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R0053-27152 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0002
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], 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).
G001 — Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], 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).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.
Before
source roles = [ci-procedure-qualification]
After
derived = [[theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement]]
Conditions: none
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[0, 1], [a, b], [[[2, 4], [2, 4]], [[2, 4], [2, 4]]], [[0, 0], [1, 1]]]
After
derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], [[a, [0, 0], 0], [b, [1, 1], 1]]]
Conditions: none
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.
Rule: CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · Uses: G001, S001
Before
CI theta0 ORIGINAL probability normalization column0 := sum(0, 1/2)
After
CI theta0 ORIGINAL probability normalization column0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
CI theta0 ORIGINAL probability normalization column1 := sum(1/2, 1/2)
After
CI theta0 ORIGINAL probability normalization column1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/2)
After
CI theta1 ORIGINAL probability normalization column0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
CI theta1 ORIGINAL probability normalization column1 := sum(1/2, 1/2)
After
CI theta1 ORIGINAL probability normalization column1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
CI theta0 ORIGINAL included coverage column0 := sum(0, 1/2)
After
CI theta0 ORIGINAL included coverage column0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [0, [a, b], [[0, 0], [1, 1]], [1/2, 1/2]]
After
derived = [[[a, 0, 0, 0, Yes, 1/2], [b, 1, 1, 0, No, 1/2]], 1/2]
Conditions: none
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.
Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S007
Before
CI theta1 ORIGINAL included coverage column1 := sum(0, 1/2)
After
CI theta1 ORIGINAL included coverage column1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [1, [a, b], [[0, 0], [1, 1]], [1/2, 1/2]]
After
derived = [[[a, 0, 0, 1, No, 1/2], [b, 1, 1, 1, Yes, 1/2]], 1/2]
Conditions: none
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.
Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S009
S010 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S009 · premise_and_result_audit — Recompute CI theta1 ORIGINAL included coverage column1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S007 · premise_and_result_audit — Recompute CI theta0 ORIGINAL included coverage column0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.([0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], Yes, total_precommitted_theta_independent_interval_procedure)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 085224b71e3ae6f0c2224114cedb76b123dc620145a0b19bad2df01c911cf0a0candidate record SHA-256: 0dc08b76620c31c4e06738ed10cd19a24ee248116ce7075bb360aa987be8ac10semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED