{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261101-r04-almost-sure-convergence-0005", "pilot_number": "R0031-16030", "number": 16030, "family_id": "qf-mod32-asconv-qualification", "title": { "zh-Hant": "第 16030 題：幾乎必然收斂", "en": "Question 16030: Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261101-r04-almost-sure-convergence-0005", "batch_id": "batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-almost-sure-convergence" ], "title_zh": "幾乎必然收斂", "stem_zh": "同一完整有限原子機率空間與明示尾端證書。操作=asconv-qualification；資料={\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}。請依 [atom_rows,exceptional_mass,positive_mass_failures,almost_sure] 欄序回答。", "answer": { "kind": "expression", "value": "([[0, 1/2, 5, 是, 是], [1, 1/2, 6, 是, 是], [2, 0, null, 否, 是]], 0, [], 是)", "explanation_zh": "Wave5根節點核對=([[0, 1/2, 5, 是, 是], [1, 1/2, 6, 是, 是], [2, 0, null, 否, 是]], 0, [], 是)；依完整明示資料重算；每個正質量原子須收斂；零質量失敗可忽略；有限prefix與依機率收斂本身不足以推出幾乎必然收斂。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "幾乎必然收斂" ] }, "variant": { "family_id": "qf-mod32-asconv-qualification", "seed": 202611014005, "parameters": { "masses": [ [ 1, 2 ], [ 1, 2 ], [ 0, 1 ] ], "settling_indices": [ 5, 6, null ], "domain": "Exact_finite_atomic_almost_sure_tail", "operation": "asconv-qualification", "semantic_template": "qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式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-20261101-r04-almost-sure-convergence-0005", "batch_id": "batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "84438f65ceccd69d5b29d3aa77ac08f6b20264c228ed47de689a8d82ae0a78ce", "family_id": "qf-mod32-asconv-qualification", "seed": 202611014005, "semantic_template": "qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}" }, "fields": { "title": "Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model", "stem": "Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.", "answer": { "kind": "expression", "value": "([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)", "explanation": "Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)." }, "skills": [ "Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model" ], "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": "同一完整有限原子機率空間與明示尾端證書。操作=asconv-qualification；資料={\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}。請依 [atom_rows,exceptional_mass,positive_mass_failures,almost_sure] 欄序回答。", "en": "Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "almost-sure full probability mass := sum(1/2, 1/2, 0)", "after": "almost-sure full probability mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位角色與已核中間量精確計算，保留分母、正負號、索引及完整載體。", "en": "Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 almost-sure full probability mass，不以小數近似替代。", "en": "Recompute almost-sure full probability mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2", "0" ], "output": "1", "label": "almost-sure full probability mass" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "same complete finite atomic space;raw masses=[[1, 2], [1, 2], [0, 1]];raw settling declarations=[5, 6, null]", "after": "normalized nonnegative masses=[1/2, 1/2, 0];total=1;integer N means equality to common limit at EVERY later index;null means model-declared nonconvergence", "relation": "implies", "rule_id": "asconv_declared_full_trajectory_model_not_missing_receipt", "conditions": [], "explanation": { "zh-Hant": "共同機率空間與對齊原子先固定；整數N是全尾段相等前提，null是模型宣告該原子不收斂，不是尚未檢查或檔案缺失。不能由一段有限觀察自行填入這些全稱宣告。", "en": "Fix the same probability space and aligned atoms.An integer N declares equality on the entire later tail;null is model-declared nonconvergence,not an unchecked or missing receipt.These universal declarations cannot be manufactured from a finite observation." }, "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 atom=0;exact mass=1/2;full-tail declaration=5", "after": "atom row=[0, 1/2, 5, 是, 是]", "relation": "implies", "rule_id": "asconv_each_positive_atom_obligation_and_null_exception", "conditions": [], "explanation": { "zh-Hant": "正質量原子必須有收斂尾證書；零質量原子的失敗只在幾乎必然命題可忽略。保留每個原子的原順序、質量與宣告，不刪去零原子後假裝處處成立。", "en": "Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "original atom=1;exact mass=1/2;full-tail declaration=6", "after": "atom row=[1, 1/2, 6, 是, 是]", "relation": "implies", "rule_id": "asconv_each_positive_atom_obligation_and_null_exception", "conditions": [], "explanation": { "zh-Hant": "正質量原子必須有收斂尾證書；零質量原子的失敗只在幾乎必然命題可忽略。保留每個原子的原順序、質量與宣告，不刪去零原子後假裝處處成立。", "en": "Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "original atom=2;exact mass=0;full-tail declaration=null", "after": "atom row=[2, 0, null, 否, 是]", "relation": "implies", "rule_id": "asconv_each_positive_atom_obligation_and_null_exception", "conditions": [], "explanation": { "zh-Hant": "正質量原子必須有收斂尾證書；零質量原子的失敗只在幾乎必然命題可忽略。保留每個原子的原順序、質量與宣告，不刪去零原子後假裝處處成立。", "en": "Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere." }, "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": "almost-sure declared nonconvergent exceptional mass := sum(0)", "after": "almost-sure declared nonconvergent exceptional mass = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位角色與已核中間量精確計算，保留分母、正負號、索引及完整載體。", "en": "Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 almost-sure declared nonconvergent exceptional mass，不以小數近似替代。", "en": "Recompute almost-sure declared nonconvergent exceptional mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0" ], "output": "0", "label": "almost-sure declared nonconvergent exceptional mass" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "all model-declared nonconvergent atom indices=[2];their total mass=0", "after": "positive-mass failures=[];almost-sure qualification=是", "relation": "implies", "rule_id": "asconv_probability_one_not_every_outcome", "conditions": [], "explanation": { "zh-Hant": "有限且完整的原子空間中，壞集合質量為0恰當且僅當沒有正質量壞原子；這才給機率1。它不把零質量壞原子的實際不收斂改成收斂。", "en": "On this complete finite atomic space,the bad set has mass0 exactly when it contains no positive-mass atom.That gives probability1 without changing actual nonconvergence on a null atom." }, "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": [ [ [ 0, "1/2", 5, true, true ], [ 1, "1/2", 6, true, true ], [ 2, "0", null, false, true ] ], "0", [], true ] }, "conclusion": { "canonical_value": "([[0, 1/2, 5, 是, 是], [1, 1/2, 6, 是, 是], [2, 0, null, 否, 是]], 0, [], 是)", "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-20261101-r04-almost-sure-convergence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations/questions.jsonl", "batch_sha256": "35523ec3fefabfb7385cb0a76555cf7a68bda123292323ebe8723ccdf66bce10", "original_line": 5, "original_line_sha256": "cd6e64104822996d51972ff40ea3f8069efc33cd57c779024a53a485b2c6c392", "question_compact_sha256": "84438f65ceccd69d5b29d3aa77ac08f6b20264c228ed47de689a8d82ae0a78ce", "english_row_compact_sha256": "79de0b5260c4edecac53302fcc0a68f56f7c25021b3c5544ecb7963c7133bd27", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations.en.jsonl" }, "mko_bindings": { "mko-almost-sure-convergence": { "sha256": "a848e75d5e3643fc87b620fc61c2385c8ed080ba4eac3a8573d9cb5e575f0b70", "english_sha256": "7fe83946a8db77910d7a5c8f69cd807c4ac79a2b6c29e835673de60af2a605ef", "original_path": "public/data/mko/almost-sure-convergence.json", "english_path": "public/data/i18n/editions/mko-almost-sure-convergence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "633793a5f242dece0a240bd92f77a33c587d12463e0beaedd8f70a14c70cf9e7", "original_candidate_sha256": "4c442db34ab02ec7ddd67406e508f1e6f8f8ca34fd908a93f56facbcdd040828", "original_candidate_line_sha256": "4c442db34ab02ec7ddd67406e508f1e6f8f8ca34fd908a93f56facbcdd040828" }, "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-20261026-007-p32", "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": "55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a", "question_first_witness_sha256": "754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "372a6262761e8dddf46e81506a7a6e8662032422ed55a0ff73cee15de07d551b", "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." } }