# Question 626: A Total Statistic Partitions the Complete Sample Carrier into Fibers [Return to catalogue page 4](https://ocme.evemisslab.com/en/questions/page/4/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations/#q-20261015-advance-20261116-r05-finite-sufficient-statistic-0001) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P1000-0626 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0001 ## Original question and givens Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate). - G001 — Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate). ## Definition and scope The statistic is total on the sample carrier; the fibers by value are disjoint and cover every sample. - [mko-finite-sufficient-statistic](https://ocme.evemisslab.com/en/mko/mko-finite-sufficient-statistic/) ## Step-by-step derivation S001 · impliesBefore Θ=[a0, b0]; Ω=[0, 1, 2, 3] After [[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]] Conditions: none Each row is one parameter law and each column an outcome; reorder probabilities together with labels. Step identifiers and dependenciesRule: full_parametric_model · Uses: G001 S002 · calculationBefore P_a0(Ω) := sum(2/5, 1/10, 2/5, 1/10) After P_a0(Ω) = 1 Conditions: none Check this law's complete total mass first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore P_b0(Ω) := sum(1/10, 2/5, 1/10, 2/5) After P_b0(Ω) = 1 Conditions: none Check this law's complete total mass first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore sample=0 After T(0)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S003 S005 · impliesBefore sample=1 After T(1)=1 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S004 S006 · impliesBefore sample=2 After T(2)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S005 S007 · impliesBefore sample=3 After T(3)=1 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S006 S008 · impliesBefore [(0, 0), (1, 1), (2, 0), (3, 1)] After fibers=[[0, [0, 2]], [1, [1, 3]]] Conditions: none Grouping by value produces disjoint fibers whose union is the complete carrier. Step identifiers and dependenciesRule: fiber_partition · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S003 · premise_and_result_audit — Recompute P_b0(Ω) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute P_a0(Ω) with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. ## Result ([0, 1, 2, 3], [0, 1, 0, 1], [[0, [0, 2]], [1, [1, 3]]], Yes, total_statistic_fibers_partition_sample_space) The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 94c8f829d9fa34a312f4cc502eac1da230bb74efe9be81aa301c44568dad9b1f - candidate record SHA-256: 8d9473c287d62e894010b6b3fe80b166728453d83956c22cf4c4f468447e29f2 - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r05-finite-sufficient-statistic-0001.json)