R0053-27125 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0100
完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"average_only","model":{"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[25,27],[2,27]],[[2,27],[25,27]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"average_only","model":{"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[25,27],[2,27]],[[2,27],[25,27]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
操作前
source roles = [ci-interpretation-boundary]
操作後
derived = [[mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary]]
條件：無
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。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[0, 1], [a, b], [[[25, 27], [2, 27]], [[2, 27], [25, 27]]], [[0, 0], [0, 0]]]
操作後
derived = [[0, 1], [a, b], [[0, 0], [0, 0]], [[25/27, 2/27], [2/27, 25/27]], [[a, [0, 0], 0], [b, [0, 0], 1]]]
條件：無
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。
規則：CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · 依賴：G001, S001
操作前
CI theta0 ORIGINAL probability normalization column0 := sum(0, 25/27)
操作後
CI theta0 ORIGINAL probability normalization column0 = 25/27
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S002
操作前
CI theta0 ORIGINAL probability normalization column1 := sum(25/27, 2/27)
操作後
CI theta0 ORIGINAL probability normalization column1 = 1
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S003
操作前
CI theta1 ORIGINAL probability normalization column0 := sum(0, 2/27)
操作後
CI theta1 ORIGINAL probability normalization column0 = 2/27
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S004
操作前
CI theta1 ORIGINAL probability normalization column1 := sum(2/27, 25/27)
操作後
CI theta1 ORIGINAL probability normalization column1 = 1
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S005
操作前
CI theta0 ORIGINAL included coverage column0 := sum(0, 25/27)
操作後
CI theta0 ORIGINAL included coverage column0 = 25/27
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
CI theta0 ORIGINAL included coverage column1 := sum(25/27, 2/27)
操作後
CI theta0 ORIGINAL included coverage column1 = 1
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
source roles = [0, [a, b], [[0, 0], [0, 0]], [25/27, 2/27]]
操作後
derived = [[[a, 0, 0, 0, 是, 25/27], [b, 0, 0, 0, 是, 2/27]], 1]
條件：無
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。
規則：CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · 依賴：G001, S008
操作前
source roles = [1, [a, b], [[0, 0], [0, 0]], [2/27, 25/27]]
操作後
derived = [[[a, 0, 0, 1, 否, 2/27], [b, 0, 0, 1, 否, 25/27]], 0]
條件：無
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。
規則：CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · 依賴：G001, S009
操作前
CI average ORIGINAL normalized weight0 := sum(0, 1/3)
操作後
CI average ORIGINAL normalized weight0 = 1/3
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
CI average ORIGINAL normalized weight1 := sum(1/3, 2/3)
操作後
CI average ORIGINAL normalized weight1 = 1
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S011
操作前
source roles = [CI average, [0, 1], [[1, 3], [2, 3]]]
操作後
derived = [[(0, 1/3), (1, 2/3)], 1]
條件：無
prior或平均權重必先對原始theta綁定，再依相同numeric theta順序移動；不能把索引位址當theta，也不能只排序權重。
規則：CI_ORIGINAL_theta_weight_pairs_NOT_slot_substitution · 依賴：G001, S012
操作前
CI ORIGINAL coverage times theta weight0 := mul(1, 1/3)
操作後
CI ORIGINAL coverage times theta weight0 = 1/3
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
CI ORIGINAL weighted average running sum0 := sum(0, 1/3)
操作後
CI ORIGINAL weighted average running sum0 = 1/3
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S014
操作前
CI ORIGINAL coverage times theta weight1 := mul(0, 2/3)
操作後
CI ORIGINAL coverage times theta weight1 = 0
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
CI ORIGINAL weighted average running sum1 := sum(1/3, 0)
操作後
CI ORIGINAL weighted average running sum1 = 1/3
條件：無
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。
規則：exact_rational_arithmetic · 依賴：G001, S016
操作前
source roles = [[[0, 1], [1, 0]], [1/3, 2/3], 1/3]
操作後
derived = [1/3, 0, 是, 是, 是, 否]
條件：無
實際coverage不同，且加權平均>=target>minimum，三個條件都核對，才構成平均掩蓋uniform失敗的反例。不能把相同coverages或沒有嚴格gap的資料稱為此見證。
規則：CI_ACTUAL_average_masks_uniform_failure_distinct_strict_witness · 依賴：G001, S017
S018 · premise_and_result_audit — 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。S017 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL weighted average running sum1，不以小數近似替代。S016 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL coverage times theta weight1，不以小數近似替代。S015 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL weighted average running sum0，不以小數近似替代。S014 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL coverage times theta weight0，不以小數近似替代。S013 · premise_and_result_audit — 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。S012 · premise_and_result_audit — 以精確有理數重算 CI average ORIGINAL normalized weight1，不以小數近似替代。S011 · premise_and_result_audit — 以精確有理數重算 CI average ORIGINAL normalized weight0，不以小數近似替代。S010 · premise_and_result_audit — 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。S009 · premise_and_result_audit — 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。S008 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL included coverage column1，不以小數近似替代。S007 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL included coverage column0，不以小數近似替代。S006 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column1，不以小數近似替代。S005 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column0，不以小數近似替代。S004 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column1，不以小數近似替代。S003 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column0，不以小數近似替代。S002 · premise_and_result_audit — 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。S001 · premise_and_result_audit — 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。(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, 是, 是, 是, 否, average_coverage_can_mask_failure_of_uniform_target)
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。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f21d476771595f981eb11382b17bb56ab9fade06ff75b4ff545fcb021bf89105candidate record SHA-256: 94ce208e1a963876618ffb3d4b62273f75b29925647c4a3ec53fc60943e1e897semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED