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=24. Evaluate the algebraic expression x+4; this is an expression, not an equation.","answer":{"kind":"numeric","value":28,"explanation":"Substitution gives 24+4=28; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0010","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"569a4e0eb05a29fa477f99d89dd520aef767ad2e96fc107eae29d6ffc1519e81","family_id":"qf-variable-expression-evaluation","seed":20260943,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":27,\"constant\":5,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=27. Evaluate the algebraic expression x+5; this is an expression, not an equation.","answer":{"kind":"numeric","value":32,"explanation":"Substitution gives 27+5=32; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0011","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f14a45b7edd59e2c6a60d267eb6acdd8a8320dba38d969e9cc6f61edef9d8ad","family_id":"qf-variable-expression-evaluation","seed":20260944,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":30,\"constant\":1,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=30. Evaluate the algebraic expression x+1; this is an expression, not an equation.","answer":{"kind":"numeric","value":31,"explanation":"Substitution gives 30+1=31; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0012","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0a42329aa534021f32d8496f10c6643dc35286fa5836a8c0fcb7ad19cb8701a","family_id":"qf-variable-expression-evaluation","seed":20260945,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":33,\"constant\":2,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=33. Evaluate the algebraic expression x+2; this is an expression, not an equation.","answer":{"kind":"numeric","value":35,"explanation":"Substitution gives 33+2=35; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0013","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8fcebb8094ed137123d58d52a5f1aa78511bc9acffdb8b8f1ea9bddf6262d61","family_id":"qf-variable-expression-evaluation","seed":20260946,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":36,\"constant\":3,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=36. Evaluate the algebraic expression x+3; this is an expression, not an equation.","answer":{"kind":"numeric","value":39,"explanation":"Substitution gives 36+3=39; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0014","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad79e4962c1f63e8920bd634b5cef413d8177b9fad7e0f4be7e8ce9745207ba9","family_id":"qf-variable-expression-evaluation","seed":20260947,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":39,\"constant\":4,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=39. Evaluate the algebraic expression x+4; this is an expression, not an equation.","answer":{"kind":"numeric","value":43,"explanation":"Substitution gives 39+4=43; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0015","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e4d0476858cbdbbec2b23ced0024d815ace66f8b2725ac82be464c6034c3629","family_id":"qf-variable-expression-evaluation","seed":20260948,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":42,\"constant\":5,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=42. Evaluate the algebraic expression x+5; this is an expression, not an equation.","answer":{"kind":"numeric","value":47,"explanation":"Substitution gives 42+5=47; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0016","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"500a52b41b08402d7a61e0fcc2a7995819cea4a95194e1b9e67c012eac90db1f","family_id":"qf-variable-expression-evaluation","seed":20260949,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":45,\"constant\":1,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=45. Evaluate the algebraic expression x+1; this is an expression, not an equation.","answer":{"kind":"numeric","value":46,"explanation":"Substitution gives 45+1=46; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0017","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b007c9c49578302973743a19fc363a6b4568d787c0514a71aaf98d9db2430862","family_id":"qf-variable-expression-evaluation","seed":20260950,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":48,\"constant\":2,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=48. Evaluate the algebraic expression x+2; this is an expression, not an equation.","answer":{"kind":"numeric","value":50,"explanation":"Substitution gives 48+2=50; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0018","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cfd4e0eeeb38665497b09bb7feece760b7d5ebc765d0fe9b6cb990ff92eb718","family_id":"qf-variable-expression-evaluation","seed":20260951,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":51,\"constant\":3,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=51. Evaluate the algebraic expression x+3; this is an expression, not an equation.","answer":{"kind":"numeric","value":54,"explanation":"Substitution gives 51+3=54; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0019","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"078e0633c72d9af684bf68d1963baac39973434394dfbf2eddd1fdde229a342e","family_id":"qf-variable-expression-evaluation","seed":20260952,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":54,\"constant\":4,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=54. Evaluate the algebraic expression x+4; this is an expression, not an equation.","answer":{"kind":"numeric","value":58,"explanation":"Substitution gives 54+4=58; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-variable-expression-evaluation-0020","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c46133e066758431546e040dba8e66b675e2644b48419cb4098db7114d813620","family_id":"qf-variable-expression-evaluation","seed":20260953,"semantic_template":"qf-variable-expression-evaluation/{\"assigned_value\":57,\"constant\":5,\"expression_form\":\"x_plus_constant\",\"number_system\":\"N\",\"operator\":\"+\",\"variable\":\"x\"}"},"fields":{"title":"Explicit Evaluation of an Algebraic Expression","stem":"In N, let x=57. Evaluate the algebraic expression x+5; this is an expression, not an equation.","answer":{"kind":"numeric","value":62,"explanation":"Substitution gives 57+5=62; only the explicitly stated addition is used."},"skills":["algebraic expressions","evaluation by substitution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0001","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ce50bbd61471b967a5b8fb4c286eb788ed9ad13ec38a7213cfbdfe2e9374bcc","family_id":"qf-finite-sample-space-completeness","seed":20260954,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記1\",\"標記2\"],\"experiment\":\"一次讀取明示標記卡 標記1、標記2\",\"listed_outcomes\":[\"標記1\",\"標記2\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 1, label 2. At this observation scale the candidate sample space is S={label 1, label 2}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0002","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b284287ccd3171112f1c2ecf74158e71d39f69719ce5a15ba3ae2a44f1a5457b","family_id":"qf-finite-sample-space-completeness","seed":20260955,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記11\",\"標記12\",\"標記13\"],\"experiment\":\"一次讀取明示標記卡 標記11、標記12、標記13\",\"listed_outcomes\":[\"標記11\",\"標記13\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 11, label 12, label 13. At this observation scale the candidate sample space is S={label 11, label 13}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0003","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"573f08783f5ba420d09fa7634affe4d5edca14d48715598e22738317baf328e2","family_id":"qf-finite-sample-space-completeness","seed":20260956,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記21\",\"標記22\",\"標記23\",\"標記24\"],\"experiment\":\"一次讀取明示標記卡 標記21、標記22、標記23、標記24\",\"listed_outcomes\":[\"標記21\",\"標記22\",\"標記23\",\"標記24\",\"非此試驗結果3\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 21, label 22, label 23, label 24. At this observation scale the candidate sample space is S={label 21, label 22, label 23, label 24, outcome outside this experiment 3}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0004","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d90cf72b700a92234a2703e37d64fcbf2db0441446a5cddd025a03bfa26d7873","family_id":"qf-finite-sample-space-completeness","seed":20260957,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記31\",\"標記32\",\"標記33\",\"標記34\",\"標記35\"],\"experiment\":\"一次讀取明示標記卡 標記31、標記32、標記33、標記34、標記35\",\"listed_outcomes\":[\"標記31\",\"標記32\",\"標記33\",\"標記34\",\"標記35\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 31, label 32, label 33, label 34, label 35. At this observation scale the candidate sample space is S={label 31, label 32, label 33, label 34, label 35}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0005","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe4ae2822154c0c04f868348cd74b00ddc605f007ecdff70b3e6347f4d7f9b0","family_id":"qf-finite-sample-space-completeness","seed":20260958,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記41\",\"標記42\"],\"experiment\":\"一次讀取明示標記卡 標記41、標記42\",\"listed_outcomes\":[\"標記42\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 41, label 42. At this observation scale the candidate sample space is S={label 42}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0006","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b78a793fcc995ff9d50bd85fa86b885a728be7b9b57736071804729613a6878","family_id":"qf-finite-sample-space-completeness","seed":20260959,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記51\",\"標記52\",\"標記53\"],\"experiment\":\"一次讀取明示標記卡 標記51、標記52、標記53\",\"listed_outcomes\":[\"標記51\",\"標記52\",\"標記53\",\"非此試驗結果6\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 51, label 52, label 53. At this observation scale the candidate sample space is S={label 51, label 52, label 53, outcome outside this experiment 6}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0007","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1bc976b3c80727ee4b8da70e9c7005868b75e32263bbb396025f66379dbdba6","family_id":"qf-finite-sample-space-completeness","seed":20260960,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記61\",\"標記62\",\"標記63\",\"標記64\"],\"experiment\":\"一次讀取明示標記卡 標記61、標記62、標記63、標記64\",\"listed_outcomes\":[\"標記61\",\"標記62\",\"標記63\",\"標記64\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 61, label 62, label 63, label 64. At this observation scale the candidate sample space is S={label 61, label 62, label 63, label 64}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0008","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ad8be61c7fee1ec5e512a75a8a03090b50761ad22b144b370e75f824141bc5","family_id":"qf-finite-sample-space-completeness","seed":20260961,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記71\",\"標記72\",\"標記73\",\"標記74\",\"標記75\"],\"experiment\":\"一次讀取明示標記卡 標記71、標記72、標記73、標記74、標記75\",\"listed_outcomes\":[\"標記71\",\"標記72\",\"標記74\",\"標記75\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 71, label 72, label 73, label 74, label 75. At this observation scale the candidate sample space is S={label 71, label 72, label 74, label 75}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0009","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"346b46d4bc343e81a8e99e87487ffd4637a5d0dee72dbecf9ebb8222ec29a7b3","family_id":"qf-finite-sample-space-completeness","seed":20260962,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記81\",\"標記82\"],\"experiment\":\"一次讀取明示標記卡 標記81、標記82\",\"listed_outcomes\":[\"標記81\",\"標記82\",\"非此試驗結果9\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 81, label 82. At this observation scale the candidate sample space is S={label 81, label 82, outcome outside this experiment 9}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0010","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9732bfdc409dc6f7eda20fb52a81edfcb68785727c03ee91cf10fbaef79bd5","family_id":"qf-finite-sample-space-completeness","seed":20260963,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記91\",\"標記92\",\"標記93\"],\"experiment\":\"一次讀取明示標記卡 標記91、標記92、標記93\",\"listed_outcomes\":[\"標記91\",\"標記92\",\"標記93\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 91, label 92, label 93. At this observation scale the candidate sample space is S={label 91, label 92, label 93}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0011","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5073642630fa4a2e6935d1e4bafdb7e3bc04cf4c397259110674ec67ff041e9","family_id":"qf-finite-sample-space-completeness","seed":20260964,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記101\",\"標記102\",\"標記103\",\"標記104\"],\"experiment\":\"一次讀取明示標記卡 標記101、標記102、標記103、標記104\",\"listed_outcomes\":[\"標記101\",\"標記102\",\"標記104\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 101, label 102, label 103, label 104. At this observation scale the candidate sample space is S={label 101, label 102, label 104}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0012","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f21da92579c472e6b46d0fe773defe013c6e0e7a1a64fc98d89089135b6e7d7","family_id":"qf-finite-sample-space-completeness","seed":20260965,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記111\",\"標記112\",\"標記113\",\"標記114\",\"標記115\"],\"experiment\":\"一次讀取明示標記卡 標記111、標記112、標記113、標記114、標記115\",\"listed_outcomes\":[\"標記111\",\"標記112\",\"標記113\",\"標記114\",\"標記115\",\"非此試驗結果12\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 111, label 112, label 113, label 114, label 115. At this observation scale the candidate sample space is S={label 111, label 112, label 113, label 114, label 115, outcome outside this experiment 12}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0013","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fb8cd638c7a39779f792cc08eb8133aad48ed3458cb149ed588fc865bee4d2d","family_id":"qf-finite-sample-space-completeness","seed":20260966,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記121\",\"標記122\"],\"experiment\":\"一次讀取明示標記卡 標記121、標記122\",\"listed_outcomes\":[\"標記121\",\"標記122\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 121, label 122. At this observation scale the candidate sample space is S={label 121, label 122}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0014","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c4abf780f63e4355bef0914a82d5c524f5c876a81b3bcc979e609090a08e83","family_id":"qf-finite-sample-space-completeness","seed":20260967,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記131\",\"標記132\",\"標記133\"],\"experiment\":\"一次讀取明示標記卡 標記131、標記132、標記133\",\"listed_outcomes\":[\"標記131\",\"標記133\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 131, label 132, label 133. At this observation scale the candidate sample space is S={label 131, label 133}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0015","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016d0eb2771cb01a053fbdf64e34e1bfa9d432d471824fd044f80f8ed2c0e937","family_id":"qf-finite-sample-space-completeness","seed":20260968,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記141\",\"標記142\",\"標記143\",\"標記144\"],\"experiment\":\"一次讀取明示標記卡 標記141、標記142、標記143、標記144\",\"listed_outcomes\":[\"標記141\",\"標記142\",\"標記143\",\"標記144\",\"非此試驗結果15\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 141, label 142, label 143, label 144. At this observation scale the candidate sample space is S={label 141, label 142, label 143, label 144, outcome outside this experiment 15}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0016","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c8b4f2b98d35df19bfb7b2774b1cb40f98fcc3a7a12ac4f198d40ba3edf9292","family_id":"qf-finite-sample-space-completeness","seed":20260969,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記151\",\"標記152\",\"標記153\",\"標記154\",\"標記155\"],\"experiment\":\"一次讀取明示標記卡 標記151、標記152、標記153、標記154、標記155\",\"listed_outcomes\":[\"標記151\",\"標記152\",\"標記153\",\"標記154\",\"標記155\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 151, label 152, label 153, label 154, label 155. At this observation scale the candidate sample space is S={label 151, label 152, label 153, label 154, label 155}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0017","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e348e3261b64e960d3d97646ba107bcb300835f3e7be24267338f902f6297851","family_id":"qf-finite-sample-space-completeness","seed":20260970,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記161\",\"標記162\"],\"experiment\":\"一次讀取明示標記卡 標記161、標記162\",\"listed_outcomes\":[\"標記162\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 161, label 162. At this observation scale the candidate sample space is S={label 162}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0018","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b08f6bc822d335fb3dbbbd36f582df442b743d3acb55573aa80e155a097df929","family_id":"qf-finite-sample-space-completeness","seed":20260971,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記171\",\"標記172\",\"標記173\"],\"experiment\":\"一次讀取明示標記卡 標記171、標記172、標記173\",\"listed_outcomes\":[\"標記171\",\"標記172\",\"標記173\",\"非此試驗結果18\"],\"mode\":\"extra_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 171, label 172, label 173. At this observation scale the candidate sample space is S={label 171, label 172, label 173, outcome outside this experiment 18}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate includes an outcome outside the explicitly stated experiment, so it is not the exact sample space."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0019","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590a399405d786b5285770ff96661dea23d5b40e120588952d431decb0be1d4e","family_id":"qf-finite-sample-space-completeness","seed":20260972,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記181\",\"標記182\",\"標記183\",\"標記184\"],\"experiment\":\"一次讀取明示標記卡 標記181、標記182、標記183、標記184\",\"listed_outcomes\":[\"標記181\",\"標記182\",\"標記183\",\"標記184\"],\"mode\":\"complete\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 181, label 182, label 183, label 184. At this observation scale the candidate sample space is S={label 181, label 182, label 183, label 184}. Is it complete?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"The enumeration exactly matches all explicitly stated elementary outcomes. This conclusion concerns completeness only and assigns no probabilities or equal likelihood."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-finite-sample-space-completeness-0020","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b808b3c0122db84ef3f428aa117b47bd6a5cc838cecc5070f03a8bdb3e6f8c3","family_id":"qf-finite-sample-space-completeness","seed":20260973,"semantic_template":"qf-finite-sample-space-completeness/{\"expected_outcomes\":[\"標記191\",\"標記192\",\"標記193\",\"標記194\",\"標記195\"],\"experiment\":\"一次讀取明示標記卡 標記191、標記192、標記193、標記194、標記195\",\"listed_outcomes\":[\"標記191\",\"標記192\",\"標記193\",\"標記194\"],\"mode\":\"missing_outcome\",\"observation_scale\":\"一次明示有限試驗\"}"},"fields":{"title":"Completeness of a Finite Sample Space","stem":"The experiment is one draw from explicitly labelled cards with elementary outcomes label 191, label 192, label 193, label 194, label 195. At this observation scale the candidate sample space is S={label 191, label 192, label 193, label 194}. Is it complete?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"The candidate omits at least one explicitly possible elementary outcome, so it is incomplete."},"skills":["sample spaces","finite enumeration"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0001","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6361e2017eae1d14e1bbe9204450c8f22df818acb79a95ca9bfb73549062bc3","family_id":"qf-induction-proof-obligation","seed":20260974,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":1,\"mode\":\"missing_role\",\"provided_roles\":[\"induction_hypothesis\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 1 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the base case. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0002","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4497134501edc8a866daed7501b80102c72702ea492e202efea97b3aab706dbd","family_id":"qf-induction-proof-obligation","seed":20260975,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":2,\"mode\":\"identify_role\",\"provided_roles\":[\"base_case\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the base case; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0003","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac29d20970642429da3b5a8fed3502a728abe3917938379c619cd906052f0ba","family_id":"qf-induction-proof-obligation","seed":20260976,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":3,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 3 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0004","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ca5ec92f646aad6108c064c33c548f49570bc9122c1b0e600d8a6c0ac342bb6","family_id":"qf-induction-proof-obligation","seed":20260977,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":4,\"mode\":\"identify_role\",\"provided_roles\":[\"induction_hypothesis\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0005","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cdcf83ce1a3b6b39cdab4f8f350d3ff4f948a1cf8380e57160eed2a0bd65e36","family_id":"qf-induction-proof-obligation","seed":20260978,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":5,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_hypothesis\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_step\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, conclusion and checks 5 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"induction step","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the induction step. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0006","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8926f78d3fa78a964226abec21cef2fbe9fc14348813651356bc2017ba0c7544","family_id":"qf-induction-proof-obligation","seed":20260979,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":6,\"mode\":\"identify_role\",\"provided_roles\":[\"induction_step\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_step\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “induction step”?","answer":{"kind":"choice","value":"induction step","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the induction step; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0007","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48cd039cfad90c5aec8d3d6ce5b0842718e8fa556bf5d7e1fb087cc0e8003de","family_id":"qf-induction-proof-obligation","seed":20260980,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":7,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_hypothesis\",\"induction_step\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"conclusion\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 7 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"conclusion","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the conclusion. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0008","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cdf769e49ea90485e310b191b71792e5ab16d786fdf82a890113e1ea4e09351","family_id":"qf-induction-proof-obligation","seed":20260981,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":8,\"mode\":\"identify_role\",\"provided_roles\":[\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"conclusion\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “conclusion”?","answer":{"kind":"choice","value":"conclusion","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the conclusion; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0009","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e985838490ba3909ff9ecb198631d1a202a5bf7ad2af57a94ffd1d2d6021ae","family_id":"qf-induction-proof-obligation","seed":20260982,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":9,\"mode\":\"missing_role\",\"provided_roles\":[\"induction_hypothesis\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 9 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the base case. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0010","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa03d9bc312a9afdc6eb69d520be1392e6a211296a3172a988c5fdb51ac4418","family_id":"qf-induction-proof-obligation","seed":20260983,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":10,\"mode\":\"identify_role\",\"provided_roles\":[\"base_case\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the base case; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0011","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56609684814a33afe8343ea704f380f65d6e7eb67c3a90682479cbadcebfd085","family_id":"qf-induction-proof-obligation","seed":20260984,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":11,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 11 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0012","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5ba9132fc492a2cc3436f844b49dee00df71e7a1b6abbd14be98480c1fe1bde","family_id":"qf-induction-proof-obligation","seed":20260985,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":12,\"mode\":\"identify_role\",\"provided_roles\":[\"induction_hypothesis\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0013","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16953c27d32f297f0a057b24362cc29e46826acbd8eede19f5394aee8ec48aec","family_id":"qf-induction-proof-obligation","seed":20260986,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":13,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_hypothesis\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_step\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, conclusion and checks 13 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"induction step","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the induction step. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0014","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ff805a0b0fe8c4e4303736957a4c16329b6f0788a5e25890ed87ad94401c5a","family_id":"qf-induction-proof-obligation","seed":20260987,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":14,\"mode\":\"identify_role\",\"provided_roles\":[\"induction_step\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_step\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “induction step”?","answer":{"kind":"choice","value":"induction step","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the induction step; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0015","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e97d271f5372cbb07c5c9ec6c37db0d4173600ec231d38277f644e9582c6cde","family_id":"qf-induction-proof-obligation","seed":20260988,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":15,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_hypothesis\",\"induction_step\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"conclusion\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 15 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"conclusion","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the conclusion. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0016","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c4dcb748048969ac60887fdb4fa71254066051fae0e9a627dcbe95e6412452","family_id":"qf-induction-proof-obligation","seed":20260989,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":16,\"mode\":\"identify_role\",\"provided_roles\":[\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"conclusion\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “conclusion”?","answer":{"kind":"choice","value":"conclusion","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the conclusion; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0017","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91d1b2282eb9d05804bb9022609c1f6d0cc82a7edbd976c7d0c90f4028fa8564","family_id":"qf-induction-proof-obligation","seed":20260990,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":17,\"mode\":\"missing_role\",\"provided_roles\":[\"induction_hypothesis\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies induction hypothesis, induction step, conclusion and checks 17 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the base case. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0018","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770e0db1d16ae67433a365f10c6e6a4962919ac9092dfca7dc99ab19996eed39","family_id":"qf-induction-proof-obligation","seed":20260991,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":18,\"mode\":\"identify_role\",\"provided_roles\":[\"base_case\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"base_case\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “base case”?","answer":{"kind":"choice","value":"base case","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the base case; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0019","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd5d2dc744a96cc69f4e0c11484647d8de96e24a50015d184c92caf9edf870f","family_id":"qf-induction-proof-obligation","seed":20260992,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":19,\"mode\":\"missing_role\",\"provided_roles\":[\"base_case\",\"induction_step\",\"conclusion\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 19 finite example(s). Which induction role is missing?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The missing role is the induction hypothesis. Finite examples do not discharge that proof obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-induction-proof-obligation-0020","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e13b97c2f74b3b2f434d91781517e96d519b4a716c7f9a93d2508d7d0c6c2a3b","family_id":"qf-induction-proof-obligation","seed":20260993,"semantic_template":"qf-induction-proof-obligation/{\"base_index\":0,\"finite_examples_checked\":20,\"mode\":\"identify_role\",\"provided_roles\":[\"induction_hypothesis\"],\"step_relation\":\"P(n)_implies_P(n+1)\",\"target_role\":\"induction_hypothesis\"}"},"fields":{"title":"Induction Proof Obligations","stem":"For induction on P(n) from 0, which proof role is represented by the supplied clause “induction hypothesis”?","answer":{"kind":"choice","value":"induction hypothesis","choices":["base case","induction hypothesis","induction step","conclusion"],"explanation":"The clause has the role of the induction hypothesis; each induction role has a distinct scope and obligation."},"skills":["mathematical induction","proof obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0001","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7203d3309c0c9e23881e7218a5cce7bd3697c0470e29c3a83c75b3d02c78ec7","family_id":"qf-basic-counting-principle","seed":20260994,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[2,3],\"mode\":\"addition_disjoint\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A task can be completed through two explicitly disjoint branches with 2 and 3 outcomes. What is the total?","answer":{"kind":"numeric","value":5,"explanation":"The branches are explicitly disjoint, so 2+3=5."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0002","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2115f9d2020a49bac2b0a524070e63dca26e308adc150330cb404f447c8e6277","family_id":"qf-basic-counting-principle","seed":20260995,"semantic_template":"qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,5]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 5 second-stage choices. How many combinations are there?","answer":{"kind":"numeric","value":15,"explanation":"Every first-stage choice has a fixed 5 continuations, so 3×5=15."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0003","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abdec127a76c9d37a704c890b507289bb498c32847a0b96b83add075ad293a0e","family_id":"qf-basic-counting-principle","seed":20260996,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[4,5],\"mode\":\"addition_overlap\",\"overlap_witness\":\"共用結果3\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"Two branches report 4 and 5 outcomes, but both include shared outcome 3. May their counts be added unconditionally?","answer":{"kind":"choice","value":"Cannot add unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The shared outcome belongs to both branches, so direct addition would count it twice."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0004","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"509e82b156472f019afd07c3f097715fda89fd4b4e27161bb69ce8b98f5785c1","family_id":"qf-basic-counting-principle","seed":20260997,"semantic_template":"qf-basic-counting-principle/{\"conditional_stage_witness\":[4,5],\"mode\":\"multiplication_conditional\",\"stages\":[5,4]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 4 or 5. May one fixed product be used unconditionally?","answer":{"kind":"choice","value":"Cannot multiply unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The second-stage count changes with the first choice, so one unconditional fixed product is invalid."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0005","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b382b132fe57f9fd1ebe1fba066abc11632be5c881e5932a555230c54801fc8","family_id":"qf-basic-counting-principle","seed":20260998,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[6,7],\"mode\":\"addition_disjoint\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A task can be completed through two explicitly disjoint branches with 6 and 7 outcomes. What is the total?","answer":{"kind":"numeric","value":13,"explanation":"The branches are explicitly disjoint, so 6+7=13."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0006","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93bab3d5cfdfbc202d67c4c86a274c346c203c51bc307262ef8fc7afb1adedf5","family_id":"qf-basic-counting-principle","seed":20260999,"semantic_template":"qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,3]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 3 second-stage choices. How many combinations are there?","answer":{"kind":"numeric","value":9,"explanation":"Every first-stage choice has a fixed 3 continuations, so 3×3=9."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0007","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31578165fe0dab9f2deea59ea501d6f0adcf4aecdd254b24503ab299439c91cf","family_id":"qf-basic-counting-principle","seed":20261000,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[8,3],\"mode\":\"addition_overlap\",\"overlap_witness\":\"共用結果7\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"Two branches report 8 and 3 outcomes, but both include shared outcome 7. May their counts be added unconditionally?","answer":{"kind":"choice","value":"Cannot add unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The shared outcome belongs to both branches, so direct addition would count it twice."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0008","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71e7bece9f2a634af1e5c9344781b30ab0fc7131701c822741ff172a8bcbf68","family_id":"qf-basic-counting-principle","seed":20261001,"semantic_template":"qf-basic-counting-principle/{\"conditional_stage_witness\":[7,8],\"mode\":\"multiplication_conditional\",\"stages\":[5,7]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 7 or 8. May one fixed product be used unconditionally?","answer":{"kind":"choice","value":"Cannot multiply unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The second-stage count changes with the first choice, so one unconditional fixed product is invalid."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0009","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"335952b15958e2c6828007b9aa371e43ff3ad3ef23de06be91e4b034ceb87c0b","family_id":"qf-basic-counting-principle","seed":20261002,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[10,5],\"mode\":\"addition_disjoint\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A task can be completed through two explicitly disjoint branches with 10 and 5 outcomes. What is the total?","answer":{"kind":"numeric","value":15,"explanation":"The branches are explicitly disjoint, so 10+5=15."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0010","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9613b592a9c9fbe1f1ae57b46a43e563d68465904fdbad1af8c896b78bdc06b6","family_id":"qf-basic-counting-principle","seed":20261003,"semantic_template":"qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,6]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 6 second-stage choices. How many combinations are there?","answer":{"kind":"numeric","value":18,"explanation":"Every first-stage choice has a fixed 6 continuations, so 3×6=18."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0011","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48079f58f5629b73a288d9b029d356308e6c97c2693a16d6b20939201fe08cb7","family_id":"qf-basic-counting-principle","seed":20261004,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[12,7],\"mode\":\"addition_overlap\",\"overlap_witness\":\"共用結果11\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"Two branches report 12 and 7 outcomes, but both include shared outcome 11. May their counts be added unconditionally?","answer":{"kind":"choice","value":"Cannot add unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The shared outcome belongs to both branches, so direct addition would count it twice."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0012","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"002d035de35b6cb42c4765937815e18ec2829b3907b8c9df89888d7f0ba7a8c3","family_id":"qf-basic-counting-principle","seed":20261005,"semantic_template":"qf-basic-counting-principle/{\"conditional_stage_witness\":[5,6],\"mode\":\"multiplication_conditional\",\"stages\":[5,5]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 5 or 6. May one fixed product be used unconditionally?","answer":{"kind":"choice","value":"Cannot multiply unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The second-stage count changes with the first choice, so one unconditional fixed product is invalid."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0013","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c7dbf6e5fa7f167b26cac8e82f903b25df9306f979c8bb8d950a5c043ee4d5e","family_id":"qf-basic-counting-principle","seed":20261006,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[14,3],\"mode\":\"addition_disjoint\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A task can be completed through two explicitly disjoint branches with 14 and 3 outcomes. What is the total?","answer":{"kind":"numeric","value":17,"explanation":"The branches are explicitly disjoint, so 14+3=17."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0014","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ef3554bb41635ac23a2c29efc1d597605bdfc14dbacd450ad33417d1d7009df","family_id":"qf-basic-counting-principle","seed":20261007,"semantic_template":"qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,4]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 4 second-stage choices. How many combinations are there?","answer":{"kind":"numeric","value":12,"explanation":"Every first-stage choice has a fixed 4 continuations, so 3×4=12."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0015","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2e00abf2268f36b60184a1d13d102aa7997f41bfa72c18cd228b450e9754c4d","family_id":"qf-basic-counting-principle","seed":20261008,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[16,5],\"mode\":\"addition_overlap\",\"overlap_witness\":\"共用結果15\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"Two branches report 16 and 5 outcomes, but both include shared outcome 15. May their counts be added unconditionally?","answer":{"kind":"choice","value":"Cannot add unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The shared outcome belongs to both branches, so direct addition would count it twice."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0016","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6235b78e4eb9357d5f36d92a7fe145cfd88e12ec942b0444bb5e741682b5a01","family_id":"qf-basic-counting-principle","seed":20261009,"semantic_template":"qf-basic-counting-principle/{\"conditional_stage_witness\":[3,4],\"mode\":\"multiplication_conditional\",\"stages\":[5,3]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 3 or 4. May one fixed product be used unconditionally?","answer":{"kind":"choice","value":"Cannot multiply unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The second-stage count changes with the first choice, so one unconditional fixed product is invalid."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0017","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e65abe078567781b55ab8e60f7e23a013b3100a42ba73d181fbbe0cd4c533111","family_id":"qf-basic-counting-principle","seed":20261010,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[18,7],\"mode\":\"addition_disjoint\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A task can be completed through two explicitly disjoint branches with 18 and 7 outcomes. What is the total?","answer":{"kind":"numeric","value":25,"explanation":"The branches are explicitly disjoint, so 18+7=25."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0018","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fa3a5827c332884ebce8631f4257cc28ecce233ab2a19ad104b236aed94d7f9","family_id":"qf-basic-counting-principle","seed":20261011,"semantic_template":"qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,7]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 7 second-stage choices. How many combinations are there?","answer":{"kind":"numeric","value":21,"explanation":"Every first-stage choice has a fixed 7 continuations, so 3×7=21."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0019","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"779e309fed1e74ecd0c08d136abda5794d3029cbdc50c144e5b61c9b75da35be","family_id":"qf-basic-counting-principle","seed":20261012,"semantic_template":"qf-basic-counting-principle/{\"branch_counts\":[20,3],\"mode\":\"addition_overlap\",\"overlap_witness\":\"共用結果19\"}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"Two branches report 20 and 3 outcomes, but both include shared outcome 19. May their counts be added unconditionally?","answer":{"kind":"choice","value":"Cannot add unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The shared outcome belongs to both branches, so direct addition would count it twice."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260914-basic-counting-principle-0020","batch_id":"batch-20260914-domain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5169a04fce6185f0e51ffdbab8c58bf6255eeb54ba4f849756f17b15637987a2","family_id":"qf-basic-counting-principle","seed":20261013,"semantic_template":"qf-basic-counting-principle/{\"conditional_stage_witness\":[6,7],\"mode\":\"multiplication_conditional\",\"stages\":[5,6]}"},"fields":{"title":"Boundary of the Basic Counting Principles","stem":"A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 6 or 7. May one fixed product be used unconditionally?","answer":{"kind":"choice","value":"Cannot multiply unconditionally","choices":["Cannot add unconditionally","Cannot multiply unconditionally"],"explanation":"The second-stage count changes with the first choice, so one unconditional fixed product is invalid."},"skills":["basic counting principles","condition recognition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}