{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0001","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":0,\"k\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(0, 0, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095001,"parameters":{"n":0,"k":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":0,\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0002","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":1,\"k\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(1, 1, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095002,"parameters":{"n":1,"k":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":1,\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0003","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":2,\"k\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(2, 2, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095003,"parameters":{"n":2,"k":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":2,\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0004","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":3,\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(3, 3, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095004,"parameters":{"n":3,"k":3,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":3,\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0005","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":4,\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(4, 4, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095005,"parameters":{"n":4,"k":4,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":4,\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0006","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":5,\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(5, 5, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095006,"parameters":{"n":5,"k":5,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":5,\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0007","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":6,\"k\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(6, 6, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095007,"parameters":{"n":6,"k":6,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":6,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0008","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":7,\"k\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(7, 7, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095008,"parameters":{"n":7,"k":7,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":7,\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0009","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":8,\"k\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(8, 8, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095009,"parameters":{"n":8,"k":8,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":8,\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0010","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":9,\"k\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(9, 9, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095010,"parameters":{"n":9,"k":9,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":9,\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0011","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":10,\"k\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(10, 10, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095011,"parameters":{"n":10,"k":10,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":10,\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0012","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":11,\"k\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 11, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(11, 11, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095012,"parameters":{"n":11,"k":11,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":11,\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0013","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":12,\"k\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(12, 12, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095013,"parameters":{"n":12,"k":12,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":12,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0014","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":13,\"k\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 13, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(13, 13, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095014,"parameters":{"n":13,"k":13,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":13,\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0015","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":14,\"k\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 14, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(14, 14, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095015,"parameters":{"n":14,"k":14,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":14,\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0016","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":15,\"k\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 15, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(15, 15, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095016,"parameters":{"n":15,"k":15,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":15,\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0017","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":16,\"k\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(16, 16, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095017,"parameters":{"n":16,"k":16,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":16,\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0018","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":17,\"k\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 17, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(17, 17, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095018,"parameters":{"n":17,"k":17,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":17,\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0019","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":18,\"k\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(18, 18, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095019,"parameters":{"n":18,"k":18,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":18,\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0020","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":19,\"k\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 19, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(19, 19, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095020,"parameters":{"n":19,"k":19,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":19,\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0021","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":20,\"k\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 20, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(20, 20, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095021,"parameters":{"n":20,"k":20,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":20,\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0022","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":21,\"k\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 21, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(21, 21, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095022,"parameters":{"n":21,"k":21,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":21,\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0023","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":22,\"k\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 22, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(22, 22, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095023,"parameters":{"n":22,"k":22,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":22,\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0024","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":23,\"k\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 23, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(23, 23, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095024,"parameters":{"n":23,"k":23,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":23,\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0025","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":24,\"k\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 24, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(24, 24, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-coefficient","seed":202611095025,"parameters":{"n":24,"k":24,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-coefficient","semantic_template":"qf-mod40-binomial-coefficient/{\"k\":24,\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0026","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 1, 0, 0]], constant_one_boundary)","explanation_zh":"Wave6第三層核對=(0, [[0, 1, 0, 0]], constant_one_boundary)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095026,"parameters":{"n":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0027","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095027,"parameters":{"n":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0028","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095028,"parameters":{"n":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0029","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095029,"parameters":{"n":3,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0030","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095030,"parameters":{"n":4,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0031","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095031,"parameters":{"n":5,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0032","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095032,"parameters":{"n":6,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0033","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095033,"parameters":{"n":7,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0034","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095034,"parameters":{"n":8,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0035","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095035,"parameters":{"n":9,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0036","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095036,"parameters":{"n":10,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0037","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095037,"parameters":{"n":11,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0038","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095038,"parameters":{"n":12,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0039","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095039,"parameters":{"n":13,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0040","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095040,"parameters":{"n":14,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0041","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095041,"parameters":{"n":15,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0042","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095042,"parameters":{"n":16,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0043","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095043,"parameters":{"n":17,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0044","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095044,"parameters":{"n":18,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0045","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095045,"parameters":{"n":19,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0046","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095046,"parameters":{"n":20,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0047","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095047,"parameters":{"n":21,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0048","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095048,"parameters":{"n":22,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0049","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095049,"parameters":{"n":23,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0050","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-expansion；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients)","explanation_zh":"Wave6第三層核對=(24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-expansion","seed":202611095050,"parameters":{"n":24,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-expansion","semantic_template":"qf-mod40-binomial-expansion/{\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0051","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":0,\"x\":-3,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, -3, 2, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(0, -3, 2, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095051,"parameters":{"n":0,"x":-3,"y":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":0,\"x\":-3,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0052","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":1,\"x\":-2,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -2, 1, -1, -1, 是)","explanation_zh":"Wave6第三層核對=(1, -2, 1, -1, -1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095052,"parameters":{"n":1,"x":-2,"y":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":1,\"x\":-2,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0053","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":2,\"x\":-1,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -1, 0, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(2, -1, 0, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095053,"parameters":{"n":2,"x":-1,"y":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":2,\"x\":-1,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0054","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":3,\"x\":0,\"y\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, -1, -1, -1, 是)","explanation_zh":"Wave6第三層核對=(3, 0, -1, -1, -1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095054,"parameters":{"n":3,"x":0,"y":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":3,\"x\":0,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0055","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":4,\"x\":1,\"y\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1, -2, 1, 1, 是)","explanation_zh":"Wave6第三層核對=(4, 1, -2, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095055,"parameters":{"n":4,"x":1,"y":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":4,\"x\":1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0056","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":5,\"x\":2,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 2, 1024, 1024, 是)","explanation_zh":"Wave6第三層核對=(5, 2, 2, 1024, 1024, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095056,"parameters":{"n":5,"x":2,"y":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":5,\"x\":2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0057","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":6,\"x\":3,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 1, 4096, 4096, 是)","explanation_zh":"Wave6第三層核對=(6, 3, 1, 4096, 4096, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095057,"parameters":{"n":6,"x":3,"y":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":6,\"x\":3,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0058","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":7,\"x\":-3,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, -3, 0, -2187, -2187, 是)","explanation_zh":"Wave6第三層核對=(7, -3, 0, -2187, -2187, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095058,"parameters":{"n":7,"x":-3,"y":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":7,\"x\":-3,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0059","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":8,\"x\":-2,\"y\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, -2, -1, 6561, 6561, 是)","explanation_zh":"Wave6第三層核對=(8, -2, -1, 6561, 6561, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095059,"parameters":{"n":8,"x":-2,"y":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":8,\"x\":-2,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0060","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":9,\"x\":-1,\"y\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, -1, -2, -19683, -19683, 是)","explanation_zh":"Wave6第三層核對=(9, -1, -2, -19683, -19683, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095060,"parameters":{"n":9,"x":-1,"y":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":9,\"x\":-1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0061","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":10,\"x\":0,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 2, 1024, 1024, 是)","explanation_zh":"Wave6第三層核對=(10, 0, 2, 1024, 1024, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095061,"parameters":{"n":10,"x":0,"y":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":10,\"x\":0,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0062","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":11,\"x\":1,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 1, 1, 2048, 2048, 是)","explanation_zh":"Wave6第三層核對=(11, 1, 1, 2048, 2048, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095062,"parameters":{"n":11,"x":1,"y":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":11,\"x\":1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0063","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":12,\"x\":2,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 0, 4096, 4096, 是)","explanation_zh":"Wave6第三層核對=(12, 2, 0, 4096, 4096, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095063,"parameters":{"n":12,"x":2,"y":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":12,\"x\":2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0064","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":13,\"x\":3,\"y\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 3, -1, 8192, 8192, 是)","explanation_zh":"Wave6第三層核對=(13, 3, -1, 8192, 8192, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095064,"parameters":{"n":13,"x":3,"y":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":13,\"x\":3,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0065","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":14,\"x\":-3,\"y\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, -3, -2, 6103515625, 6103515625, 是)","explanation_zh":"Wave6第三層核對=(14, -3, -2, 6103515625, 6103515625, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095065,"parameters":{"n":14,"x":-3,"y":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":14,\"x\":-3,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0066","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":15,\"x\":-2,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, -2, 2, 0, 0, 是)","explanation_zh":"Wave6第三層核對=(15, -2, 2, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095066,"parameters":{"n":15,"x":-2,"y":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":15,\"x\":-2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0067","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":16,\"x\":-1,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, -1, 1, 0, 0, 是)","explanation_zh":"Wave6第三層核對=(16, -1, 1, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095067,"parameters":{"n":16,"x":-1,"y":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":16,\"x\":-1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0068","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":17,\"x\":0,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 0, 0, 0, 0, 是)","explanation_zh":"Wave6第三層核對=(17, 0, 0, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095068,"parameters":{"n":17,"x":0,"y":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":17,\"x\":0,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0069","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":18,\"x\":1,\"y\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 1, -1, 0, 0, 是)","explanation_zh":"Wave6第三層核對=(18, 1, -1, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095069,"parameters":{"n":18,"x":1,"y":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":18,\"x\":1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0070","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":19,\"x\":2,\"y\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 2, -2, 0, 0, 是)","explanation_zh":"Wave6第三層核對=(19, 2, -2, 0, 0, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095070,"parameters":{"n":19,"x":2,"y":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":19,\"x\":2,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0071","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":20,\"x\":3,\"y\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 3, 2, 95367431640625, 95367431640625, 是)","explanation_zh":"Wave6第三層核對=(20, 3, 2, 95367431640625, 95367431640625, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095071,"parameters":{"n":20,"x":3,"y":2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":20,\"x\":3,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0072","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":21,\"x\":-3,\"y\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, -3, 1, -2097152, -2097152, 是)","explanation_zh":"Wave6第三層核對=(21, -3, 1, -2097152, -2097152, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095072,"parameters":{"n":21,"x":-3,"y":1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":21,\"x\":-3,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0073","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":22,\"x\":-2,\"y\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, -2, 0, 4194304, 4194304, 是)","explanation_zh":"Wave6第三層核對=(22, -2, 0, 4194304, 4194304, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095073,"parameters":{"n":22,"x":-2,"y":0,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":22,\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0074","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":23,\"x\":-1,\"y\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, -1, -1, -8388608, -8388608, 是)","explanation_zh":"Wave6第三層核對=(23, -1, -1, -8388608, -8388608, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095074,"parameters":{"n":23,"x":-1,"y":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":23,\"x\":-1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0075","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-evaluation-identity；資料={\"n\":24,\"x\":0,\"y\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 0, -2, 16777216, 16777216, 是)","explanation_zh":"Wave6第三層核對=(24, 0, -2, 16777216, 16777216, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095075,"parameters":{"n":24,"x":0,"y":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-evaluation-identity","semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":24,\"x\":0,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0076","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-1, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095076,"parameters":{"exponent":-1,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0077","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-2, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095077,"parameters":{"exponent":-2,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0078","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-3, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095078,"parameters":{"exponent":-3,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0079","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-4, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095079,"parameters":{"exponent":-4,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0080","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-5, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095080,"parameters":{"exponent":-5,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0081","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-6, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095081,"parameters":{"exponent":-6,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0082","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-7, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095082,"parameters":{"exponent":-7,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0083","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-8, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095083,"parameters":{"exponent":-8,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0084","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-9, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095084,"parameters":{"exponent":-9,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0085","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-10, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095085,"parameters":{"exponent":-10,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0086","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-11, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095086,"parameters":{"exponent":-11,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0087","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-12, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095087,"parameters":{"exponent":-12,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0088","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-13, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095088,"parameters":{"exponent":-13,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0089","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-14, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095089,"parameters":{"exponent":-14,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0090","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-15, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095090,"parameters":{"exponent":-15,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0091","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-16, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095091,"parameters":{"exponent":-16,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0092","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-17, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095092,"parameters":{"exponent":-17,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0093","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-18, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095093,"parameters":{"exponent":-18,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0094","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-19, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095094,"parameters":{"exponent":-19,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0095","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-20, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095095,"parameters":{"exponent":-20,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0096","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-21, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095096,"parameters":{"exponent":-21,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0097","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-22, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095097,"parameters":{"exponent":-22,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0098","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-23, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095098,"parameters":{"exponent":-23,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0099","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-24, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095099,"parameters":{"exponent":-24,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r05-binomial-theorem-0100","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-binomial-theorem"],"title_zh":"二項式定理","stem_zh":"commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation_zh":"Wave6第三層核對=(-25, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二項式定理"]},"variant":{"family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095100,"parameters":{"exponent":-25,"domain":"Exact_commutative_binomial_theorem","operation":"binomial-exponent-boundary","semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}