{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0014", "pilot_number": "R0053-27164", "number": 27164, "family_id": "qf-mod49-ci-procedure-qualification", "title": { "zh-Hant": "第 27164 題：有限模型信賴區間覆蓋率", "en": "Question 27164: A Complete Precommitted Interval Procedure Keeps Every Probability Alignment" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0014", "batch_id": "batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-confidence-interval-coverage" ], "title_zh": "有限模型信賴區間覆蓋率", "stem_zh": "完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-procedure-qualification；資料={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[14,16],[2,16]],[[2,16],[14,16]]],\"intervals\":[[0,0],[1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], 是, total_precommitted_theta_independent_interval_procedure)", "explanation_zh": "Wave6第十二層核對=([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], 是, total_precommitted_theta_independent_interval_procedure)；依完整明示資料獨立重算；pointwise coverage與worst-case minimum逐theta重算；average不等於uniform，posterior不是coverage；selection納入procedure後須重新計算。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限模型信賴區間覆蓋率" ] }, "variant": { "family_id": "qf-mod49-ci-procedure-qualification", "seed": 202611183014, "parameters": { "theta_values": [ 0, 1 ], "samples": [ "a", "b" ], "probability_rows": [ [ [ 14, 16 ], [ 2, 16 ] ], [ [ 2, 16 ], [ 14, 16 ] ] ], "intervals": [ [ 0, 0 ], [ 1, 1 ] ], "domain": "Exact_finite_confidence_coverage", "operation": "ci-procedure-qualification", "semantic_template": "ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],theta:[0,1]}" } }, "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；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0014", "batch_id": "batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "11c4e3b0922dc1246b7887457f96d35368a61760ea10510080562239033f9cd3", "family_id": "qf-mod49-ci-procedure-qualification", "seed": 202611183014, "semantic_template": "ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],theta:[0,1]}" }, "fields": { "title": "A Complete Precommitted Interval Procedure Keeps Every Probability Alignment", "stem": "Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).", "answer": { "kind": "expression", "value": "([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure)", "explanation": "Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure)." }, "skills": [ "A Complete Precommitted Interval Procedure Keeps Every Probability Alignment" ], "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": "完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-procedure-qualification；資料={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[14,16],[2,16]],[[2,16],[14,16]]],\"intervals\":[[0,0],[1,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [ci-procedure-qualification]", "after": "derived = [[theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。", "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[0, 1], [a, b], [[[14, 16], [2, 16]], [[2, 16], [14, 16]]], [[0, 0], [1, 1]]]", "after": "derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], [[a, [0, 0], 0], [b, [1, 1], 1]]]", "relation": "implies", "rule_id": "CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric", "conditions": [], "explanation": { "zh-Hant": "interval先綁原始samples，機率row先綁原始theta與原始sample欄，再連資料一起排序；Source13 theta用numeric，samples用English canonical。區間閉合且有序，每個theta law非負正規化。同一預先固定、total的I(X)不依未知theta。", "en": "Bind intervals to ORIGINAL samples and probability rows/columns to ORIGINAL theta/samples BEFORE sorting those data together.Source13 theta order is numeric and sample order English canonical.Intervals are ordered and closed;each theta law is nonnegative and normalized.The same precommitted,total I(X) does not depend on unknown theta." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "CI theta0 ORIGINAL probability normalization column0 := sum(0, 7/8)", "after": "CI theta0 ORIGINAL probability normalization column0 = 7/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta0 ORIGINAL probability normalization column0，不以小數近似替代。", "en": "Recompute CI theta0 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "7/8" ], "output": "7/8", "label": "CI theta0 ORIGINAL probability normalization column0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "CI theta0 ORIGINAL probability normalization column1 := sum(7/8, 1/8)", "after": "CI theta0 ORIGINAL probability normalization column1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta0 ORIGINAL probability normalization column1，不以小數近似替代。", "en": "Recompute CI theta0 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7/8", "1/8" ], "output": "1", "label": "CI theta0 ORIGINAL probability normalization column1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/8)", "after": "CI theta1 ORIGINAL probability normalization column0 = 1/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta1 ORIGINAL probability normalization column0，不以小數近似替代。", "en": "Recompute CI theta1 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/8" ], "output": "1/8", "label": "CI theta1 ORIGINAL probability normalization column0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "CI theta1 ORIGINAL probability normalization column1 := sum(1/8, 7/8)", "after": "CI theta1 ORIGINAL probability normalization column1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta1 ORIGINAL probability normalization column1，不以小數近似替代。", "en": "Recompute CI theta1 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/8", "7/8" ], "output": "1", "label": "CI theta1 ORIGINAL probability normalization column1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "CI theta0 ORIGINAL included coverage column0 := sum(0, 7/8)", "after": "CI theta0 ORIGINAL included coverage column0 = 7/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta0 ORIGINAL included coverage column0，不以小數近似替代。", "en": "Recompute CI theta0 ORIGINAL included coverage column0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "7/8" ], "output": "7/8", "label": "CI theta0 ORIGINAL included coverage column0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [0, [a, b], [[0, 0], [1, 1]], [7/8, 1/8]]", "after": "derived = [[[a, 0, 0, 0, 是, 7/8], [b, 1, 1, 0, 否, 1/8]], 7/8]", "relation": "implies", "rule_id": "CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage", "conditions": [], "explanation": { "zh-Hant": "固定真theta，逐所有可能samples判L(x)<=theta<=U(x)，兩端相等也算hit，只把hit的P_theta(x)加起。這是反覆抽樣coverage，不是觀測資料後theta的posterior。", "en": "Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "CI theta1 ORIGINAL included coverage column1 := sum(0, 7/8)", "after": "CI theta1 ORIGINAL included coverage column1 = 7/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 CI theta1 ORIGINAL included coverage column1，不以小數近似替代。", "en": "Recompute CI theta1 ORIGINAL included coverage column1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "7/8" ], "output": "7/8", "label": "CI theta1 ORIGINAL included coverage column1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [1, [a, b], [[0, 0], [1, 1]], [1/8, 7/8]]", "after": "derived = [[[a, 0, 0, 1, 否, 1/8], [b, 1, 1, 1, 是, 7/8]], 7/8]", "relation": "implies", "rule_id": "CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage", "conditions": [], "explanation": { "zh-Hant": "固定真theta，逐所有可能samples判L(x)<=theta<=U(x)，兩端相等也算hit，只把hit的P_theta(x)加起。這是反覆抽樣coverage，不是觀測資料後theta的posterior。", "en": "Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ [ 0, 1 ], [ "a", "b" ], [ [ "0", "0" ], [ "1", "1" ] ], [ [ "7/8", "1/8" ], [ "1/8", "7/8" ] ], true, "total_precommitted_theta_independent_interval_procedure" ] }, "conclusion": { "canonical_value": "([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], 是, total_precommitted_theta_independent_interval_procedure)", "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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations/questions.jsonl", "batch_sha256": "e32eae55169a7001013c379f42d8fb53292340b088f540d145ae9717c637db21", "original_line": 14, "original_line_sha256": "8cf495a0fc2f8aa15d20ed8efebd8bbac6f4bbf600045f2a768c57fd5a255691", "question_compact_sha256": "11c4e3b0922dc1246b7887457f96d35368a61760ea10510080562239033f9cd3", "english_row_compact_sha256": "17a5cddd22f42c45cb8f1c930791e8a3dcbb288cc0096213cfbe39d1838edce8", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-confidence-interval-coverage": { "sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363", "english_sha256": "e2602185873b2f9b57fed1ac6efd6d390016ea0a0ff7efc9dc970097b8fca2cf", "original_path": "public/data/mko/finite-confidence-interval-coverage.json", "english_path": "public/data/i18n/editions/mko-finite-confidence-interval-coverage.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283", "original_candidate_sha256": "069163255e46069f72ab8e892e24e142fbfc044999248bb2107d1953b53ba33f", "original_candidate_line_sha256": "069163255e46069f72ab8e892e24e142fbfc044999248bb2107d1953b53ba33f" }, "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-20261026-010-p54", "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": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62", "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1" }, "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": "4c67f85c078c432248777c8e658699738454e1c00745469c6ee204950a58e445", "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." } }