{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r04-improper-integral-0047", "pilot_number": "R0046-23622", "number": 23622, "family_id": "qf-mod43-improper-interior-singular-split", "title": { "zh-Hant": "第 23622 題：瑕積分", "en": "Question 23622: Both Sides of the Interior Singularity Need Independent Limits" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r04-improper-integral-0047", "batch_id": "batch-20261014-advance-20261112-r04-improper-integral-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-improper-integral" ], "title_zh": "瑕積分", "stem_zh": "題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":22,\"right_scale\":23,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, 是, 是, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)", "explanation_zh": "Wave6第六層核對=([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, 是, 是, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "瑕積分" ] }, "variant": { "family_id": "qf-mod43-improper-interior-singular-split", "seed": 202611124047, "parameters": { "left_scale": 22, "right_scale": 23, "left_power_numerator": 1, "left_power_denominator": 2, "right_power_numerator": 2, "right_power_denominator": 3, "singularity": 9, "domain": "Exact_symbolic_improper_integral", "operation": "improper-interior-singular-split", "semantic_template": "qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":22,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":23,\"singularity\":9}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261112-r04-improper-integral-0047", "batch_id": "batch-20261014-advance-20261112-r04-improper-integral-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "0d97b6264fe9bd7d8e3d05742ef16b1ca07f2e03064dd4694948def65959bdbf", "family_id": "qf-mod43-improper-interior-singular-split", "seed": 202611124047, "semantic_template": "qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":22,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":23,\"singularity\":9}" }, "fields": { "title": "Both Sides of the Interior Singularity Need Independent Limits", "stem": "Use c=9, interval [c−1,c+1], left integrand=22*(c−x)^(−1/2) and right integrand=23*(x−c)^(−2/3). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate).", "answer": { "kind": "expression", "value": "([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, Yes, Yes, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)", "explanation": "Split at c. Compute each one-sided limit as its scale/(1−exponent) before summing. Keep both convergence flags; signed cancellation cannot substitute for either required limit. The exact result, retaining the requested order, is ([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, Yes, Yes, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)." }, "skills": [ "Both Sides of the Interior Singularity Need Independent Limits" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":22,\"right_scale\":23,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use c=9, interval [c−1,c+1], left integrand=22*(c−x)^(−1/2) and right integrand=23*(x−c)^(−2/3). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=improper-interior-singular-split;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題介面快照，本頁直接列來源欄序，不需JavaScript。radius由原mode選6／11欄，不補欄；No與實際否定依來源責任分清。", "en": "Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Improper LEFT reduced exponent numerator gap := sub(2, 1)", "after": "Improper LEFT reduced exponent numerator gap = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper LEFT reduced exponent numerator gap，不以小數近似替代。", "en": "Recompute Improper LEFT reduced exponent numerator gap with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "1" ], "output": "1", "label": "Improper LEFT reduced exponent numerator gap" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Improper LEFT reciprocal of one minus exponent := div(2, 1)", "after": "Improper LEFT reciprocal of one minus exponent = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper LEFT reciprocal of one minus exponent，不以小數近似替代。", "en": "Recompute Improper LEFT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "1" ], "output": "2", "label": "Improper LEFT reciprocal of one minus exponent" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Improper LEFT scale times singular-side integral := mul(22, 2)", "after": "Improper LEFT scale times singular-side integral = 44", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper LEFT scale times singular-side integral，不以小數近似替代。", "en": "Recompute Improper LEFT scale times singular-side integral with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "22", "2" ], "output": "44", "label": "Improper LEFT scale times singular-side integral" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "ORIGINAL LEFT side at c=9;scale=22;REDUCED exponent=1/2 in[0,1)", "after": "put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=44", "relation": "implies", "rule_id": "improper_EACH_interior_singular_side_independent_limit", "conditions": [], "explanation": { "zh-Hant": "每一側各用到奇點的正距離，單獨取delta→0；0次方亦合法且0/1已約分。尺度可負／0，不能由左右相消代替任一單側極限。", "en": "Use positive distance from the singularity on each side and take its own delta->0 limit.Exponent0 is legal with reduced0/1.Signed/zero scale is allowed;left-right cancellation replaces neither one-sided limit." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Improper RIGHT reduced exponent numerator gap := sub(3, 2)", "after": "Improper RIGHT reduced exponent numerator gap = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper RIGHT reduced exponent numerator gap，不以小數近似替代。", "en": "Recompute Improper RIGHT reduced exponent numerator gap with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "2" ], "output": "1", "label": "Improper RIGHT reduced exponent numerator gap" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Improper RIGHT reciprocal of one minus exponent := div(3, 1)", "after": "Improper RIGHT reciprocal of one minus exponent = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper RIGHT reciprocal of one minus exponent，不以小數近似替代。", "en": "Recompute Improper RIGHT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3", "1" ], "output": "3", "label": "Improper RIGHT reciprocal of one minus exponent" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Improper RIGHT scale times singular-side integral := mul(23, 3)", "after": "Improper RIGHT scale times singular-side integral = 69", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper RIGHT scale times singular-side integral，不以小數近似替代。", "en": "Recompute Improper RIGHT scale times singular-side integral with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "23", "3" ], "output": "69", "label": "Improper RIGHT scale times singular-side integral" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "ORIGINAL RIGHT side at c=9;scale=23;REDUCED exponent=2/3 in[0,1)", "after": "put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=69", "relation": "implies", "rule_id": "improper_EACH_interior_singular_side_independent_limit", "conditions": [], "explanation": { "zh-Hant": "每一側各用到奇點的正距離，單獨取delta→0；0次方亦合法且0/1已約分。尺度可負／0，不能由左右相消代替任一單側極限。", "en": "Use positive distance from the singularity on each side and take its own delta->0 limit.Exponent0 is legal with reduced0/1.Signed/zero scale is allowed;left-right cancellation replaces neither one-sided limit." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Improper independent singular-side limits sum := sum(44, 69)", "after": "Improper independent singular-side limits sum = 113", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Improper independent singular-side limits sum，不以小數近似替代。", "en": "Recompute Improper independent singular-side limits sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "44", "69" ], "output": "113", "label": "Improper independent singular-side limits sum" } } ], "math_model": { "result": [ "[8,10]", "9", "left:22*(c-x)^(-1/2)", "right:23*(x-c)^(-2/3)", "44", "69", true, true, "113", "interior_singularity_is_split_and_both_one_sided_limits_converge_independently" ] }, "conclusion": { "canonical_value": "([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, 是, 是, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261112-r04-improper-integral-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r04-improper-integral-foundations/questions.jsonl", "batch_sha256": "14e78737d92e71d6c5047067508ac7f61e5b4c940d79dc67ebfd2fcb06e2e642", "original_line": 47, "original_line_sha256": "dc156d0306e455caea6d13830281865a2792ab7f6af80977d75edbb38f112c47", "question_compact_sha256": "0d97b6264fe9bd7d8e3d05742ef16b1ca07f2e03064dd4694948def65959bdbf", "english_row_compact_sha256": "f8e6fafba018a68ba55ce35b1b40feef3ac6d71c62c91c8de4b434dfaee47d8e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r04-improper-integral-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r04-improper-integral-foundations.en.jsonl" }, "mko_bindings": { "mko-improper-integral": { "sha256": "db7f78b411add242ee46cc38c5fede57caf33f99eeb955e7bbe6271da2d44dec", "english_sha256": "4395ebe9f67586ef48cc68baaec03b6a479aa7a30603cacff5d2740e4d57b731", "original_path": "public/data/mko/improper-integral.json", "english_path": "public/data/i18n/editions/mko-improper-integral.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "04d5fe00084f382c9e6853d5f11c80b558d9cf857160ceb644fa20b64948d1fe", "original_candidate_sha256": "3630bd4357b47669e267c34fcea468b6edad208bd99a303819a84f00953140f2", "original_candidate_line_sha256": "3630bd4357b47669e267c34fcea468b6edad208bd99a303819a84f00953140f2", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-improper-integral": "eca1531915ecb4012c527263b5c182660d8af9f7435a6873b08e6fd17d33b5e4" } } }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-009-p47", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1", "question_first_witness_sha256": "c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 10, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "8c8f9bc1fecc288cf2f7fbe20c9059e2317ad496717119af1305af6920f94973", "arithmetic_steps": 7 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }