{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r07-convergence-in-probability-0061", "pilot_number": "R0029-15086", "number": 15086, "family_id": "qf-mod30-convprob-finite-prefix-boundary", "title": { "zh-Hant": "第 15086 題：依機率收斂", "en": "Question 15086: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r07-convergence-in-probability-0061", "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-finite-prefix-boundary；資料={\"numerator\":1,\"shift\":5,\"prefix_length\":11}。請依 [numerator,shift,prefix_length,values,convergence_proved,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], 否, finite_prefix_not_convergence_certificate)", "explanation_zh": "Wave4第四層核對=(1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], 否, finite_prefix_not_convergence_certificate)；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": 2, "skills": [ "依機率收斂" ] }, "variant": { "family_id": "qf-mod30-convprob-finite-prefix-boundary", "seed": 202610307061, "parameters": { "numerator": 1, "shift": 5, "prefix_length": 11, "domain": "Symbolic_deviation_probability_tail", "operation": "convprob-finite-prefix-boundary", "semantic_template": "qf-mod30-convprob-finite-prefix-boundary/{\"numerator\":1,\"prefix_length\":11,\"shift\":5}" } }, "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-0061", "batch_id": "batch-20261014-advance-20261030-r07-convergence-in-probability-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "443106cc36d3972f630969fb71fbedb4b774b45e8f60ef7fbc9aebff44aac5d4", "family_id": "qf-mod30-convprob-finite-prefix-boundary", "seed": 202610307061, "semantic_template": "qf-mod30-convprob-finite-prefix-boundary/{\"numerator\":1,\"prefix_length\":11,\"shift\":5}" }, "fields": { "title": "A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate", "stem": "Use only the explicitly supplied finite list p_n=1/(n+5) for n=0 through prefix_length-1, with prefix_length=11. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).", "answer": { "kind": "expression", "value": "(1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate)", "explanation": "List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate)." }, "skills": [ "A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate" ], "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": "共同機率空間與forall epsilon tail責任。操作=convprob-finite-prefix-boundary；資料={\"numerator\":1,\"shift\":5,\"prefix_length\":11}。請依 [numerator,shift,prefix_length,values,convergence_proved,boundary] 欄序回答。", "en": "Use only the explicitly supplied finite list p_n=1/(n+5) for n=0 through prefix_length-1, with prefix_length=11. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "positive profile parameters c=1;s=5;raw fraction=c/(n+s)", "after": "nonnegative valid probability indices start at=0;only there0