{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081", "pilot_number": "R0046-23881", "number": 23881, "family_id": "qf-mod43-radius-zero-infinity-prefix-boundary", "title": { "zh-Hant": "第 23881 題：冪級數收斂半徑", "en": "Question 23881: Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081", "batch_id": "batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-power-series-radius-of-convergence" ], "title_zh": "冪級數收斂半徑", "stem_zh": "明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={\"mode\":\"finite_prefix\",\"scale\":6,\"prefix_length\":2,\"first_ratio\":[1,2],\"second_ratio\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)", "explanation_zh": "Wave6第六層核對=(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "冪級數收斂半徑" ] }, "variant": { "family_id": "qf-mod43-radius-zero-infinity-prefix-boundary", "seed": 202611125081, "parameters": { "mode": "finite_prefix", "scale": 6, "prefix_length": 2, "first_ratio": [ 1, 2 ], "second_ratio": [ 1, 4 ], "domain": "Exact_power_series_radius", "operation": "radius-zero-infinity-prefix-boundary", "semantic_template": "qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":2,\"scale\":6,\"second_ratio\":[1,4]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、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-r05-power-series-radius-of-convergence-0081", "batch_id": "batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8a4b41e59c78875f4a01a5252877c78822a4b3bd1f1f9ff6e3f0de38ec7c616b", "family_id": "qf-mod43-radius-zero-infinity-prefix-boundary", "seed": 202611125081, "semantic_template": "qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":2,\"scale\":6,\"second_ratio\":[1,4]}" }, "fields": { "title": "Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii", "stem": "Use mode=finite_prefix, positive scale=6, source prefix_length=2, first_ratio=1/2, second_ratio=1/4. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).", "answer": { "kind": "expression", "value": "(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)", "explanation": "zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)." }, "skills": [ "Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii" ], "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": "明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={\"mode\":\"finite_prefix\",\"scale\":6,\"prefix_length\":2,\"first_ratio\":[1,2],\"second_ratio\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=finite_prefix, positive scale=6, source prefix_length=2, first_ratio=1/2, second_ratio=1/4. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=radius-zero-infinity-prefix-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, 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": "Radius ORIGINAL first complete-tail reciprocal := div(1, 1/2)", "after": "Radius ORIGINAL first complete-tail reciprocal = 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 Radius ORIGINAL first complete-tail reciprocal，不以小數近似替代。", "en": "Recompute Radius ORIGINAL first complete-tail reciprocal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1/2" ], "output": "2", "label": "Radius ORIGINAL first complete-tail reciprocal" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Radius ORIGINAL second complete-tail reciprocal := div(1, 1/4)", "after": "Radius ORIGINAL second complete-tail reciprocal = 4", "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 Radius ORIGINAL second complete-tail reciprocal，不以小數近似替代。", "en": "Recompute Radius ORIGINAL second complete-tail reciprocal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1/4" ], "output": "4", "label": "Radius ORIGINAL second complete-tail reciprocal" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "ORIGINAL prefix_length parameter N=2;positive scale=6;distinct positive tail ratios=[1/2, 1/4]", "after": "COMMON zero coefficients at ALL n0..N=[0, 0, 0];there are N+1=3 zeros;for n>N laws are scale*q^(n-N-1),first tail term scale;radii=[2, 4] DIFFER", "relation": "implies", "rule_id": "radius_SHARED_prefix_LAST_index_Nplus1_not_lengthN", "conditions": [], "explanation": { "zh-Hant": "source參數是共同zero prefix的最後index，不是列表長度。兩條完整尾律的ratio分別q1、q2，有限修改／位移不改radius；不由共同有限prefix決定無窮tail。", "en": "The source parameter is the last shared zero index,not list length.The complete tails have ratios q1 and q2;finite changes/shifts do not change radius.The common finite prefix does not determine an infinite tail." }, "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": [ "finite_prefix", 6, 2, [ 0, 0, 0 ], "a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2", "b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2", "2", "4", true, true, "two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii" ] }, "conclusion": { "canonical_value": "(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)", "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-r05-power-series-radius-of-convergence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/questions.jsonl", "batch_sha256": "75ea72f67b08f0d64822fea279c968391c8a15e19b1c10230cf82af81dabe3f4", "original_line": 81, "original_line_sha256": "4bad50a9347caca585b05e796d5ffc9733bfa89a9eb9970d58c08ee0a8b12e51", "question_compact_sha256": "8a4b41e59c78875f4a01a5252877c78822a4b3bd1f1f9ff6e3f0de38ec7c616b", "english_row_compact_sha256": "3b30a9568628e1ad1197dc2816dc392a022367dd18472a1895904d7564f598ac", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations.en.jsonl" }, "mko_bindings": { "mko-power-series-radius-of-convergence": { "sha256": "6c1fd0862cf0aff9deb2796ea76f2af6aa6acd82d07eb53b87a10aab859cde1f", "english_sha256": "f732982ca9bde1233312a021b49651f496d8732ee6037036ac799752c9694e68", "original_path": "public/data/mko/power-series-radius-of-convergence.json", "english_path": "public/data/i18n/editions/mko-power-series-radius-of-convergence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "04d5fe00084f382c9e6853d5f11c80b558d9cf857160ceb644fa20b64948d1fe", "original_candidate_sha256": "f7caef3912c08bdf109a16adce07f88799c9d3f401fd9baba40988f43438c3ac", "original_candidate_line_sha256": "f7caef3912c08bdf109a16adce07f88799c9d3f401fd9baba40988f43438c3ac", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-power-series-radius-of-convergence": "1905cc950ea16efa2bba777857906fac8a5629d35e8fe96a78832a5f09343413" } } }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1edd6f48c5b09f84af5b1835e69118935e7a47ce109e77fbad0783330f350e74", "arithmetic_steps": 2 }, "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." } }