{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r07-martingale-0034", "pilot_number": "R0066-33759", "number": 33759, "family_id": "qf-mod31-martdeep-difference-mean", "title": { "zh-Hant": "第 33759 題：Martingale深度", "en": "Question 33759: Zero Total Mean Is Not Enough: Check Each Information Block" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r07-martingale-0034", "batch_id": "batch-20261014-advance-20261031-r07-martingale-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-martingale" ], "title_zh": "Martingale深度", "stem_zh": "同一有限機率空間與filtration的精確模型。操作=martdeep-difference-mean；資料={\"increments\":[9,-9,10,-10],\"blocks\":[0,0,1,1]}。請依 [conditional_rows,all_zero,classification] 欄序回答。", "answer": { "kind": "expression", "value": "([[0, [9, -9], 0], [1, [10, -10], 0]], 是, conditional_mean_zero_difference)", "explanation_zh": "Wave4收束核對=([[0, [9, -9], 0], [1, [10, -10], 0]], 是, conditional_mean_zero_difference)；依完整明示資料重算；常期望不足；差分須逐資訊block條件均值為零；不引入optional stopping、收斂、Markov或獨立增量聲明。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "Martingale深度" ] }, "variant": { "family_id": "qf-mod31-martdeep-difference-mean", "seed": 202610317034, "parameters": { "increments": [ 9, -9, 10, -10 ], "blocks": [ 0, 0, 1, 1 ], "domain": "Exact_finite_martingale_construction", "operation": "martdeep-difference-mean", "semantic_template": "qf-mod31-martdeep-difference-mean/{\"blocks\":[0,0,1,1],\"increments\":[9,-9,10,-10]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_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、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、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-20261031-r07-martingale-0034", "batch_id": "batch-20261014-advance-20261031-r07-martingale-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3775ba77a0120cc7fcfe3fc76bfb654b342a9f2ece309b46c5d376d530a46f17", "family_id": "qf-mod31-martdeep-difference-mean", "seed": 202610317034, "semantic_template": "qf-mod31-martdeep-difference-mean/{\"blocks\":[0,0,1,1],\"increments\":[9,-9,10,-10]}" }, "fields": { "title": "Zero Total Mean Is Not Enough: Check Each Information Block", "stem": "Use four equiprobable outcomes with aligned increments=[9, -9, 10, -10] and current information block labels=[0, 0, 1, 1]. Give (conditional_rows,all_zero?,classification), with rows [block_label,increments_in_outcome_order,exact_block_mean] sorted by block label. Classification is conditional_mean_zero_difference or difference_fails_martingale_condition.", "answer": { "kind": "expression", "value": "([[0, [9, -9], 0], [1, [10, -10], 0]], Yes, conditional_mean_zero_difference)", "explanation": "Average the increments within every nonempty information block. All block means must be zero; a globally cancelling mean alone does not establish a martingale difference. The exact result, retaining the requested order, is ([[0, [9, -9], 0], [1, [10, -10], 0]], Yes, conditional_mean_zero_difference)." }, "skills": [ "Zero Total Mean Is Not Enough: Check Each Information Block" ], "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": "同一有限機率空間與filtration的精確模型。操作=martdeep-difference-mean；資料={\"increments\":[9,-9,10,-10],\"blocks\":[0,0,1,1]}。請依 [conditional_rows,all_zero,classification] 欄序回答。", "en": "Use four equiprobable outcomes with aligned increments=[9, -9, 10, -10] and current information block labels=[0, 0, 1, 1]. Give (conditional_rows,all_zero?,classification), with rows [block_label,increments_in_outcome_order,exact_block_mean] sorted by block label. Classification is conditional_mean_zero_difference or difference_fails_martingale_condition." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod31-martdeep-difference-mean, [blocks, domain, increments, operation]]", "after": "derived = [martdeep-difference-mean, Exact_finite_martingale_construction]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Difference ORIGINAL block0 sum := sum(9, -9)", "after": "Difference ORIGINAL block0 sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Difference ORIGINAL block0 sum，不以小數近似替代。", "en": "Recompute Difference ORIGINAL block0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "-9" ], "output": "0", "label": "Difference ORIGINAL block0 sum" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [Difference block0 mean, 1000000, 0, 2]", "after": "derived = [[0, 1], 0]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Difference block0 mean ORIGINAL numerator over denominator := div(0, 2)", "after": "Difference block0 mean ORIGINAL numerator over denominator = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Difference block0 mean ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Difference block0 mean ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "2" ], "output": "0", "label": "Difference block0 mean ORIGINAL numerator over denominator" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [0, [0, 1], [9, -9]]", "after": "derived = [0]", "relation": "implies", "rule_id": "Martingale_EVERY_information_block_conditional_mean", "conditions": [], "explanation": { "zh-Hant": "四等機率outcomes按此資訊block條件化後仍等機率，故均值是本block原increments總和/個數。逐所有非空block核，不以全域相消代替。", "en": "Conditioning four equiprobable outcomes on THIS information block leaves equal weights,so its conditional mean is the ORIGINAL increment sum divided by this block’s size.Check EVERY nonempty block,not global cancellation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Difference ORIGINAL block1 sum := sum(10, -10)", "after": "Difference ORIGINAL block1 sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Difference ORIGINAL block1 sum，不以小數近似替代。", "en": "Recompute Difference ORIGINAL block1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "10", "-10" ], "output": "0", "label": "Difference ORIGINAL block1 sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Difference block1 mean, 1000000, 0, 2]", "after": "derived = [[0, 1], 0]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Difference block1 mean ORIGINAL numerator over denominator := div(0, 2)", "after": "Difference block1 mean ORIGINAL numerator over denominator = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Difference block1 mean ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Difference block1 mean ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "2" ], "output": "0", "label": "Difference block1 mean ORIGINAL numerator over denominator" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [1, [2, 3], [10, -10]]", "after": "derived = [0]", "relation": "implies", "rule_id": "Martingale_EVERY_information_block_conditional_mean", "conditions": [], "explanation": { "zh-Hant": "四等機率outcomes按此資訊block條件化後仍等機率，故均值是本block原increments總和/個數。逐所有非空block核，不以全域相消代替。", "en": "Conditioning four equiprobable outcomes on THIS information block leaves equal weights,so its conditional mean is the ORIGINAL increment sum divided by this block’s size.Check EVERY nonempty block,not global cancellation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [[9, -9, 10, -10], [0, 0, 1, 1], [[0, [9, -9], 0], [1, [10, -10], 0]]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Martingale_DIFFERENCE_all_block_means_zero_required", "conditions": [], "explanation": { "zh-Hant": "只有每一資訊block均值都0才通過conditional difference責任；全體mean0而有非零block仍失敗。", "en": "Pass the conditional-difference obligation ONLY if EVERY information-block mean is0.A globally zero mean with a nonzero block still fails." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ 0, [ 9, -9 ], "0" ], [ 1, [ 10, -10 ], "0" ] ], true, "conditional_mean_zero_difference" ] }, "conclusion": { "canonical_value": "([[0, [9, -9], 0], [1, [10, -10], 0]], 是, conditional_mean_zero_difference)", "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-20261031-r07-martingale-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r07-martingale-foundations/questions.jsonl", "batch_sha256": "ee82a8e65508794c0f2899807a2e0f55a4c717acd7a4691a97b1de21d4c4a698", "original_line": 34, "original_line_sha256": "8718374f39970f6c5f03d2bd43f278d046daae613c146cac358f28f5fa321d8e", "question_compact_sha256": "3775ba77a0120cc7fcfe3fc76bfb654b342a9f2ece309b46c5d376d530a46f17", "english_row_compact_sha256": "25759b42c0aa469cb3bf9c4067fe98b51e97a77ddf23754d3ad27fe23b902092", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r07-martingale-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r07-martingale-foundations.en.jsonl" }, "mko_bindings": { "mko-martingale": { "sha256": "25744aeb70a10daa6dcabb09f5b71bc0ee3e9f2c266412b727873f5e2abddf7b", "english_sha256": "baf1b76a7fdf33a34d203015ec0ab002e6398566cc74b3d6decc7de93b52132e", "original_path": "public/data/mko/martingale.json", "english_path": "public/data/i18n/editions/mko-martingale.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310", "original_candidate_sha256": "3bbb8ff7a0568b04a3923dc3fc87e0549e0b103f14c6951509d5ebcc8813b3f7", "original_candidate_line_sha256": "3bbb8ff7a0568b04a3923dc3fc87e0549e0b103f14c6951509d5ebcc8813b3f7" }, "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-20261026-012-p67", "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": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267", "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 10, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "97677cbef9446f8b89611eb8b9c27d5962108cfd9115cf4a87573a4a76b3a975", "arithmetic_steps": 4 }, "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." } }