R0053-27128 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0028
完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-pointwise-coverage；資料={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[3,5],[2,5]],[[2,5],[3,5]]],"intervals":[[0,0],[1,1]]}。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-pointwise-coverage；資料={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[3,5],[2,5]],[[2,5],[3,5]]],"intervals":[[0,0],[1,1]]}。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-pointwise-coverage]
操作後
derived = [[pointwise_coverage_table, coverage_definition, frequentist_scope]]
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], [[[3, 5], [2, 5]], [[2, 5], [3, 5]]], [[0, 0], [1, 1]]]
操作後
derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[3/5, 2/5], [2/5, 3/5]], [[a, [0, 0], 0], [b, [1, 1], 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, 3/5)
操作後
CI theta0 ORIGINAL probability normalization column0 = 3/5
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(3/5, 2/5)
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, 2/5)
操作後
CI theta1 ORIGINAL probability normalization column0 = 2/5
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(2/5, 3/5)
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, 3/5)
操作後
CI theta0 ORIGINAL included coverage column0 = 3/5
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
source roles = [0, [a, b], [[0, 0], [1, 1]], [3/5, 2/5]]
操作後
derived = [[[a, 0, 0, 0, 是, 3/5], [b, 1, 1, 0, 否, 2/5]], 3/5]
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, S007
操作前
CI theta1 ORIGINAL included coverage column1 := sum(0, 3/5)
操作後
CI theta1 ORIGINAL included coverage column1 = 3/5
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, S008
操作前
source roles = [1, [a, b], [[0, 0], [1, 1]], [2/5, 3/5]]
操作後
derived = [[[a, 0, 0, 1, 否, 2/5], [b, 1, 1, 1, 是, 3/5]], 3/5]
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
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 — 以精確有理數重算 CI theta1 ORIGINAL included coverage column1，不以小數近似替代。S008 · 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。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。([[0, 3/5], [1, 3/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
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: e2f3fdccb61d90de096c6e2fe2e03b068e1c378113d26d0c87c909ef7f722118candidate record SHA-256: e7dda2c3d2a5eeb90f1512a2a1f17047e485c571909bac61e80a5d2aa2329e24semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED