{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r04-improper-integral-0076", "pilot_number": "R0046-23651", "number": 23651, "family_id": "qf-mod43-improper-principal-value-boundary", "title": { "zh-Hant": "第 23651 題：瑕積分", "en": "Question 23651: A Zero Symmetric Principal Value Does Not Give Ordinary Improper Convergence" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r04-improper-integral-0076", "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-principal-value-boundary；資料={\"cutoff\":2,\"epsilon_denominator\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)", "explanation_zh": "Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；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": 1, "skills": [ "瑕積分" ] }, "variant": { "family_id": "qf-mod43-improper-principal-value-boundary", "seed": 202611124076, "parameters": { "cutoff": 2, "epsilon_denominator": 2, "domain": "Exact_symbolic_improper_integral", "operation": "improper-principal-value-boundary", "semantic_template": "qf-mod43-improper-principal-value-boundary/{\"cutoff\":2,\"epsilon_denominator\":2}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、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-0076", "batch_id": "batch-20261014-advance-20261112-r04-improper-integral-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "075c83f515a10122bfc33b9c8972c8c468663f6fbe1aa83981eaa7abffe593aa", "family_id": "qf-mod43-improper-principal-value-boundary", "seed": 202611124076, "semantic_template": "qf-mod43-improper-principal-value-boundary/{\"cutoff\":2,\"epsilon_denominator\":2}" }, "fields": { "title": "A Zero Symmetric Principal Value Does Not Give Ordinary Improper Convergence", "stem": "Use f(x)=1/x on the real line with singularity0, symmetric cutoff B=2 and exclusion epsilon=1/2. Give (integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary).", "answer": { "kind": "expression", "value": "(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, Yes, No, No, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)", "explanation": "The symmetric truncated integrals over [−B,−epsilon] and [epsilon,B] sum to0, and the symmetric principal value is0. The required one-sided limits at the singularity and infinite ends diverge, so ordinary improper convergence is No. Finitely many cutoffs cannot certify a universal convergence obligation. The exact result, retaining the requested order, is (f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, Yes, No, No, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)." }, "skills": [ "A Zero Symmetric Principal Value Does Not Give Ordinary Improper Convergence" ], "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-principal-value-boundary；資料={\"cutoff\":2,\"epsilon_denominator\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use f(x)=1/x on the real line with singularity0, symmetric cutoff B=2 and exclusion epsilon=1/2. Give (integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=improper-principal-value-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary]", "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": "ORIGINAL f(x)=1/x;B=2>=2;epsilon=1/2<=1/2", "after": "for EVERY B>epsilon>0,positive truncated integral=log(B)-log(epsilon),negative truncated integral=its negative;sum0,so symmetric principal value0", "relation": "implies", "rule_id": "improper_PV_exact_odd_symmetric_truncations_ALL_cutoffs", "conditions": [], "explanation": { "zh-Hant": "負側以x=-u代換，兩個對稱截斷積分恰好相反；這對全部有效cutoffs成立，不只原顯示B/epsilon這一組。", "en": "Substitute x=-u on the negative side to obtain the opposite truncated integral.This holds for every valid symmetric pair,not just the displayed 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": "S003", "uses": [ "G001", "S002" ], "before": "ORIGINAL whole integrand1/x on both infinite ends and both sides of singularity0", "after": "integral_epsilon^1=-log(epsilon) tends+infinity and integral_-1^-epsilon tends-infinity;integral_1^B=log(B) tends+infinity and integral_-B^-1 tends-infinity;ordinary improper convergence=否,finite_cutoffs_certificate=否", "relation": "implies", "rule_id": "improper_REQUIRED_independent_limits_diverge_despite_PV0", "conditions": [], "explanation": { "zh-Hant": "通常瑕積分要求各無界端／奇點逐側的有限極限；+infinity與−infinity不能相消定義普通積分。PV0與有限截斷0不替代這些責任。", "en": "Ordinary improper integration requires finite limits separately at every infinite end and singular side.+infinity and−infinity cannot be cancelled to define it.PV0 and finite truncations0 do not replace these obligations." }, "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": [ "f(x)=1/x", "(-infinity,infinity) with singularity at 0", 2, "1/2", 0, true, false, false, "symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence" ] }, "conclusion": { "canonical_value": "(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)", "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": 76, "original_line_sha256": "1a39588f5fac61702b7935ca4ad77b8f0fe92b1aba88de45b6c6265a6d6bc763", "question_compact_sha256": "075c83f515a10122bfc33b9c8972c8c468663f6fbe1aa83981eaa7abffe593aa", "english_row_compact_sha256": "fdbe127d22a61ff243c91cfc80aa9e93e9a16a5ecc5ff1231b0c1d2c270e8670", "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": "e70c415aa2368c3243b7e4ea319afd4269e452583c6de5202a5977ba0203521e", "original_candidate_line_sha256": "e70c415aa2368c3243b7e4ea319afd4269e452583c6de5202a5977ba0203521e", "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "f981e4769f73c3c57c89026ad09885700dea6e54274a15f87ca44b91d7a3d801", "arithmetic_steps": 0 }, "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." } }