{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r04-continuous-real-distribution-function-0016", "pilot_number": "R0044-22841", "number": 22841, "family_id": "qf-mod42-contcdf-piecewise-linear-qualification", "title": { "zh-Hant": "第 22841 題：連續實值分布函數", "en": "Question 22841: A Continuous CDF May Have Flat Intervals" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r04-continuous-real-distribution-function-0016", "batch_id": "batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-continuous-real-distribution-function" ], "title_zh": "連續實值分布函數", "stem_zh": "CDF本身continuous、nondecreasing、right-continuous且audit query_points嚴格遞增。操作=contcdf-piecewise-linear-qualification；資料={\"knots\":[-17,0,17],\"values\":[0,[16,17],1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([-17, 0, 17], [0, 16/17, 1], 是, 是, 是, 是, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)", "explanation_zh": "Wave6第五層核對=([-17, 0, 17], [0, 16/17, 1], 是, 是, 是, 是, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)；依完整明示資料獨立重算；continuous CDF無jumps所以singletons質量0，flat intervals仍允許且兩端limits為0與1；continuous或atomless不推出density或absolute continuity；singular Cantor law是boundary。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "連續實值分布函數" ] }, "variant": { "family_id": "qf-mod42-contcdf-piecewise-linear-qualification", "seed": 202611114016, "parameters": { "knots": [ -17, 0, 17 ], "values": [ 0, [ 16, 17 ], 1 ], "domain": "Exact_continuous_piecewise_linear_CDF", "operation": "contcdf-piecewise-linear-qualification", "semantic_template": "qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-17,0,17],\"values\":[0,[16,17],1]}" } }, "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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r04-continuous-real-distribution-function-0016", "batch_id": "batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b22598ebd30677866580df726cc30056a959600ae3d3b850ebd491e50397f8af", "family_id": "qf-mod42-contcdf-piecewise-linear-qualification", "seed": 202611114016, "semantic_template": "qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-17,0,17],\"values\":[0,[16,17],1]}" }, "fields": { "title": "A Continuous CDF May Have Flat Intervals", "stem": "Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).", "answer": { "kind": "expression", "value": "([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)", "explanation": "Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)." }, "skills": [ "A Continuous CDF May Have Flat Intervals" ], "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": "CDF本身continuous、nondecreasing、right-continuous且audit query_points嚴格遞增。操作=contcdf-piecewise-linear-qualification；資料={\"knots\":[-17,0,17],\"values\":[0,[16,17],1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=contcdf-piecewise-linear-qualification;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。", "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness." }, "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": "CDF segment0 positive knot width := sub(0, -17)", "after": "CDF segment0 positive knot width = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment0 positive knot width，不以小數近似替代。", "en": "Recompute CDF segment0 positive knot width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-17" ], "output": "17", "label": "CDF segment0 positive knot width" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "CDF segment0 nonnegative value increment := sub(16/17, 0)", "after": "CDF segment0 nonnegative value increment = 16/17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment0 nonnegative value increment，不以小數近似替代。", "en": "Recompute CDF segment0 nonnegative value increment with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "16/17", "0" ], "output": "16/17", "label": "CDF segment0 nonnegative value increment" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "CDF segment0 exact nonnegative slope := div(16/17, 17)", "after": "CDF segment0 exact nonnegative slope = 16/289", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment0 exact nonnegative slope，不以小數近似替代。", "en": "Recompute CDF segment0 exact nonnegative slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "16/17", "17" ], "output": "16/289", "label": "CDF segment0 exact nonnegative slope" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "CDF segment1 positive knot width := sub(17, 0)", "after": "CDF segment1 positive knot width = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment1 positive knot width，不以小數近似替代。", "en": "Recompute CDF segment1 positive knot width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "17", "0" ], "output": "17", "label": "CDF segment1 positive knot width" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "CDF segment1 nonnegative value increment := sub(1, 16/17)", "after": "CDF segment1 nonnegative value increment = 1/17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment1 nonnegative value increment，不以小數近似替代。", "en": "Recompute CDF segment1 nonnegative value increment with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "16/17" ], "output": "1/17", "label": "CDF segment1 nonnegative value increment" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "CDF segment1 exact nonnegative slope := div(1/17, 17)", "after": "CDF segment1 exact nonnegative slope = 1/289", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CDF segment1 exact nonnegative slope，不以小數近似替代。", "en": "Recompute CDF segment1 exact nonnegative slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/17", "17" ], "output": "1/289", "label": "CDF segment1 exact nonnegative slope" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL strictly increasing knots=[-17, 0, 17];aligned exact nondecreasing values=[0, 16/17, 1];ALL [width,height,slope] rows=[[17, 16/17, 16/289], [17, 1/17, 1/289]]", "after": "declare linear interpolation on EVERY segment,constant0 left and1 right;shared knot values make joins continuous;all slopes>=0 allow flats;continuous implies right-continuous and endpoints give limits0/1", "relation": "implies", "rule_id": "contcdf_WHOLE_piecewise_law_continuous_flat_monotone_tails", "conditions": [], "explanation": { "zh-Hant": "全函數性質來自明示插值law，不是有限knot或query取樣推測。各段斜率非負且接點兩側同值，故單調且連續；constant tails給兩端機率界。這個PL例可有density，不代表所有continuous CDF都如此。", "en": "These are properties of the declared whole interpolation law,not inferences from finite knot/query observations.Nonnegative slopes and matching joins give monotonicity and continuity;constant tails give the limits.This PL example may have a density without every continuous CDF having one." }, "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": [ [ -17, 0, 17 ], [ "0", "16/17", "1" ], true, true, true, true, "linear_interpolation_is_continuous_and_flat_intervals_are_allowed" ] }, "conclusion": { "canonical_value": "([-17, 0, 17], [0, 16/17, 1], 是, 是, 是, 是, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)", "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-20261111-r04-continuous-real-distribution-function-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations/questions.jsonl", "batch_sha256": "922bc00d0ce7ad011b1cc9da9f0178f8f519641b0217848966d6bb6ce2a1df29", "original_line": 16, "original_line_sha256": "6ec869b5937f1309c0a36d08e2a6e9c1aaf0b2cad6cf8a9dbb9d2e475e035e78", "question_compact_sha256": "b22598ebd30677866580df726cc30056a959600ae3d3b850ebd491e50397f8af", "english_row_compact_sha256": "8c86799482eefa45571d05b24ef105e4d41a0d04e7f6eeaa3df6525627e5cc79", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations.en.jsonl" }, "mko_bindings": { "mko-continuous-real-distribution-function": { "sha256": "1dede2c575b28e28d08e5c509fe02847a319df1bd2e2069c0c6163ebfa3ebfac", "english_sha256": "bfbdc0d972f8b891c5b2fb6f469ee997dab5a660e6ada2c36c2b8f63bda3eae1", "original_path": "public/data/mko/continuous-real-distribution-function.json", "english_path": "public/data/i18n/editions/mko-continuous-real-distribution-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4db6f09f48d29a15f89787c4aac3ae0abe7981d9f906e3fbecc88b60dd686db5", "original_candidate_sha256": "540d94cffe05682bd756e8418807833e61be2247da6b5cdaacca230bc218efb0", "original_candidate_line_sha256": "540d94cffe05682bd756e8418807833e61be2247da6b5cdaacca230bc218efb0" }, "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-p45", "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": "89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81", "question_first_witness_sha256": "b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "b373834217fc32bfc76d7ecd81f9938f5d4c43da35358b58a77bdff3b430cc76", "arithmetic_steps": 6 }, "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." } }