{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r07-convergence-in-probability-0031", "pilot_number": "R0029-15031", "number": 15031, "family_id": "qf-mod30-convprob-epsilon-witness", "title": { "zh-Hant": "第 15031 題：依機率收斂", "en": "Question 15031: A Strict Rational Tail Cutoff Is Different from a Finite Prefix" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r07-convergence-in-probability-0031", "batch_id": "batch-20261014-advance-20261030-r07-convergence-in-probability-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-convergence-in-probability" ], "title_zh": "依機率收斂", "stem_zh": "共同機率空間與forall epsilon tail責任。操作=convprob-epsilon-witness；資料={\"numerator\":1,\"shift\":7,\"delta\":[1,15]}。請依 [numerator,shift,delta,N,bound,pass,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "(1, 7, 1/15, 9, 1/16, 是, forall_n_ge_N_tail_bound)", "explanation_zh": "Wave4第四層核對=(1, 7, 1/15, 9, 1/16, 是, forall_n_ge_N_tail_bound)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；finite prefix不足且只接受symbolic tail或eventual equality；不混同almost sure Lp或distribution convergence。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "依機率收斂" ] }, "variant": { "family_id": "qf-mod30-convprob-epsilon-witness", "seed": 202610307031, "parameters": { "numerator": 1, "shift": 7, "delta": [ 1, 15 ], "domain": "Symbolic_deviation_probability_tail", "operation": "convprob-epsilon-witness", "semantic_template": "qf-mod30-convprob-epsilon-witness/{\"delta\":[1,15],\"numerator\":1,\"shift\":7}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_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、一般域／Hilbert／martingale／無限基數／一般收斂、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-20261030-r07-convergence-in-probability-0031", "batch_id": "batch-20261014-advance-20261030-r07-convergence-in-probability-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8751cd725bead8dedcdb05a6bd09c2e65f4f948c1d5d44d5546b21e073f54b19", "family_id": "qf-mod30-convprob-epsilon-witness", "seed": 202610307031, "semantic_template": "qf-mod30-convprob-epsilon-witness/{\"delta\":[1,15],\"numerator\":1,\"shift\":7}" }, "fields": { "title": "A Strict Rational Tail Cutoff Is Different from a Finite Prefix", "stem": "Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/15. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).", "answer": { "kind": "expression", "value": "(1, 7, 1/15, 9, 1/16, Yes, forall_n_ge_N_tail_bound)", "explanation": "Solve the strict inequality c/(N+s)cb/a，所以取floor(cb/a)−s+1再與0取最大。嚴格不等式在剛好等號時仍需下一索引。", "en": "The strict inequality c/(N+s)cb/a,hence floor(cb/a)−s+1 clamped only at0.Equality still requires the next index." }, "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": "strict witness denominator := sum(9, 7)", "after": "strict witness denominator = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 strict witness denominator，不以小數近似替代。", "en": "Recompute strict witness denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "7" ], "output": "16", "label": "strict witness denominator" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "strict witness valid probability bound := div(1, 16)", "after": "strict witness valid probability bound = 1/16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 strict witness valid probability bound，不以小數近似替代。", "en": "Recompute strict witness valid probability bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "16" ], "output": "1/16", "label": "strict witness valid probability bound" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "previous raw reciprocal denominator := sum(8, 7)", "after": "previous raw reciprocal denominator = 15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 previous raw reciprocal denominator，不以小數近似替代。", "en": "Recompute previous raw reciprocal denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "8", "7" ], "output": "15", "label": "previous raw reciprocal denominator" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "previous raw reciprocal value := div(1, 15)", "after": "previous raw reciprocal value = 1/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 previous raw reciprocal value，不以小數近似替代。", "en": "Recompute previous raw reciprocal value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "15" ], "output": "1/15", "label": "previous raw reciprocal value" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "N=9;current valid probability=1/16;previous RAW reciprocal=1/15;delta=1/15", "after": "current strict bound holds,previous raw threshold fails;N is minimal", "relation": "implies", "rule_id": "strict_profile_minimality_without_invalid_probability_claim", "conditions": [], "explanation": { "zh-Hant": "前一算術分式仍≥delta，所以所有更小索引都不滿足原嚴格門檻。若前索引不在機率域，這裡只稱raw reciprocal，不把大於1的值當機率；本N必已通過原英文有效域guard，不另max平移答案。", "en": "The previous arithmetic reciprocal is still≥delta,so every smaller index fails the original strict threshold.If that earlier index is outside the probability domain,it is only a raw reciprocal,not a probability above1.The returned N must pass the original English domain guard;do not max-shift the source answer." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "declared same-space deviation profile on valid tail;n>=N=9;positive denominators increase", "after": "every later profile<=bound=1/16