{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0001","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], u_xx+u_yy=source, -2, -1, -1, 否, interior_only)","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], u_xx+u_yy=source, -2, -1, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251001,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"source":-1,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0002","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251002,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"source":2,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0003","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-10,0,-1,3,1,0],\"source\":6}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251003,"parameters":{"coefficients":[-10,0,-1,3,1,0],"source":6,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-10,0,-1,3,1,0],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0004","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251004,"parameters":{"coefficients":[-9,1,0,4,-1,1],"source":10,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0005","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-8,2,1,1,0,2],\"source\":7}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], u_xx+u_yy=source, 6, 7, -1, 否, interior_only)","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], u_xx+u_yy=source, 6, 7, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251005,"parameters":{"coefficients":[-8,2,1,1,0,2],"source":7,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-8,2,1,1,0,2],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0006","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], u_xx+u_yy=source, 0, 0, 0, 是, interior_only)","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], u_xx+u_yy=source, 0, 0, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251006,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"source":0,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0007","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":4}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251007,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"source":4,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0008","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-5,0,-3,4,0,0],\"source\":8}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251008,"parameters":{"coefficients":[-5,0,-3,4,0,0],"source":8,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-5,0,-3,4,0,0],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0009","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-4,1,-2,1,1,1],\"source\":5}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], u_xx+u_yy=source, 4, 5, -1, 否, interior_only)","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], u_xx+u_yy=source, 4, 5, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251009,"parameters":{"coefficients":[-4,1,-2,1,1,1],"source":5,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-4,1,-2,1,1,1],\"source\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0010","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251010,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"source":8,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0011","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"source\":2}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251011,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"source":2,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-2,-2,0,3,0,-2],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0012","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251012,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"source":6,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0013","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[0,0,2,1,-1,0],\"source\":3}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], u_xx+u_yy=source, 2, 3, -1, 否, interior_only)","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], u_xx+u_yy=source, 2, 3, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251013,"parameters":{"coefficients":[0,0,2,1,-1,0],"source":3,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[0,0,2,1,-1,0],\"source\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0014","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[1,1,3,2,0,1],\"source\":6}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251014,"parameters":{"coefficients":[1,1,3,2,0,1],"source":6,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[1,1,3,2,0,1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0015","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[2,2,-3,3,1,2],\"source\":10}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251015,"parameters":{"coefficients":[2,2,-3,3,1,2],"source":10,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[2,2,-3,3,1,2],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0016","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251016,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"source":4,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0017","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"source\":1}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], u_xx+u_yy=source, 0, 1, -1, 否, interior_only)","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], u_xx+u_yy=source, 0, 1, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251017,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"source":1,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[4,-1,-1,1,0,-1],\"source\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0018","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[5,0,0,2,1,0],\"source\":4}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], u_xx+u_yy=source, 4, 4, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251018,"parameters":{"coefficients":[5,0,0,2,1,0],"source":4,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[5,0,0,2,1,0],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0019","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[6,1,1,3,-1,1],\"source\":8}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], u_xx+u_yy=source, 8, 8, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251019,"parameters":{"coefficients":[6,1,1,3,-1,1],"source":8,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[6,1,1,3,-1,1],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0020","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[7,2,2,4,0,2],\"source\":12}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], u_xx+u_yy=source, 12, 12, 0, 是, interior_only)","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], u_xx+u_yy=source, 12, 12, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251020,"parameters":{"coefficients":[7,2,2,4,0,2],"source":12,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[7,2,2,4,0,2],\"source\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0021","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[8,-2,3,1,1,-2],\"source\":-1}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], u_xx+u_yy=source, -2, -1, -1, 否, interior_only)","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], u_xx+u_yy=source, -2, -1, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251021,"parameters":{"coefficients":[8,-2,3,1,1,-2],"source":-1,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[8,-2,3,1,1,-2],\"source\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0022","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":2}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], u_xx+u_yy=source, 2, 2, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251022,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"source":2,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0023","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[10,0,-2,3,0,0],\"source\":6}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], u_xx+u_yy=source, 6, 6, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251023,"parameters":{"coefficients":[10,0,-2,3,0,0],"source":6,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[10,0,-2,3,0,0],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0024","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[11,1,-1,4,1,1],\"source\":10}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], u_xx+u_yy=source, 10, 10, 0, 是, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251024,"parameters":{"coefficients":[11,1,-1,4,1,1],"source":10,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0025","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-interior-pde-identity；資料={\"coefficients\":[12,2,0,1,-1,2],\"source\":7}。請依 [coefficients,equation,laplacian,source,residual,interior_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], u_xx+u_yy=source, 6, 7, -1, 否, interior_only)","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], u_xx+u_yy=source, 6, 7, -1, 否, interior_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-interior-pde-identity","seed":202610251025,"parameters":{"coefficients":[12,2,0,1,-1,2],"source":7,"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-interior-pde-identity","semantic_template":"qf-mod25-bvp-interior-pde-identity/{\"coefficients\":[12,2,0,1,-1,2],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0026","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"bounds\":[-2,2,-1,3],\"edge\":\"left\",\"boundary_coefficients\":[-3,-1,-2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-2, 2, -1, 3], left, y, [-4, -1, -2], [-3, -1, -2], 否, declared_edge_only)","explanation_zh":"收束波核對=([-2, 2, -1, 3], left, y, [-4, -1, -2], [-3, -1, -2], 否, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251026,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"edge":"left","boundary_coefficients":[-3,-1,-2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-3,-1,-2],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0027","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"bounds\":[-3,3,-2,4],\"edge\":\"right\",\"boundary_coefficients\":[4,-2,-1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], 是, declared_edge_only)","explanation_zh":"收束波核對=([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251027,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"bounds":[-3,3,-2,4],"edge":"right","boundary_coefficients":[4,-2,-1],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[4,-2,-1],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0028","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-10,0,-1,3,1,0],\"bounds\":[-4,4,-3,5],\"edge\":\"bottom\",\"boundary_coefficients\":[-7,-3,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-4, 4, -3, 5], bottom, x, [-7, -3, 3], [-7, -3, 3], 是, declared_edge_only)","explanation_zh":"收束波核對=([-4, 4, -3, 5], bottom, x, [-7, -3, 3], [-7, -3, 3], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251028,"parameters":{"coefficients":[-10,0,-1,3,1,0],"bounds":[-4,4,-3,5],"edge":"bottom","boundary_coefficients":[-7,-3,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-7,-3,3],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0029","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-9,1,0,4,-1,1],\"bounds\":[-5,5,-4,6],\"edge\":\"top\",\"boundary_coefficients\":[27,-5,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-5, 5, -4, 6], top, x, [27, -5, 4], [27, -5, 4], 是, declared_edge_only)","explanation_zh":"收束波核對=([-5, 5, -4, 6], top, x, [27, -5, 4], [27, -5, 4], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251029,"parameters":{"coefficients":[-9,1,0,4,-1,1],"bounds":[-5,5,-4,6],"edge":"top","boundary_coefficients":[27,-5,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[27,-5,4],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0030","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-8,2,1,1,0,2],\"bounds\":[-6,6,-5,7],\"edge\":\"left\",\"boundary_coefficients\":[16,1,2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-6, 6, -5, 7], left, y, [16, 1, 2], [16, 1, 2], 是, declared_edge_only)","explanation_zh":"收束波核對=([-6, 6, -5, 7], left, y, [16, 1, 2], [16, 1, 2], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251030,"parameters":{"coefficients":[-8,2,1,1,0,2],"bounds":[-6,6,-5,7],"edge":"left","boundary_coefficients":[16,1,2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[16,1,2],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0031","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"bounds\":[-7,7,-6,8],\"edge\":\"right\",\"boundary_coefficients\":[78,9,-2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-7, 7, -6, 8], right, y, [77, 9, -2], [78, 9, -2], 否, declared_edge_only)","explanation_zh":"收束波核對=([-7, 7, -6, 8], right, y, [77, 9, -2], [78, 9, -2], 否, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251031,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"bounds":[-7,7,-6,8],"edge":"right","boundary_coefficients":[78,9,-2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[78,9,-2],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0032","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"bounds\":[-8,8,-7,9],\"edge\":\"bottom\",\"boundary_coefficients\":[-76,6,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-8, 8, -7, 9], bottom, x, [-76, 6, 3], [-76, 6, 3], 是, declared_edge_only)","explanation_zh":"收束波核對=([-8, 8, -7, 9], bottom, x, [-76, 6, 3], [-76, 6, 3], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251032,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"bounds":[-8,8,-7,9],"edge":"bottom","boundary_coefficients":[-76,6,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-76,6,3],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0033","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-5,0,-3,4,0,0],\"bounds\":[-9,9,-8,10],\"edge\":\"top\",\"boundary_coefficients\":[-35,0,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-9, 9, -8, 10], top, x, [-35, 0, 4], [-35, 0, 4], 是, declared_edge_only)","explanation_zh":"收束波核對=([-9, 9, -8, 10], top, x, [-35, 0, 4], [-35, 0, 4], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251033,"parameters":{"coefficients":[-5,0,-3,4,0,0],"bounds":[-9,9,-8,10],"edge":"top","boundary_coefficients":[-35,0,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-35,0,4],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0034","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-4,1,-2,1,1,1],\"bounds\":[-10,10,-9,11],\"edge\":\"left\",\"boundary_coefficients\":[86,-12,1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-10, 10, -9, 11], left, y, [86, -12, 1], [86, -12, 1], 是, declared_edge_only)","explanation_zh":"收束波核對=([-10, 10, -9, 11], left, y, [86, -12, 1], [86, -12, 1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251034,"parameters":{"coefficients":[-4,1,-2,1,1,1],"bounds":[-10,10,-9,11],"edge":"left","boundary_coefficients":[86,-12,1],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[86,-12,1],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0035","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"bounds\":[-11,11,-10,12],\"edge\":\"right\",\"boundary_coefficients\":[261,-12,2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-11, 11, -10, 12], right, y, [261, -12, 2], [261, -12, 2], 是, declared_edge_only)","explanation_zh":"收束波核對=([-11, 11, -10, 12], right, y, [261, -12, 2], [261, -12, 2], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251035,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"bounds":[-11,11,-10,12],"edge":"right","boundary_coefficients":[261,-12,2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[261,-12,2],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0036","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"bounds\":[-12,12,-11,13],\"edge\":\"bottom\",\"boundary_coefficients\":[-243,-2,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-12, 12, -11, 13], bottom, x, [-244, -2, 3], [-243, -2, 3], 否, declared_edge_only)","explanation_zh":"收束波核對=([-12, 12, -11, 13], bottom, x, [-244, -2, 3], [-243, -2, 3], 否, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251036,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"bounds":[-12,12,-11,13],"edge":"bottom","boundary_coefficients":[-243,-2,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-243,-2,3],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0037","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"bounds\":[-13,13,-12,14],\"edge\":\"top\",\"boundary_coefficients\":[-183,13,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-13, 13, -12, 14], top, x, [-183, 13, 4], [-183, 13, 4], 是, declared_edge_only)","explanation_zh":"收束波核對=([-13, 13, -12, 14], top, x, [-183, 13, 4], [-183, 13, 4], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251037,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"bounds":[-13,13,-12,14],"edge":"top","boundary_coefficients":[-183,13,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-183,13,4],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0038","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[0,0,2,1,-1,0],\"bounds\":[-14,14,-13,15],\"edge\":\"left\",\"boundary_coefficients\":[196,16,0]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-14, 14, -13, 15], left, y, [196, 16, 0], [196, 16, 0], 是, declared_edge_only)","explanation_zh":"收束波核對=([-14, 14, -13, 15], left, y, [196, 16, 0], [196, 16, 0], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251038,"parameters":{"coefficients":[0,0,2,1,-1,0],"bounds":[-14,14,-13,15],"edge":"left","boundary_coefficients":[196,16,0],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[196,16,0],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0039","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[1,1,3,2,0,1],\"bounds\":[-15,15,-14,16],\"edge\":\"right\",\"boundary_coefficients\":[466,3,1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-15, 15, -14, 16], right, y, [466, 3, 1], [466, 3, 1], 是, declared_edge_only)","explanation_zh":"收束波核對=([-15, 15, -14, 16], right, y, [466, 3, 1], [466, 3, 1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251039,"parameters":{"coefficients":[1,1,3,2,0,1],"bounds":[-15,15,-14,16],"edge":"right","boundary_coefficients":[466,3,1],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[466,3,1],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0040","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[2,2,-3,3,1,2],\"bounds\":[-16,16,-15,17],\"edge\":\"bottom\",\"boundary_coefficients\":[497,-13,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-16, 16, -15, 17], bottom, x, [497, -13, 3], [497, -13, 3], 是, declared_edge_only)","explanation_zh":"收束波核對=([-16, 16, -15, 17], bottom, x, [497, -13, 3], [497, -13, 3], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251040,"parameters":{"coefficients":[2,2,-3,3,1,2],"bounds":[-16,16,-15,17],"edge":"bottom","boundary_coefficients":[497,-13,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[497,-13,3],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0041","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"bounds\":[-17,17,-16,18],\"edge\":\"top\",\"boundary_coefficients\":[-680,-20,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-17, 17, -16, 18], top, x, [-681, -20, 4], [-680, -20, 4], 否, declared_edge_only)","explanation_zh":"收束波核對=([-17, 17, -16, 18], top, x, [-681, -20, 4], [-680, -20, 4], 否, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251041,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"bounds":[-17,17,-16,18],"edge":"top","boundary_coefficients":[-680,-20,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[-680,-20,4],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0042","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"bounds\":[-18,18,-17,19],\"edge\":\"left\",\"boundary_coefficients\":[346,-1,-1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-18, 18, -17, 19], left, y, [346, -1, -1], [346, -1, -1], 是, declared_edge_only)","explanation_zh":"收束波核對=([-18, 18, -17, 19], left, y, [346, -1, -1], [346, -1, -1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251042,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"bounds":[-18,18,-17,19],"edge":"left","boundary_coefficients":[346,-1,-1],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[346,-1,-1],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0043","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[5,0,0,2,1,0],\"bounds\":[-19,19,-18,20],\"edge\":\"right\",\"boundary_coefficients\":[727,19,0]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-19, 19, -18, 20], right, y, [727, 19, 0], [727, 19, 0], 是, declared_edge_only)","explanation_zh":"收束波核對=([-19, 19, -18, 20], right, y, [727, 19, 0], [727, 19, 0], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251043,"parameters":{"coefficients":[5,0,0,2,1,0],"bounds":[-19,19,-18,20],"edge":"right","boundary_coefficients":[727,19,0],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[727,19,0],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0044","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[6,1,1,3,-1,1],\"bounds\":[-20,20,-19,21],\"edge\":\"bottom\",\"boundary_coefficients\":[348,20,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-20, 20, -19, 21], bottom, x, [348, 20, 3], [348, 20, 3], 是, declared_edge_only)","explanation_zh":"收束波核對=([-20, 20, -19, 21], bottom, x, [348, 20, 3], [348, 20, 3], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251044,"parameters":{"coefficients":[6,1,1,3,-1,1],"bounds":[-20,20,-19,21],"edge":"bottom","boundary_coefficients":[348,20,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[348,20,3],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0045","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[7,2,2,4,0,2],\"bounds\":[-21,21,-20,22],\"edge\":\"top\",\"boundary_coefficients\":[1019,2,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-21, 21, -20, 22], top, x, [1019, 2, 4], [1019, 2, 4], 是, declared_edge_only)","explanation_zh":"收束波核對=([-21, 21, -20, 22], top, x, [1019, 2, 4], [1019, 2, 4], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251045,"parameters":{"coefficients":[7,2,2,4,0,2],"bounds":[-21,21,-20,22],"edge":"top","boundary_coefficients":[1019,2,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[1019,2,4],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0046","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[8,-2,3,1,1,-2],\"bounds\":[-22,22,-21,23],\"edge\":\"left\",\"boundary_coefficients\":[537,-19,-2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-22, 22, -21, 23], left, y, [536, -19, -2], [537, -19, -2], 否, declared_edge_only)","explanation_zh":"收束波核對=([-22, 22, -21, 23], left, y, [536, -19, -2], [537, -19, -2], 否, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251046,"parameters":{"coefficients":[8,-2,3,1,1,-2],"bounds":[-22,22,-21,23],"edge":"left","boundary_coefficients":[537,-19,-2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[537,-19,-2],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0047","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"bounds\":[-23,23,-22,24],\"edge\":\"right\",\"boundary_coefficients\":[1044,-26,-1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-23, 23, -22, 24], right, y, [1044, -26, -1], [1044, -26, -1], 是, declared_edge_only)","explanation_zh":"收束波核對=([-23, 23, -22, 24], right, y, [1044, -26, -1], [1044, -26, -1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251047,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"bounds":[-23,23,-22,24],"edge":"right","boundary_coefficients":[1044,-26,-1],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[1044,-26,-1],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"edge\":\"right\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0048","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[10,0,-2,3,0,0],\"bounds\":[-24,24,-23,25],\"edge\":\"bottom\",\"boundary_coefficients\":[56,0,3]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-24, 24, -23, 25], bottom, x, [56, 0, 3], [56, 0, 3], 是, declared_edge_only)","explanation_zh":"收束波核對=([-24, 24, -23, 25], bottom, x, [56, 0, 3], [56, 0, 3], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251048,"parameters":{"coefficients":[10,0,-2,3,0,0],"bounds":[-24,24,-23,25],"edge":"bottom","boundary_coefficients":[56,0,3],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[56,0,3],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"edge\":\"bottom\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0049","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[11,1,-1,4,1,1],\"bounds\":[-25,25,-24,26],\"edge\":\"top\",\"boundary_coefficients\":[661,27,4]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-25, 25, -24, 26], top, x, [661, 27, 4], [661, 27, 4], 是, declared_edge_only)","explanation_zh":"收束波核對=([-25, 25, -24, 26], top, x, [661, 27, 4], [661, 27, 4], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251049,"parameters":{"coefficients":[11,1,-1,4,1,1],"bounds":[-25,25,-24,26],"edge":"top","boundary_coefficients":[661,27,4],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[661,27,4],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"edge\":\"top\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0050","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[12,2,0,1,-1,2],\"bounds\":[-26,26,-25,27],\"edge\":\"left\",\"boundary_coefficients\":[636,26,2]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。","answer":{"kind":"expression","value":"([-26, 26, -25, 27], left, y, [636, 26, 2], [636, 26, 2], 是, declared_edge_only)","explanation_zh":"收束波核對=([-26, 26, -25, 27], left, y, [636, 26, 2], [636, 26, 2], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-dirichlet-edge-identity","seed":202610251050,"parameters":{"coefficients":[12,2,0,1,-1,2],"bounds":[-26,26,-25,27],"edge":"left","boundary_coefficients":[636,26,2],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-dirichlet-edge-identity","semantic_template":"qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[636,26,2],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"edge\":\"left\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0051","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"bounds\":[-2,2,-1,3],\"source\":-1,\"boundary_data\":[[\"left\",[-3,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, -1, 否, [[left, y, [-4, -1, -2], [-3, -1, -2], 否], [right, y, [-12, -5, -2], [-12, -5, -2], 是], [bottom, x, [-11, -1, 1], [-11, -1, 1], 是], [top, x, [-39, -5, 1], [-39, -5, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], -2, -1, 否, [[left, y, [-4, -1, -2], [-3, -1, -2], 否], [right, y, [-12, -5, -2], [-12, -5, -2], 是], [bottom, x, [-11, -1, 1], [-11, -1, 1], 是], [top, x, [-39, -5, 1], [-39, -5, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251051,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"source":-1,"boundary_data":[["left",[-3,-1,-2]],["right",[-12,-5,-2]],["bottom",[-11,-1,1]],["top",[-39,-5,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[-3,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0052","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"bounds\":[-3,3,-2,4],\"source\":2,\"boundary_data\":[[\"left\",[10,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], 2, 2, 是, [[left, y, [10, -2, -1], [10, -2, -1], 是], [right, y, [4, -2, -1], [4, -2, -1], 是], [bottom, x, [-11, -1, 2], [-11, -1, 2], 是], [top, x, [-35, -1, 2], [-35, -1, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], 2, 2, 是, [[left, y, [10, -2, -1], [10, -2, -1], 是], [right, y, [4, -2, -1], [4, -2, -1], 是], [bottom, x, [-11, -1, 2], [-11, -1, 2], 是], [top, x, [-35, -1, 2], [-35, -1, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251052,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"bounds":[-3,3,-2,4],"source":2,"boundary_data":[["left",[10,-2,-1]],["right",[4,-2,-1]],["bottom",[-11,-1,2]],["top",[-35,-1,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[10,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0053","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-10,0,-1,3,1,0],\"bounds\":[-4,4,-3,5],\"source\":6,\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 6, 6, 是, [[left, y, [38, -5, 0], [38, -5, 0], 是], [right, y, [38, 3, 0], [38, 3, 0], 是], [bottom, x, [-7, -3, 3], [-7, -3, 3], 是], [top, x, [-15, 5, 3], [-15, 5, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], 6, 6, 是, [[left, y, [38, -5, 0], [38, -5, 0], 是], [right, y, [38, 3, 0], [38, 3, 0], 是], [bottom, x, [-7, -3, 3], [-7, -3, 3], 是], [top, x, [-15, 5, 3], [-15, 5, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251053,"parameters":{"coefficients":[-10,0,-1,3,1,0],"bounds":[-4,4,-3,5],"source":6,"boundary_data":[["left",[38,-5,0]],["right",[38,3,0]],["bottom",[-7,-3,3]],["top",[-15,5,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0054","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-9,1,0,4,-1,1],\"bounds\":[-5,5,-4,6],\"source\":10,\"boundary_data\":[[\"left\",[86,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 10, 10, 是, [[left, y, [86, 5, 1], [86, 5, 1], 是], [right, y, [96, -5, 1], [96, -5, 1], 是], [bottom, x, [7, 5, 4], [7, 5, 4], 是], [top, x, [27, -5, 4], [27, -5, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], 10, 10, 是, [[left, y, [86, 5, 1], [86, 5, 1], 是], [right, y, [96, -5, 1], [96, -5, 1], 是], [bottom, x, [7, 5, 4], [7, 5, 4], 是], [top, x, [27, -5, 4], [27, -5, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251054,"parameters":{"coefficients":[-9,1,0,4,-1,1],"bounds":[-5,5,-4,6],"source":10,"boundary_data":[["left",[86,5,1]],["right",[96,-5,1]],["bottom",[7,5,4]],["top",[27,-5,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[86,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0055","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-8,2,1,1,0,2],\"bounds\":[-6,6,-5,7],\"source\":6,\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 6, 6, 是, [[left, y, [16, 1, 2], [16, 1, 2], 是], [right, y, [40, 1, 2], [40, 1, 2], 是], [bottom, x, [37, 2, 1], [37, 2, 1], 是], [top, x, [97, 2, 1], [97, 2, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], 6, 6, 是, [[left, y, [16, 1, 2], [16, 1, 2], 是], [right, y, [40, 1, 2], [40, 1, 2], 是], [bottom, x, [37, 2, 1], [37, 2, 1], 是], [top, x, [97, 2, 1], [97, 2, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251055,"parameters":{"coefficients":[-8,2,1,1,0,2],"bounds":[-6,6,-5,7],"source":6,"boundary_data":[["left",[16,1,2]],["right",[40,1,2]],["bottom",[37,2,1]],["top",[97,2,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0056","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"bounds\":[-7,7,-6,8],\"source\":0,\"boundary_data\":[[\"left\",[105,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], 0, 0, 是, [[left, y, [105, -5, -2], [105, -5, -2], 是], [right, y, [77, 9, -2], [77, 9, -2], 是], [bottom, x, [-91, -8, 2], [-91, -8, 2], 是], [top, x, [-119, 6, 2], [-119, 6, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], 0, 0, 是, [[left, y, [105, -5, -2], [105, -5, -2], 是], [right, y, [77, 9, -2], [77, 9, -2], 是], [bottom, x, [-91, -8, 2], [-91, -8, 2], 是], [top, x, [-119, 6, 2], [-119, 6, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251056,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"bounds":[-7,7,-6,8],"source":0,"boundary_data":[["left",[105,-5,-2]],["right",[77,9,-2]],["bottom",[-91,-8,2]],["top",[-119,6,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[105,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0057","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"bounds\":[-8,8,-7,9],\"source\":4,\"boundary_data\":[[\"left\",[195,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], 4, 4, 是, [[left, y, [194, 11, -1], [195, 11, -1], 否], [right, y, [178, -5, -1], [178, -5, -1], 是], [bottom, x, [-76, 6, 3], [-76, 6, 3], 是], [top, x, [-60, -10, 3], [-60, -10, 3], 是]], 否, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], 4, 4, 是, [[left, y, [194, 11, -1], [195, 11, -1], 否], [right, y, [178, -5, -1], [178, -5, -1], 是], [bottom, x, [-76, 6, 3], [-76, 6, 3], 是], [top, x, [-60, -10, 3], [-60, -10, 3], 是]], 否, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251057,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"bounds":[-8,8,-7,9],"source":4,"boundary_data":[["left",[195,11,-1]],["right",[178,-5,-1]],["bottom",[-76,6,3]],["top",[-60,-10,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[195,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0058","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-5,0,-3,4,0,0],\"bounds\":[-9,9,-8,10],\"source\":9,\"boundary_data\":[[\"left\",[319,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 8, 9, 否, [[left, y, [319, -3, 0], [319, -3, 0], 是], [right, y, [319, -3, 0], [319, -3, 0], 是], [bottom, x, [19, 0, 4], [19, 0, 4], 是], [top, x, [-35, 0, 4], [-35, 0, 4], 是]], 是, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], 8, 9, 否, [[left, y, [319, -3, 0], [319, -3, 0], 是], [right, y, [319, -3, 0], [319, -3, 0], 是], [bottom, x, [19, 0, 4], [19, 0, 4], 是], [top, x, [-35, 0, 4], [-35, 0, 4], 是]], 是, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251058,"parameters":{"coefficients":[-5,0,-3,4,0,0],"bounds":[-9,9,-8,10],"source":9,"boundary_data":[["left",[319,-3,0]],["right",[319,-3,0]],["bottom",[19,0,4]],["top",[-35,0,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[319,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"source\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0059","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-4,1,-2,1,1,1],\"bounds\":[-10,10,-9,11],\"source\":4,\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 4, 4, 是, [[left, y, [86, -12, 1], [86, -12, 1], 是], [right, y, [106, 8, 1], [106, 8, 1], 是], [bottom, x, [95, -8, 1], [95, -8, 1], 是], [top, x, [95, 12, 1], [95, 12, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], 4, 4, 是, [[left, y, [86, -12, 1], [86, -12, 1], 是], [right, y, [106, 8, 1], [106, 8, 1], 是], [bottom, x, [95, -8, 1], [95, -8, 1], 是], [top, x, [95, 12, 1], [95, 12, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251059,"parameters":{"coefficients":[-4,1,-2,1,1,1],"bounds":[-10,10,-9,11],"source":4,"boundary_data":[["left",[86,-12,1]],["right",[106,8,1]],["bottom",[95,-8,1]],["top",[95,12,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0060","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"bounds\":[-11,11,-10,12],\"source\":8,\"boundary_data\":[[\"left\",[217,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 8, 8, 是, [[left, y, [217, 10, 2], [217, 10, 2], 是], [right, y, [261, -12, 2], [261, -12, 2], 是], [bottom, x, [207, 12, 2], [207, 12, 2], 是], [top, x, [273, -10, 2], [273, -10, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], 8, 8, 是, [[left, y, [217, 10, 2], [217, 10, 2], 是], [right, y, [261, -12, 2], [261, -12, 2], 是], [bottom, x, [207, 12, 2], [207, 12, 2], 是], [top, x, [273, -10, 2], [273, -10, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251060,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"bounds":[-11,11,-10,12],"source":8,"boundary_data":[["left",[217,10,2]],["right",[261,-12,2]],["bottom",[207,12,2]],["top",[273,-10,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[217,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0061","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"bounds\":[-12,12,-11,13],\"source\":2,\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], 2, 2, 是, [[left, y, [454, 0, -2], [454, 0, -2], 是], [right, y, [406, 0, -2], [406, 0, -2], 是], [bottom, x, [-244, -2, 3], [-244, -2, 3], 是], [top, x, [-340, -2, 3], [-340, -2, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], 2, 2, 是, [[left, y, [454, 0, -2], [454, 0, -2], 是], [right, y, [406, 0, -2], [406, 0, -2], 是], [bottom, x, [-244, -2, 3], [-244, -2, 3], 是], [top, x, [-340, -2, 3], [-340, -2, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251061,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"bounds":[-12,12,-11,13],"source":2,"boundary_data":[["left",[454,0,-2]],["right",[406,0,-2]],["bottom",[-244,-2,3]],["top",[-340,-2,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0062","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"bounds\":[-13,13,-12,14],\"source\":6,\"boundary_data\":[[\"left\",[688,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], 6, 6, 是, [[left, y, [688, -12, -1], [688, -12, -1], 是], [right, y, [662, 14, -1], [662, 14, -1], 是], [bottom, x, [-157, -13, 4], [-157, -13, 4], 是], [top, x, [-183, 13, 4], [-183, 13, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], 6, 6, 是, [[left, y, [688, -12, -1], [688, -12, -1], 是], [right, y, [662, 14, -1], [662, 14, -1], 是], [bottom, x, [-157, -13, 4], [-157, -13, 4], 是], [top, x, [-183, 13, 4], [-183, 13, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251062,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"bounds":[-13,13,-12,14],"source":6,"boundary_data":[["left",[688,-12,-1]],["right",[662,14,-1]],["bottom",[-157,-13,4]],["top",[-183,13,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[688,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0063","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[0,0,2,1,-1,0],\"bounds\":[-14,14,-13,15],\"source\":2,\"boundary_data\":[[\"left\",[197,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 2, 2, 是, [[left, y, [196, 16, 0], [197, 16, 0], 否], [right, y, [196, -12, 0], [196, -12, 0], 是], [bottom, x, [-26, 13, 1], [-26, 13, 1], 是], [top, x, [30, -15, 1], [30, -15, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], 2, 2, 是, [[left, y, [196, 16, 0], [197, 16, 0], 否], [right, y, [196, -12, 0], [196, -12, 0], 是], [bottom, x, [-26, 13, 1], [-26, 13, 1], 是], [top, x, [30, -15, 1], [30, -15, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251063,"parameters":{"coefficients":[0,0,2,1,-1,0],"bounds":[-14,14,-13,15],"source":2,"boundary_data":[["left",[197,16,0]],["right",[196,-12,0]],["bottom",[-26,13,1]],["top",[30,-15,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[197,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0064","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[1,1,3,2,0,1],\"bounds\":[-15,15,-14,16],\"source\":6,\"boundary_data\":[[\"left\",[436,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 6, 6, 是, [[left, y, [436, 3, 1], [436, 3, 1], 是], [right, y, [466, 3, 1], [466, 3, 1], 是], [bottom, x, [155, 1, 2], [155, 1, 2], 是], [top, x, [305, 1, 2], [305, 1, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], 6, 6, 是, [[left, y, [436, 3, 1], [436, 3, 1], 是], [right, y, [466, 3, 1], [466, 3, 1], 是], [bottom, x, [155, 1, 2], [155, 1, 2], 是], [top, x, [305, 1, 2], [305, 1, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251064,"parameters":{"coefficients":[1,1,3,2,0,1],"bounds":[-15,15,-14,16],"source":6,"boundary_data":[["left",[436,3,1]],["right",[466,3,1]],["bottom",[155,1,2]],["top",[305,1,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[436,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0065","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[2,2,-3,3,1,2],\"bounds\":[-16,16,-15,17],\"source\":11,\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 10, 11, 否, [[left, y, [738, -19, 2], [738, -19, 2], 是], [right, y, [802, 13, 2], [802, 13, 2], 是], [bottom, x, [497, -13, 3], [497, -13, 3], 是], [top, x, [529, 19, 3], [529, 19, 3], 是]], 是, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], 10, 11, 否, [[left, y, [738, -19, 2], [738, -19, 2], 是], [right, y, [802, 13, 2], [802, 13, 2], 是], [bottom, x, [497, -13, 3], [497, -13, 3], 是], [top, x, [529, 19, 3], [529, 19, 3], 是]], 是, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251065,"parameters":{"coefficients":[2,2,-3,3,1,2],"bounds":[-16,16,-15,17],"source":11,"boundary_data":[["left",[738,-19,2]],["right",[802,13,2]],["bottom",[497,-13,3]],["top",[529,19,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"source\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0066","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"bounds\":[-17,17,-16,18],\"source\":4,\"boundary_data\":[[\"left\",[1193,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], 4, 4, 是, [[left, y, [1193, 15, -2], [1193, 15, -2], 是], [right, y, [1125, -19, -2], [1125, -19, -2], 是], [bottom, x, [-477, 14, 4], [-477, 14, 4], 是], [top, x, [-681, -20, 4], [-681, -20, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], 4, 4, 是, [[left, y, [1193, 15, -2], [1193, 15, -2], 是], [right, y, [1125, -19, -2], [1125, -19, -2], 是], [bottom, x, [-477, 14, 4], [-477, 14, 4], 是], [top, x, [-681, -20, 4], [-681, -20, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251066,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"bounds":[-17,17,-16,18],"source":4,"boundary_data":[["left",[1193,15,-2]],["right",[1125,-19,-2]],["bottom",[-477,14,4]],["top",[-681,-20,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1193,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0067","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"bounds\":[-18,18,-17,19],\"source\":0,\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], 0, 0, 是, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], 0, 0, 是, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251067,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"bounds":[-18,18,-17,19],"source":0,"boundary_data":[["left",[346,-1,-1]],["right",[310,-1,-1]],["bottom",[-268,-1,1]],["top",[-376,-1,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0068","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[5,0,0,2,1,0],\"bounds\":[-19,19,-18,20],\"source\":4,\"boundary_data\":[[\"left\",[727,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 4, 4, 是, [[left, y, [727, -19, 0], [727, -19, 0], 是], [right, y, [727, 19, 0], [727, 19, 0], 是], [bottom, x, [5, -18, 2], [5, -18, 2], 是], [top, x, [5, 20, 2], [5, 20, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], 4, 4, 是, [[left, y, [727, -19, 0], [727, -19, 0], 是], [right, y, [727, 19, 0], [727, 19, 0], 是], [bottom, x, [5, -18, 2], [5, -18, 2], 是], [top, x, [5, 20, 2], [5, 20, 2], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251068,"parameters":{"coefficients":[5,0,0,2,1,0],"bounds":[-19,19,-18,20],"source":4,"boundary_data":[["left",[727,-19,0]],["right",[727,19,0]],["bottom",[5,-18,2]],["top",[5,20,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[727,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0069","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[6,1,1,3,-1,1],\"bounds\":[-20,20,-19,21],\"source\":8,\"boundary_data\":[[\"left\",[1187,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 8, 8, 是, [[left, y, [1186, 21, 1], [1187, 21, 1], 否], [right, y, [1226, -19, 1], [1226, -19, 1], 是], [bottom, x, [348, 20, 3], [348, 20, 3], 是], [top, x, [468, -20, 3], [468, -20, 3], 是]], 否, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], 8, 8, 是, [[left, y, [1186, 21, 1], [1187, 21, 1], 否], [right, y, [1226, -19, 1], [1226, -19, 1], 是], [bottom, x, [348, 20, 3], [348, 20, 3], 是], [top, x, [468, -20, 3], [468, -20, 3], 是]], 否, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251069,"parameters":{"coefficients":[6,1,1,3,-1,1],"bounds":[-20,20,-19,21],"source":8,"boundary_data":[["left",[1187,21,1]],["right",[1226,-19,1]],["bottom",[348,20,3]],["top",[468,-20,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1187,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0070","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[7,2,2,4,0,2],\"bounds\":[-21,21,-20,22],\"source\":12,\"boundary_data\":[[\"left\",[1729,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 12, 12, 是, [[left, y, [1729, 2, 2], [1729, 2, 2], 是], [right, y, [1813, 2, 2], [1813, 2, 2], 是], [bottom, x, [767, 2, 4], [767, 2, 4], 是], [top, x, [1019, 2, 4], [1019, 2, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], 12, 12, 是, [[left, y, [1729, 2, 2], [1729, 2, 2], 是], [right, y, [1813, 2, 2], [1813, 2, 2], 是], [bottom, x, [767, 2, 4], [767, 2, 4], 是], [top, x, [1019, 2, 4], [1019, 2, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251070,"parameters":{"coefficients":[7,2,2,4,0,2],"bounds":[-21,21,-20,22],"source":12,"boundary_data":[["left",[1729,2,2]],["right",[1813,2,2]],["bottom",[767,2,4]],["top",[1019,2,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1729,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"source\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0071","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[8,-2,3,1,1,-2],\"bounds\":[-22,22,-21,23],\"source\":-2,\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, -2, 是, [[left, y, [536, -19, -2], [536, -19, -2], 是], [right, y, [448, 25, -2], [448, 25, -2], 是], [bottom, x, [-937, -23, 1], [-937, -23, 1], 是], [top, x, [-981, 21, 1], [-981, 21, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], -2, -2, 是, [[left, y, [536, -19, -2], [536, -19, -2], 是], [right, y, [448, 25, -2], [448, 25, -2], 是], [bottom, x, [-937, -23, 1], [-937, -23, 1], 是], [top, x, [-981, 21, 1], [-981, 21, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251071,"parameters":{"coefficients":[8,-2,3,1,1,-2],"bounds":[-22,22,-21,23],"source":-2,"boundary_data":[["left",[536,-19,-2]],["right",[448,25,-2]],["bottom",[-937,-23,1]],["top",[-981,21,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"source\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0072","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"bounds\":[-23,23,-22,24],\"source\":3,\"boundary_data\":[[\"left\",[1090,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], 2, 3, 否, [[left, y, [1090, 20, -1], [1090, 20, -1], 是], [right, y, [1044, -26, -1], [1044, -26, -1], 是], [bottom, x, [-409, 21, 2], [-409, 21, 2], 是], [top, x, [-639, -25, 2], [-639, -25, 2], 是]], 是, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], 2, 3, 否, [[left, y, [1090, 20, -1], [1090, 20, -1], 是], [right, y, [1044, -26, -1], [1044, -26, -1], 是], [bottom, x, [-409, 21, 2], [-409, 21, 2], 是], [top, x, [-639, -25, 2], [-639, -25, 2], 是]], 是, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251072,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"bounds":[-23,23,-22,24],"source":3,"boundary_data":[["left",[1090,20,-1]],["right",[1044,-26,-1]],["bottom",[-409,21,2]],["top",[-639,-25,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1090,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0073","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[10,0,-2,3,0,0],\"bounds\":[-24,24,-23,25],\"source\":6,\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 6, 6, 是, [[left, y, [1738, -2, 0], [1738, -2, 0], 是], [right, y, [1738, -2, 0], [1738, -2, 0], 是], [bottom, x, [56, 0, 3], [56, 0, 3], 是], [top, x, [-40, 0, 3], [-40, 0, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], 6, 6, 是, [[left, y, [1738, -2, 0], [1738, -2, 0], 是], [right, y, [1738, -2, 0], [1738, -2, 0], 是], [bottom, x, [56, 0, 3], [56, 0, 3], 是], [top, x, [-40, 0, 3], [-40, 0, 3], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251073,"parameters":{"coefficients":[10,0,-2,3,0,0],"bounds":[-24,24,-23,25],"source":6,"boundary_data":[["left",[1738,-2,0]],["right",[1738,-2,0]],["bottom",[56,0,3]],["top",[-40,0,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0074","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[11,1,-1,4,1,1],\"bounds\":[-25,25,-24,26],\"source\":10,\"boundary_data\":[[\"left\",[2486,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 10, 10, 是, [[left, y, [2486, -26, 1], [2486, -26, 1], 是], [right, y, [2536, 24, 1], [2536, 24, 1], 是], [bottom, x, [611, -23, 4], [611, -23, 4], 是], [top, x, [661, 27, 4], [661, 27, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], 10, 10, 是, [[left, y, [2486, -26, 1], [2486, -26, 1], 是], [right, y, [2536, 24, 1], [2536, 24, 1], 是], [bottom, x, [611, -23, 4], [611, -23, 4], 是], [top, x, [661, 27, 4], [661, 27, 4], 是]], 是, 是, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251074,"parameters":{"coefficients":[11,1,-1,4,1,1],"bounds":[-25,25,-24,26],"source":10,"boundary_data":[["left",[2486,-26,1]],["right",[2536,24,1]],["bottom",[611,-23,4]],["top",[661,27,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[2486,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0075","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-joint-candidate-audit；資料={\"coefficients\":[12,2,0,1,-1,2],\"bounds\":[-26,26,-25,27],\"source\":6,\"boundary_data\":[[\"left\",[637,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 6, 6, 是, [[left, y, [636, 26, 2], [637, 26, 2], 否], [right, y, [740, -26, 2], [740, -26, 2], 是], [bottom, x, [1262, 27, 1], [1262, 27, 1], 是], [top, x, [1470, -25, 1], [1470, -25, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], 6, 6, 是, [[left, y, [636, 26, 2], [637, 26, 2], 否], [right, y, [740, -26, 2], [740, -26, 2], 是], [bottom, x, [1262, 27, 1], [1262, 27, 1], 是], [top, x, [1470, -25, 1], [1470, -25, 1], 是]], 否, 否, candidate_satisfies_declared_BVP_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-joint-candidate-audit","seed":202610251075,"parameters":{"coefficients":[12,2,0,1,-1,2],"bounds":[-26,26,-25,27],"source":6,"boundary_data":[["left",[637,26,2]],["right",[740,-26,2]],["bottom",[1262,27,1]],["top",[1470,-25,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-joint-candidate-audit","semantic_template":"qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[637,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0076","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"bounds\":[-2,2,-1,3],\"source\":-1,\"boundary_data\":[[\"left\",[-4,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(-2, -1, 否, [[left, y, [-4, -1, -2], [-4, -1, -2], 是], [right, y, [-12, -5, -2], [-12, -5, -2], 是], [bottom, x, [-11, -1, 1], [-11, -1, 1], 是], [top, x, [-39, -5, 1], [-39, -5, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(-2, -1, 否, [[left, y, [-4, -1, -2], [-4, -1, -2], 是], [right, y, [-12, -5, -2], [-12, -5, -2], 是], [bottom, x, [-11, -1, 1], [-11, -1, 1], 是], [top, x, [-39, -5, 1], [-39, -5, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251076,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"source":-1,"boundary_data":[["left",[-4,-1,-2]],["right",[-12,-5,-2]],["bottom",[-11,-1,1]],["top",[-39,-5,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[-4,-1,-2]],[\"right\",[-12,-5,-2]],[\"bottom\",[-11,-1,1]],[\"top\",[-39,-5,1]]],\"bounds\":[-2,2,-1,3],\"coefficients\":[-12,-2,-3,1,-1,-2],\"source\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0077","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"bounds\":[-3,3,-2,4],\"source\":2,\"boundary_data\":[[\"left\",[11,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, [[left, y, [10, -2, -1], [11, -2, -1], 否], [right, y, [4, -2, -1], [4, -2, -1], 是], [bottom, x, [-11, -1, 2], [-11, -1, 2], 是], [top, x, [-35, -1, 2], [-35, -1, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(2, 2, 是, [[left, y, [10, -2, -1], [11, -2, -1], 否], [right, y, [4, -2, -1], [4, -2, -1], 是], [bottom, x, [-11, -1, 2], [-11, -1, 2], 是], [top, x, [-35, -1, 2], [-35, -1, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251077,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"bounds":[-3,3,-2,4],"source":2,"boundary_data":[["left",[11,-2,-1]],["right",[4,-2,-1]],["bottom",[-11,-1,2]],["top",[-35,-1,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[11,-2,-1]],[\"right\",[4,-2,-1]],[\"bottom\",[-11,-1,2]],[\"top\",[-35,-1,2]]],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0078","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-10,0,-1,3,1,0],\"bounds\":[-4,4,-3,5],\"source\":7,\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 否, [[left, y, [38, -5, 0], [38, -5, 0], 是], [right, y, [38, 3, 0], [38, 3, 0], 是], [bottom, x, [-7, -3, 3], [-7, -3, 3], 是], [top, x, [-15, 5, 3], [-15, 5, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(6, 7, 否, [[left, y, [38, -5, 0], [38, -5, 0], 是], [right, y, [38, 3, 0], [38, 3, 0], 是], [bottom, x, [-7, -3, 3], [-7, -3, 3], 是], [top, x, [-15, 5, 3], [-15, 5, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251078,"parameters":{"coefficients":[-10,0,-1,3,1,0],"bounds":[-4,4,-3,5],"source":7,"boundary_data":[["left",[38,-5,0]],["right",[38,3,0]],["bottom",[-7,-3,3]],["top",[-15,5,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[38,-5,0]],[\"right\",[38,3,0]],[\"bottom\",[-7,-3,3]],[\"top\",[-15,5,3]]],\"bounds\":[-4,4,-3,5],\"coefficients\":[-10,0,-1,3,1,0],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0079","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-9,1,0,4,-1,1],\"bounds\":[-5,5,-4,6],\"source\":10,\"boundary_data\":[[\"left\",[87,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 是, [[left, y, [86, 5, 1], [87, 5, 1], 否], [right, y, [96, -5, 1], [96, -5, 1], 是], [bottom, x, [7, 5, 4], [7, 5, 4], 是], [top, x, [27, -5, 4], [27, -5, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(10, 10, 是, [[left, y, [86, 5, 1], [87, 5, 1], 否], [right, y, [96, -5, 1], [96, -5, 1], 是], [bottom, x, [7, 5, 4], [7, 5, 4], 是], [top, x, [27, -5, 4], [27, -5, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251079,"parameters":{"coefficients":[-9,1,0,4,-1,1],"bounds":[-5,5,-4,6],"source":10,"boundary_data":[["left",[87,5,1]],["right",[96,-5,1]],["bottom",[7,5,4]],["top",[27,-5,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[87,5,1]],[\"right\",[96,-5,1]],[\"bottom\",[7,5,4]],[\"top\",[27,-5,4]]],\"bounds\":[-5,5,-4,6],\"coefficients\":[-9,1,0,4,-1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0080","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-8,2,1,1,0,2],\"bounds\":[-6,6,-5,7],\"source\":7,\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 否, [[left, y, [16, 1, 2], [16, 1, 2], 是], [right, y, [40, 1, 2], [40, 1, 2], 是], [bottom, x, [37, 2, 1], [37, 2, 1], 是], [top, x, [97, 2, 1], [97, 2, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(6, 7, 否, [[left, y, [16, 1, 2], [16, 1, 2], 是], [right, y, [40, 1, 2], [40, 1, 2], 是], [bottom, x, [37, 2, 1], [37, 2, 1], 是], [top, x, [97, 2, 1], [97, 2, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251080,"parameters":{"coefficients":[-8,2,1,1,0,2],"bounds":[-6,6,-5,7],"source":7,"boundary_data":[["left",[16,1,2]],["right",[40,1,2]],["bottom",[37,2,1]],["top",[97,2,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[16,1,2]],[\"right\",[40,1,2]],[\"bottom\",[37,2,1]],[\"top\",[97,2,1]]],\"bounds\":[-6,6,-5,7],\"coefficients\":[-8,2,1,1,0,2],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0081","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"bounds\":[-7,7,-6,8],\"source\":0,\"boundary_data\":[[\"left\",[106,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 是, [[left, y, [105, -5, -2], [106, -5, -2], 否], [right, y, [77, 9, -2], [77, 9, -2], 是], [bottom, x, [-91, -8, 2], [-91, -8, 2], 是], [top, x, [-119, 6, 2], [-119, 6, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(0, 0, 是, [[left, y, [105, -5, -2], [106, -5, -2], 否], [right, y, [77, 9, -2], [77, 9, -2], 是], [bottom, x, [-91, -8, 2], [-91, -8, 2], 是], [top, x, [-119, 6, 2], [-119, 6, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251081,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"bounds":[-7,7,-6,8],"source":0,"boundary_data":[["left",[106,-5,-2]],["right",[77,9,-2]],["bottom",[-91,-8,2]],["top",[-119,6,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[106,-5,-2]],[\"right\",[77,9,-2]],[\"bottom\",[-91,-8,2]],[\"top\",[-119,6,2]]],\"bounds\":[-7,7,-6,8],\"coefficients\":[-7,-2,2,2,1,-2],\"source\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0082","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"bounds\":[-8,8,-7,9],\"source\":5,\"boundary_data\":[[\"left\",[194,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 否, [[left, y, [194, 11, -1], [194, 11, -1], 是], [right, y, [178, -5, -1], [178, -5, -1], 是], [bottom, x, [-76, 6, 3], [-76, 6, 3], 是], [top, x, [-60, -10, 3], [-60, -10, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(4, 5, 否, [[left, y, [194, 11, -1], [194, 11, -1], 是], [right, y, [178, -5, -1], [178, -5, -1], 是], [bottom, x, [-76, 6, 3], [-76, 6, 3], 是], [top, x, [-60, -10, 3], [-60, -10, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251082,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"bounds":[-8,8,-7,9],"source":5,"boundary_data":[["left",[194,11,-1]],["right",[178,-5,-1]],["bottom",[-76,6,3]],["top",[-60,-10,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[194,11,-1]],[\"right\",[178,-5,-1]],[\"bottom\",[-76,6,3]],[\"top\",[-60,-10,3]]],\"bounds\":[-8,8,-7,9],\"coefficients\":[-6,-1,3,3,-1,-1],\"source\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0083","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-5,0,-3,4,0,0],\"bounds\":[-9,9,-8,10],\"source\":8,\"boundary_data\":[[\"left\",[320,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 是, [[left, y, [319, -3, 0], [320, -3, 0], 否], [right, y, [319, -3, 0], [319, -3, 0], 是], [bottom, x, [19, 0, 4], [19, 0, 4], 是], [top, x, [-35, 0, 4], [-35, 0, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(8, 8, 是, [[left, y, [319, -3, 0], [320, -3, 0], 否], [right, y, [319, -3, 0], [319, -3, 0], 是], [bottom, x, [19, 0, 4], [19, 0, 4], 是], [top, x, [-35, 0, 4], [-35, 0, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251083,"parameters":{"coefficients":[-5,0,-3,4,0,0],"bounds":[-9,9,-8,10],"source":8,"boundary_data":[["left",[320,-3,0]],["right",[319,-3,0]],["bottom",[19,0,4]],["top",[-35,0,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[320,-3,0]],[\"right\",[319,-3,0]],[\"bottom\",[19,0,4]],[\"top\",[-35,0,4]]],\"bounds\":[-9,9,-8,10],\"coefficients\":[-5,0,-3,4,0,0],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0084","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-4,1,-2,1,1,1],\"bounds\":[-10,10,-9,11],\"source\":5,\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 否, [[left, y, [86, -12, 1], [86, -12, 1], 是], [right, y, [106, 8, 1], [106, 8, 1], 是], [bottom, x, [95, -8, 1], [95, -8, 1], 是], [top, x, [95, 12, 1], [95, 12, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(4, 5, 否, [[left, y, [86, -12, 1], [86, -12, 1], 是], [right, y, [106, 8, 1], [106, 8, 1], 是], [bottom, x, [95, -8, 1], [95, -8, 1], 是], [top, x, [95, 12, 1], [95, 12, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251084,"parameters":{"coefficients":[-4,1,-2,1,1,1],"bounds":[-10,10,-9,11],"source":5,"boundary_data":[["left",[86,-12,1]],["right",[106,8,1]],["bottom",[95,-8,1]],["top",[95,12,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[86,-12,1]],[\"right\",[106,8,1]],[\"bottom\",[95,-8,1]],[\"top\",[95,12,1]]],\"bounds\":[-10,10,-9,11],\"coefficients\":[-4,1,-2,1,1,1],\"source\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0085","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"bounds\":[-11,11,-10,12],\"source\":8,\"boundary_data\":[[\"left\",[218,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 是, [[left, y, [217, 10, 2], [218, 10, 2], 否], [right, y, [261, -12, 2], [261, -12, 2], 是], [bottom, x, [207, 12, 2], [207, 12, 2], 是], [top, x, [273, -10, 2], [273, -10, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(8, 8, 是, [[left, y, [217, 10, 2], [218, 10, 2], 否], [right, y, [261, -12, 2], [261, -12, 2], 是], [bottom, x, [207, 12, 2], [207, 12, 2], 是], [top, x, [273, -10, 2], [273, -10, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251085,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"bounds":[-11,11,-10,12],"source":8,"boundary_data":[["left",[218,10,2]],["right",[261,-12,2]],["bottom",[207,12,2]],["top",[273,-10,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[218,10,2]],[\"right\",[261,-12,2]],[\"bottom\",[207,12,2]],[\"top\",[273,-10,2]]],\"bounds\":[-11,11,-10,12],\"coefficients\":[-3,2,-1,2,-1,2],\"source\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0086","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"bounds\":[-12,12,-11,13],\"source\":3,\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 否, [[left, y, [454, 0, -2], [454, 0, -2], 是], [right, y, [406, 0, -2], [406, 0, -2], 是], [bottom, x, [-244, -2, 3], [-244, -2, 3], 是], [top, x, [-340, -2, 3], [-340, -2, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(2, 3, 否, [[left, y, [454, 0, -2], [454, 0, -2], 是], [right, y, [406, 0, -2], [406, 0, -2], 是], [bottom, x, [-244, -2, 3], [-244, -2, 3], 是], [top, x, [-340, -2, 3], [-340, -2, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251086,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"bounds":[-12,12,-11,13],"source":3,"boundary_data":[["left",[454,0,-2]],["right",[406,0,-2]],["bottom",[-244,-2,3]],["top",[-340,-2,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[454,0,-2]],[\"right\",[406,0,-2]],[\"bottom\",[-244,-2,3]],[\"top\",[-340,-2,3]]],\"bounds\":[-12,12,-11,13],\"coefficients\":[-2,-2,0,3,0,-2],\"source\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0087","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"bounds\":[-13,13,-12,14],\"source\":6,\"boundary_data\":[[\"left\",[689,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, [[left, y, [688, -12, -1], [689, -12, -1], 否], [right, y, [662, 14, -1], [662, 14, -1], 是], [bottom, x, [-157, -13, 4], [-157, -13, 4], 是], [top, x, [-183, 13, 4], [-183, 13, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(6, 6, 是, [[left, y, [688, -12, -1], [689, -12, -1], 否], [right, y, [662, 14, -1], [662, 14, -1], 是], [bottom, x, [-157, -13, 4], [-157, -13, 4], 是], [top, x, [-183, 13, 4], [-183, 13, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251087,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"bounds":[-13,13,-12,14],"source":6,"boundary_data":[["left",[689,-12,-1]],["right",[662,14,-1]],["bottom",[-157,-13,4]],["top",[-183,13,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[689,-12,-1]],[\"right\",[662,14,-1]],[\"bottom\",[-157,-13,4]],[\"top\",[-183,13,4]]],\"bounds\":[-13,13,-12,14],\"coefficients\":[-1,-1,1,4,1,-1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0088","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[0,0,2,1,-1,0],\"bounds\":[-14,14,-13,15],\"source\":3,\"boundary_data\":[[\"left\",[196,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 否, [[left, y, [196, 16, 0], [196, 16, 0], 是], [right, y, [196, -12, 0], [196, -12, 0], 是], [bottom, x, [-26, 13, 1], [-26, 13, 1], 是], [top, x, [30, -15, 1], [30, -15, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(2, 3, 否, [[left, y, [196, 16, 0], [196, 16, 0], 是], [right, y, [196, -12, 0], [196, -12, 0], 是], [bottom, x, [-26, 13, 1], [-26, 13, 1], 是], [top, x, [30, -15, 1], [30, -15, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251088,"parameters":{"coefficients":[0,0,2,1,-1,0],"bounds":[-14,14,-13,15],"source":3,"boundary_data":[["left",[196,16,0]],["right",[196,-12,0]],["bottom",[-26,13,1]],["top",[30,-15,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[196,16,0]],[\"right\",[196,-12,0]],[\"bottom\",[-26,13,1]],[\"top\",[30,-15,1]]],\"bounds\":[-14,14,-13,15],\"coefficients\":[0,0,2,1,-1,0],\"source\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0089","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[1,1,3,2,0,1],\"bounds\":[-15,15,-14,16],\"source\":6,\"boundary_data\":[[\"left\",[437,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, [[left, y, [436, 3, 1], [437, 3, 1], 否], [right, y, [466, 3, 1], [466, 3, 1], 是], [bottom, x, [155, 1, 2], [155, 1, 2], 是], [top, x, [305, 1, 2], [305, 1, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(6, 6, 是, [[left, y, [436, 3, 1], [437, 3, 1], 否], [right, y, [466, 3, 1], [466, 3, 1], 是], [bottom, x, [155, 1, 2], [155, 1, 2], 是], [top, x, [305, 1, 2], [305, 1, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251089,"parameters":{"coefficients":[1,1,3,2,0,1],"bounds":[-15,15,-14,16],"source":6,"boundary_data":[["left",[437,3,1]],["right",[466,3,1]],["bottom",[155,1,2]],["top",[305,1,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[437,3,1]],[\"right\",[466,3,1]],[\"bottom\",[155,1,2]],[\"top\",[305,1,2]]],\"bounds\":[-15,15,-14,16],\"coefficients\":[1,1,3,2,0,1],\"source\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0090","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[2,2,-3,3,1,2],\"bounds\":[-16,16,-15,17],\"source\":11,\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(10, 11, 否, [[left, y, [738, -19, 2], [738, -19, 2], 是], [right, y, [802, 13, 2], [802, 13, 2], 是], [bottom, x, [497, -13, 3], [497, -13, 3], 是], [top, x, [529, 19, 3], [529, 19, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(10, 11, 否, [[left, y, [738, -19, 2], [738, -19, 2], 是], [right, y, [802, 13, 2], [802, 13, 2], 是], [bottom, x, [497, -13, 3], [497, -13, 3], 是], [top, x, [529, 19, 3], [529, 19, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251090,"parameters":{"coefficients":[2,2,-3,3,1,2],"bounds":[-16,16,-15,17],"source":11,"boundary_data":[["left",[738,-19,2]],["right",[802,13,2]],["bottom",[497,-13,3]],["top",[529,19,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[738,-19,2]],[\"right\",[802,13,2]],[\"bottom\",[497,-13,3]],[\"top\",[529,19,3]]],\"bounds\":[-16,16,-15,17],\"coefficients\":[2,2,-3,3,1,2],\"source\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0091","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"bounds\":[-17,17,-16,18],\"source\":4,\"boundary_data\":[[\"left\",[1194,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 是, [[left, y, [1193, 15, -2], [1194, 15, -2], 否], [right, y, [1125, -19, -2], [1125, -19, -2], 是], [bottom, x, [-477, 14, 4], [-477, 14, 4], 是], [top, x, [-681, -20, 4], [-681, -20, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(4, 4, 是, [[left, y, [1193, 15, -2], [1194, 15, -2], 否], [right, y, [1125, -19, -2], [1125, -19, -2], 是], [bottom, x, [-477, 14, 4], [-477, 14, 4], 是], [top, x, [-681, -20, 4], [-681, -20, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251091,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"bounds":[-17,17,-16,18],"source":4,"boundary_data":[["left",[1194,15,-2]],["right",[1125,-19,-2]],["bottom",[-477,14,4]],["top",[-681,-20,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1194,15,-2]],[\"right\",[1125,-19,-2]],[\"bottom\",[-477,14,4]],[\"top\",[-681,-20,4]]],\"bounds\":[-17,17,-16,18],\"coefficients\":[3,-2,-2,4,-1,-2],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0092","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"bounds\":[-18,18,-17,19],\"source\":1,\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 否, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(0, 1, 否, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251092,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"bounds":[-18,18,-17,19],"source":1,"boundary_data":[["left",[346,-1,-1]],["right",[310,-1,-1]],["bottom",[-268,-1,1]],["top",[-376,-1,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0093","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[5,0,0,2,1,0],\"bounds\":[-19,19,-18,20],\"source\":4,\"boundary_data\":[[\"left\",[728,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 是, [[left, y, [727, -19, 0], [728, -19, 0], 否], [right, y, [727, 19, 0], [727, 19, 0], 是], [bottom, x, [5, -18, 2], [5, -18, 2], 是], [top, x, [5, 20, 2], [5, 20, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(4, 4, 是, [[left, y, [727, -19, 0], [728, -19, 0], 否], [right, y, [727, 19, 0], [727, 19, 0], 是], [bottom, x, [5, -18, 2], [5, -18, 2], 是], [top, x, [5, 20, 2], [5, 20, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251093,"parameters":{"coefficients":[5,0,0,2,1,0],"bounds":[-19,19,-18,20],"source":4,"boundary_data":[["left",[728,-19,0]],["right",[727,19,0]],["bottom",[5,-18,2]],["top",[5,20,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[728,-19,0]],[\"right\",[727,19,0]],[\"bottom\",[5,-18,2]],[\"top\",[5,20,2]]],\"bounds\":[-19,19,-18,20],\"coefficients\":[5,0,0,2,1,0],\"source\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0094","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[6,1,1,3,-1,1],\"bounds\":[-20,20,-19,21],\"source\":9,\"boundary_data\":[[\"left\",[1186,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(8, 9, 否, [[left, y, [1186, 21, 1], [1186, 21, 1], 是], [right, y, [1226, -19, 1], [1226, -19, 1], 是], [bottom, x, [348, 20, 3], [348, 20, 3], 是], [top, x, [468, -20, 3], [468, -20, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(8, 9, 否, [[left, y, [1186, 21, 1], [1186, 21, 1], 是], [right, y, [1226, -19, 1], [1226, -19, 1], 是], [bottom, x, [348, 20, 3], [348, 20, 3], 是], [top, x, [468, -20, 3], [468, -20, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251094,"parameters":{"coefficients":[6,1,1,3,-1,1],"bounds":[-20,20,-19,21],"source":9,"boundary_data":[["left",[1186,21,1]],["right",[1226,-19,1]],["bottom",[348,20,3]],["top",[468,-20,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1186,21,1]],[\"right\",[1226,-19,1]],[\"bottom\",[348,20,3]],[\"top\",[468,-20,3]]],\"bounds\":[-20,20,-19,21],\"coefficients\":[6,1,1,3,-1,1],\"source\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0095","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[7,2,2,4,0,2],\"bounds\":[-21,21,-20,22],\"source\":12,\"boundary_data\":[[\"left\",[1730,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 是, [[left, y, [1729, 2, 2], [1730, 2, 2], 否], [right, y, [1813, 2, 2], [1813, 2, 2], 是], [bottom, x, [767, 2, 4], [767, 2, 4], 是], [top, x, [1019, 2, 4], [1019, 2, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(12, 12, 是, [[left, y, [1729, 2, 2], [1730, 2, 2], 否], [right, y, [1813, 2, 2], [1813, 2, 2], 是], [bottom, x, [767, 2, 4], [767, 2, 4], 是], [top, x, [1019, 2, 4], [1019, 2, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251095,"parameters":{"coefficients":[7,2,2,4,0,2],"bounds":[-21,21,-20,22],"source":12,"boundary_data":[["left",[1730,2,2]],["right",[1813,2,2]],["bottom",[767,2,4]],["top",[1019,2,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1730,2,2]],[\"right\",[1813,2,2]],[\"bottom\",[767,2,4]],[\"top\",[1019,2,4]]],\"bounds\":[-21,21,-20,22],\"coefficients\":[7,2,2,4,0,2],\"source\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0096","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[8,-2,3,1,1,-2],\"bounds\":[-22,22,-21,23],\"source\":-1,\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(-2, -1, 否, [[left, y, [536, -19, -2], [536, -19, -2], 是], [right, y, [448, 25, -2], [448, 25, -2], 是], [bottom, x, [-937, -23, 1], [-937, -23, 1], 是], [top, x, [-981, 21, 1], [-981, 21, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(-2, -1, 否, [[left, y, [536, -19, -2], [536, -19, -2], 是], [right, y, [448, 25, -2], [448, 25, -2], 是], [bottom, x, [-937, -23, 1], [-937, -23, 1], 是], [top, x, [-981, 21, 1], [-981, 21, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251096,"parameters":{"coefficients":[8,-2,3,1,1,-2],"bounds":[-22,22,-21,23],"source":-1,"boundary_data":[["left",[536,-19,-2]],["right",[448,25,-2]],["bottom",[-937,-23,1]],["top",[-981,21,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[536,-19,-2]],[\"right\",[448,25,-2]],[\"bottom\",[-937,-23,1]],[\"top\",[-981,21,1]]],\"bounds\":[-22,22,-21,23],\"coefficients\":[8,-2,3,1,1,-2],\"source\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0097","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"bounds\":[-23,23,-22,24],\"source\":2,\"boundary_data\":[[\"left\",[1091,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, [[left, y, [1090, 20, -1], [1091, 20, -1], 否], [right, y, [1044, -26, -1], [1044, -26, -1], 是], [bottom, x, [-409, 21, 2], [-409, 21, 2], 是], [top, x, [-639, -25, 2], [-639, -25, 2], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(2, 2, 是, [[left, y, [1090, 20, -1], [1091, 20, -1], 否], [right, y, [1044, -26, -1], [1044, -26, -1], 是], [bottom, x, [-409, 21, 2], [-409, 21, 2], 是], [top, x, [-639, -25, 2], [-639, -25, 2], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251097,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"bounds":[-23,23,-22,24],"source":2,"boundary_data":[["left",[1091,20,-1]],["right",[1044,-26,-1]],["bottom",[-409,21,2]],["top",[-639,-25,2]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1091,20,-1]],[\"right\",[1044,-26,-1]],[\"bottom\",[-409,21,2]],[\"top\",[-639,-25,2]]],\"bounds\":[-23,23,-22,24],\"coefficients\":[9,-1,-3,2,-1,-1],\"source\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0098","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[10,0,-2,3,0,0],\"bounds\":[-24,24,-23,25],\"source\":7,\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 否, [[left, y, [1738, -2, 0], [1738, -2, 0], 是], [right, y, [1738, -2, 0], [1738, -2, 0], 是], [bottom, x, [56, 0, 3], [56, 0, 3], 是], [top, x, [-40, 0, 3], [-40, 0, 3], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(6, 7, 否, [[left, y, [1738, -2, 0], [1738, -2, 0], 是], [right, y, [1738, -2, 0], [1738, -2, 0], 是], [bottom, x, [56, 0, 3], [56, 0, 3], 是], [top, x, [-40, 0, 3], [-40, 0, 3], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251098,"parameters":{"coefficients":[10,0,-2,3,0,0],"bounds":[-24,24,-23,25],"source":7,"boundary_data":[["left",[1738,-2,0]],["right",[1738,-2,0]],["bottom",[56,0,3]],["top",[-40,0,3]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[1738,-2,0]],[\"right\",[1738,-2,0]],[\"bottom\",[56,0,3]],[\"top\",[-40,0,3]]],\"bounds\":[-24,24,-23,25],\"coefficients\":[10,0,-2,3,0,0],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0099","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[11,1,-1,4,1,1],\"bounds\":[-25,25,-24,26],\"source\":10,\"boundary_data\":[[\"left\",[2487,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 是, [[left, y, [2486, -26, 1], [2487, -26, 1], 否], [right, y, [2536, 24, 1], [2536, 24, 1], 是], [bottom, x, [611, -23, 4], [611, -23, 4], 是], [top, x, [661, 27, 4], [661, 27, 4], 是]], 否, interior_pass_boundary_fail)","explanation_zh":"收束波核對=(10, 10, 是, [[left, y, [2486, -26, 1], [2487, -26, 1], 否], [right, y, [2536, 24, 1], [2536, 24, 1], 是], [bottom, x, [611, -23, 4], [611, -23, 4], 是], [top, x, [661, 27, 4], [661, 27, 4], 是]], 否, interior_pass_boundary_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251099,"parameters":{"coefficients":[11,1,-1,4,1,1],"bounds":[-25,25,-24,26],"source":10,"boundary_data":[["left",[2487,-26,1]],["right",[2536,24,1]],["bottom",[611,-23,4]],["top",[661,27,4]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[2487,-26,1]],[\"right\",[2536,24,1]],[\"bottom\",[611,-23,4]],[\"top\",[661,27,4]]],\"bounds\":[-25,25,-24,26],\"coefficients\":[11,1,-1,4,1,1],\"source\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r01-boundary-value-problem-0100","batch_id":"batch-20261013-advance-20261025-r01-boundary-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-boundary-value-problem"],"title_zh":"邊界值問題","stem_zh":"內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={\"coefficients\":[12,2,0,1,-1,2],\"bounds\":[-26,26,-25,27],\"source\":7,\"boundary_data\":[[\"left\",[636,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 否, [[left, y, [636, 26, 2], [636, 26, 2], 是], [right, y, [740, -26, 2], [740, -26, 2], 是], [bottom, x, [1262, 27, 1], [1262, 27, 1], 是], [top, x, [1470, -25, 1], [1470, -25, 1], 是]], 是, boundary_pass_interior_fail)","explanation_zh":"收束波核對=(6, 7, 否, [[left, y, [636, 26, 2], [636, 26, 2], 是], [right, y, [740, -26, 2], [740, -26, 2], 是], [bottom, x, [1262, 27, 1], [1262, 27, 1], 是], [top, x, [1470, -25, 1], [1470, -25, 1], 是]], 是, boundary_pass_interior_fail)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["邊界值問題"]},"variant":{"family_id":"qf-mod25-bvp-obligation-separation","seed":202610251100,"parameters":{"coefficients":[12,2,0,1,-1,2],"bounds":[-26,26,-25,27],"source":7,"boundary_data":[["left",[636,26,2]],["right",[740,-26,2]],["bottom",[1262,27,1]],["top",[1470,-25,1]]],"domain":"Polynomial_rectangle_boundary_value_certificates","operation":"bvp-obligation-separation","semantic_template":"qf-mod25-bvp-obligation-separation/{\"boundary_data\":[[\"left\",[636,26,2]],[\"right\",[740,-26,2]],[\"bottom\",[1262,27,1]],[\"top\",[1470,-25,1]]],\"bounds\":[-26,26,-25,27],\"coefficients\":[12,2,0,1,-1,2],\"source\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}