# Question 672: Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber [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-0072) 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. P1000-0672 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0072 ## Original question and givens Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], 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 (fiber_invariance_rows, sufficient, certificate). - G001 — Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], 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 (fiber_invariance_rows, sufficient, certificate). ## Definition and scope Conditional laws are defined only on positive-mass fibers; sufficiency needs every fiber to be parameter-invariant. - [mko-finite-sufficient-statistic](https://ocme.evemisslab.com/en/mko/mko-finite-sufficient-statistic/) ## Step-by-step derivation S001 · impliesBefore Θ=[a21, b21]; Ω=[210, 211, 212, 213] 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_a21(Ω) := sum(2/5, 1/10, 2/5, 1/10) After P_a21(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_b21(Ω) := sum(1/10, 2/5, 1/10, 2/5) After P_b21(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=210 After T(210)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S003 S005 · impliesBefore sample=211 After T(211)=1 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S004 S006 · impliesBefore sample=212 After T(212)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S005 S007 · impliesBefore sample=213 After T(213)=1 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S006 S008 · impliesBefore [(210, 0), (211, 1), (212, 0), (213, 1)] After fibers=[[0, [210, 212]], [1, [211, 213]]] Conditions: none Grouping by value produces disjoint fibers whose union is the complete carrier. Step identifiers and dependenciesRule: fiber_partition · Uses: G001, S007 S009 · calculationBefore P_a21(T=0) := sum(2/5, 2/5) After P_a21(T=0) = 4/5 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore P_a21(210|T=0) := div(2/5, 4/5) After P_a21(210|T=0) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore reconstruct atom 210 under a21 := mul(1/2, 4/5) After reconstruct atom 210 under a21 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore P_a21(212|T=0) := div(2/5, 4/5) After P_a21(212|T=0) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore reconstruct atom 212 under a21 := mul(1/2, 4/5) After reconstruct atom 212 under a21 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore conditional total (a21,0) := sum(1/2, 1/2) After conditional total (a21,0) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore P_a21(T=1) := sum(1/10, 1/10) After P_a21(T=1) = 1/5 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore P_a21(211|T=1) := div(1/10, 1/5) After P_a21(211|T=1) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore reconstruct atom 211 under a21 := mul(1/2, 1/5) After reconstruct atom 211 under a21 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore P_a21(213|T=1) := div(1/10, 1/5) After P_a21(213|T=1) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore reconstruct atom 213 under a21 := mul(1/2, 1/5) After reconstruct atom 213 under a21 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore conditional total (a21,1) := sum(1/2, 1/2) After conditional total (a21,1) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore P_b21(T=0) := sum(1/10, 1/10) After P_b21(T=0) = 1/5 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore P_b21(210|T=0) := div(1/10, 1/5) After P_b21(210|T=0) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore reconstruct atom 210 under b21 := mul(1/2, 1/5) After reconstruct atom 210 under b21 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore P_b21(212|T=0) := div(1/10, 1/5) After P_b21(212|T=0) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore reconstruct atom 212 under b21 := mul(1/2, 1/5) After reconstruct atom 212 under b21 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore conditional total (b21,0) := sum(1/2, 1/2) After conditional total (b21,0) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore P_b21(T=1) := sum(2/5, 2/5) After P_b21(T=1) = 4/5 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore P_b21(211|T=1) := div(2/5, 4/5) After P_b21(211|T=1) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore reconstruct atom 211 under b21 := mul(1/2, 4/5) After reconstruct atom 211 under b21 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore P_b21(213|T=1) := div(2/5, 4/5) After P_b21(213|T=1) = 1/2 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore reconstruct atom 213 under b21 := mul(1/2, 4/5) After reconstruct atom 213 under b21 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore conditional total (b21,1) := sum(1/2, 1/2) After conditional total (b21,1) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore valid positive-mass laws for T=0: [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]] After parameter-invariant: Yes Conditions: none Compare full defined conditional laws; parameter invariance requires every sample probability to agree. Step identifiers and dependenciesRule: conditional_law_comparison · Uses: G001, S032 S034 · impliesBefore valid positive-mass laws for T=1: [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]] After parameter-invariant: Yes Conditions: none Compare full defined conditional laws; parameter invariance requires every sample probability to agree. Step identifiers and dependenciesRule: conditional_law_comparison · Uses: G001, S033 S035 · impliesBefore [[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]] After all fibers invariant: Yes Conditions: none Sufficiency under this finite contract requires every fiber to pass, not just one. Step identifiers and dependenciesRule: finite_sufficiency_definition · Uses: G001, S034 ## Reverse checks - S035 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S034 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S033 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S032 · premise_and_result_audit — Recompute conditional total (b21,1) with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute reconstruct atom 213 under b21 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute P_b21(213|T=1) with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute reconstruct atom 211 under b21 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute P_b21(211|T=1) with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute P_b21(T=1) with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute conditional total (b21,0) with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute reconstruct atom 212 under b21 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute P_b21(212|T=0) with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute reconstruct atom 210 under b21 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute P_b21(210|T=0) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute P_b21(T=0) with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute conditional total (a21,1) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute reconstruct atom 213 under a21 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute P_a21(213|T=1) with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute reconstruct atom 211 under a21 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute P_a21(211|T=1) with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute P_a21(T=1) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute conditional total (a21,0) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute reconstruct atom 212 under a21 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute P_a21(212|T=0) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute reconstruct atom 210 under a21 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute P_a21(210|T=0) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute P_a21(T=0) with exact rational arithmetic, not rounded decimals. - 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_b21(Ω) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute P_a21(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, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell) 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: b358c3fb5206856bb9077a1cc067054d3c523990d9e833854f7aa96daa95fd22 - candidate record SHA-256: ff392a56d42b9bb3724d74bd38a187b0e5a5ea69579bfa8827eece2f63463259 - 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-0072.json)