{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r04-improper-integral-0073", "pilot_number": "R0046-23698", "number": 23698, "family_id": "qf-mod43-improper-two-sided-independent", "title": { "zh-Hant": "第 23698 題：瑕積分", "en": "Question 23698: Two Infinite Tails Must Converge Separately before Their Values Are Added" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r04-improper-integral-0073", "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-two-sided-independent；資料={\"left_scale\":23,\"right_scale\":24,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(x<=-1:23/abs(x)^4;-1=1:24/x^4, (-infinity,infinity), 23/3, 8, 是, 是, 47/3, both_one_sided_tail_integrals_converge_independently)", "explanation_zh": "Wave6第六層核對=(x<=-1:23/abs(x)^4;-1=1:24/x^4, (-infinity,infinity), 23/3, 8, 是, 是, 47/3, both_one_sided_tail_integrals_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-two-sided-independent", "seed": 202611124073, "parameters": { "left_scale": 23, "right_scale": 24, "exponent": 4, "domain": "Exact_symbolic_improper_integral", "operation": "improper-two-sided-independent", "semantic_template": "qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":23,\"right_scale\":24}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、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-0073", "batch_id": "batch-20261014-advance-20261112-r04-improper-integral-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3cc90f85f9c6d2473248e20943cc27dd5b660396754050c3df49f300b8eb74a1", "family_id": "qf-mod43-improper-two-sided-independent", "seed": 202611124073, "semantic_template": "qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":23,\"right_scale\":24}" }, "fields": { "title": "Two Infinite Tails Must Converge Separately before Their Values Are Added", "stem": "Use f(x)=23/abs(x)^4 for x≤−1, f(x)=0 for −1=1:24/x^4, (-infinity,infinity), 23/3, 8, Yes, Yes, 47/3, both_one_sided_tail_integrals_converge_independently)", "explanation": "Take the negative and positive infinite-tail limits independently. The zero middle contributes0; each tail equals its own scale/(exponent−1). Even if their finite values cancel, ordinary convergence comes from both separate obligations, not symmetry. The exact result, retaining the requested order, is (x<=-1:23/abs(x)^4;-1=1:24/x^4, (-infinity,infinity), 23/3, 8, Yes, Yes, 47/3, both_one_sided_tail_integrals_converge_independently)." }, "skills": [ "Two Infinite Tails Must Converge Separately before Their Values Are Added" ], "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-two-sided-independent；資料={\"left_scale\":23,\"right_scale\":24,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use f(x)=23/abs(x)^4 for x≤−1, f(x)=0 for −11,lower=1>=1;integrand scale/x^exponent", "after": "for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=23/3", "relation": "implies", "rule_id": "improper_WHOLE_one_sided_tail_antiderivative_limit_not_samples", "conditions": [], "explanation": { "zh-Hant": "k>1使B的負次冪趨0；絕對尾誤差是|scale|/((k−1)B^(k−1))，對任意epsilon可選足夠B。signed或0 scale皆可，這是全稱limit，不靠幾個cutoff。", "en": "Because k>1,the negative power tends0.The absolute tail error is |scale|/((k−1)B^(k−1)),which is below any epsilon for sufficiently large B.Signed or zero scale is allowed;this is a universal limit,not sampled cutoffs." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "Improper ORIGINAL positive tail lower positive power step1 := mul(1, 1)", "after": "Improper ORIGINAL positive tail lower positive power step1 = 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 ORIGINAL positive tail lower positive power step1，不以小數近似替代。", "en": "Recompute Improper ORIGINAL positive tail lower positive power step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Improper ORIGINAL positive tail lower positive power step1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Improper ORIGINAL positive tail lower positive power step2 := mul(1, 1)", "after": "Improper ORIGINAL positive tail lower positive power step2 = 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 ORIGINAL positive tail lower positive power step2，不以小數近似替代。", "en": "Recompute Improper ORIGINAL positive tail lower positive power step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Improper ORIGINAL positive tail lower positive power step2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Improper ORIGINAL positive tail lower positive power step3 := mul(1, 1)", "after": "Improper ORIGINAL positive tail lower positive power step3 = 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 ORIGINAL positive tail lower positive power step3，不以小數近似替代。", "en": "Recompute Improper ORIGINAL positive tail lower positive power step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Improper ORIGINAL positive tail lower positive power step3" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(3, 1)", "after": "Improper ORIGINAL positive tail exponent-minus-one times lower-power = 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 ORIGINAL positive tail exponent-minus-one times lower-power，不以小數近似替代。", "en": "Recompute Improper ORIGINAL positive tail exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "Improper ORIGINAL positive tail exponent-minus-one times lower-power" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Improper ORIGINAL positive tail exact antiderivative limit value := div(24, 3)", "after": "Improper ORIGINAL positive tail exact antiderivative limit value = 8", "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 ORIGINAL positive tail exact antiderivative limit value，不以小數近似替代。", "en": "Recompute Improper ORIGINAL positive tail exact antiderivative limit value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "24", "3" ], "output": "8", "label": "Improper ORIGINAL positive tail exact antiderivative limit value" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Improper ORIGINAL positive tail:ORIGINAL scale=24,exponent=4>1,lower=1>=1;integrand scale/x^exponent", "after": "for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=8", "relation": "implies", "rule_id": "improper_WHOLE_one_sided_tail_antiderivative_limit_not_samples", "conditions": [], "explanation": { "zh-Hant": "k>1使B的負次冪趨0；絕對尾誤差是|scale|/((k−1)B^(k−1))，對任意epsilon可選足夠B。signed或0 scale皆可，這是全稱limit，不靠幾個cutoff。", "en": "Because k>1,the negative power tends0.The absolute tail error is |scale|/((k−1)B^(k−1)),which is below any epsilon for sufficiently large B.Signed or zero scale is allowed;this is a universal limit,not sampled cutoffs." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "Improper independently convergent tails plus zero middle := sum(23/3, 8)", "after": "Improper independently convergent tails plus zero middle = 47/3", "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 independently convergent tails plus zero middle，不以小數近似替代。", "en": "Recompute Improper independently convergent tails plus zero middle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "23/3", "8" ], "output": "47/3", "label": "Improper independently convergent tails plus zero middle" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "ORIGINAL piecewise scales left=23,right=24,exponent=4;middle(-1,1) integrand0", "after": "negative-tail limit=23/3;positive-tail limit=8;BOTH independently finite,then ordinary sum=47/3", "relation": "implies", "rule_id": "improper_TWO_independent_infinite_tails_before_signed_sum", "conditions": [], "explanation": { "zh-Hant": "負尾先用u=-x轉到[1,infinity)，不多帶負號；中段0。即使兩值相消成0，收斂理由仍是兩側分別的limit證書，不是對稱性。", "en": "Substitute u=-x to turn the negative tail into[1,infinity),without an extra sign;the middle contributes0.Even if the values cancel,convergence follows from two independent limit certificates,not symmetry." }, "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 } ], "math_model": { "result": [ "x<=-1:23/abs(x)^4;-1=1:24/x^4", "(-infinity,infinity)", "23/3", "8", true, true, "47/3", "both_one_sided_tail_integrals_converge_independently" ] }, "conclusion": { "canonical_value": "(x<=-1:23/abs(x)^4;-1=1:24/x^4, (-infinity,infinity), 23/3, 8, 是, 是, 47/3, both_one_sided_tail_integrals_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": 73, "original_line_sha256": "d45be8f3dd6f4f0d5a4ba50f1cfb68cd4915669060ab9844051dfd5e3fd1d670", "question_compact_sha256": "3cc90f85f9c6d2473248e20943cc27dd5b660396754050c3df49f300b8eb74a1", "english_row_compact_sha256": "59d37105052609b53360bc26a1fd8eea6758b6f201980e18adb79fce48658e1e", "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": "4d764a927119d8e21de58b5893494eec18eb8d6b084019f8aea93b754cc7ed46", "original_candidate_line_sha256": "4d764a927119d8e21de58b5893494eec18eb8d6b084019f8aea93b754cc7ed46", "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": 15, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1b2ff848228f1c0c31ade91a8ffd40c6a2a81f34462f0ec5051e351f856118ca", "arithmetic_steps": 11 }, "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." } }