# Question 699: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks [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-0099) 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. P1000-0699 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0099 ## Original question and givens Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary). - G001 — Use complete theta labels=[a23, b23], samples=[230, 231, 232, 233], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary). ## Definition and scope One positive-mass fiber with differing conditional laws refutes sufficiency. An injective statistic is sufficient; a constant one exactly when all laws agree; zero-mass cells stay null. - [mko-finite-sufficient-statistic](https://ocme.evemisslab.com/en/mko/mko-finite-sufficient-statistic/) ## Step-by-step derivation S001 · impliesBefore Θ=[a23, b23]; Ω=[230, 231, 232, 233] 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_a23(Ω) := sum(2/5, 1/10, 2/5, 1/10) After P_a23(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_b23(Ω) := sum(1/10, 2/5, 1/10, 2/5) After P_b23(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=230 After T(230)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S003 S005 · impliesBefore sample=231 After T(231)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S004 S006 · impliesBefore sample=232 After T(232)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S005 S007 · impliesBefore sample=233 After T(233)=0 Conditions: none Every sample has exactly one statistic value. Step identifiers and dependenciesRule: total_statistic_map · Uses: G001, S006 S008 · impliesBefore [(230, 0), (231, 0), (232, 0), (233, 0)] After fibers=[[0, [230, 231, 232, 233]]] 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_a23(T=0) := sum(2/5, 1/10, 2/5, 1/10) After P_a23(T=0) = 1 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore P_a23(230|T=0) := div(2/5, 1) After P_a23(230|T=0) = 2/5 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 230 under a23 := mul(2/5, 1) After reconstruct atom 230 under a23 = 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_a23(231|T=0) := div(1/10, 1) After P_a23(231|T=0) = 1/10 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 231 under a23 := mul(1/10, 1) After reconstruct atom 231 under a23 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore P_a23(232|T=0) := div(2/5, 1) After P_a23(232|T=0) = 2/5 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore reconstruct atom 232 under a23 := mul(2/5, 1) After reconstruct atom 232 under a23 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore P_a23(233|T=0) := div(1/10, 1) After P_a23(233|T=0) = 1/10 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 233 under a23 := mul(1/10, 1) After reconstruct atom 233 under a23 = 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 conditional total (a23,0) := sum(2/5, 1/10, 2/5, 1/10) After conditional total (a23,0) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore P_b23(T=0) := sum(1/10, 2/5, 1/10, 2/5) After P_b23(T=0) = 1 Conditions: none First compute the conditioning event's full mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore P_b23(230|T=0) := div(1/10, 1) After P_b23(230|T=0) = 1/10 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore reconstruct atom 230 under b23 := mul(1/10, 1) After reconstruct atom 230 under b23 = 1/10 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore P_b23(231|T=0) := div(2/5, 1) After P_b23(231|T=0) = 2/5 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 231 under b23 := mul(2/5, 1) After reconstruct atom 231 under b23 = 2/5 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_b23(232|T=0) := div(1/10, 1) After P_b23(232|T=0) = 1/10 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 232 under b23 := mul(1/10, 1) After reconstruct atom 232 under b23 = 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 P_b23(233|T=0) := div(2/5, 1) After P_b23(233|T=0) = 2/5 Conditions: none Divide the same law's atom mass by its positive fiber mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore reconstruct atom 233 under b23 := mul(2/5, 1) After reconstruct atom 233 under b23 = 2/5 Conditions: none Multiply by fiber mass to check the original atom probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore conditional total (b23,0) := sum(1/10, 2/5, 1/10, 2/5) After conditional total (b23,0) = 1 Conditions: none The complete conditional law must have total mass1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore valid positive-mass laws for T=0: [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]] After parameter-invariant: No Conditions: none Compare full defined conditional laws; parameter invariance requires every sample probability to agree. Step identifiers and dependenciesRule: conditional_law_comparison · Uses: G001, S028 S030 · impliesBefore [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]] After all fibers invariant: No Conditions: none Sufficiency under this finite contract requires every fiber to pass, not just one. Step identifiers and dependenciesRule: finite_sufficiency_definition · Uses: G001, S029 S031 · impliesBefore the only fiber is Ω with probability1 After conditional sample law=P_theta; sufficient iff all rows agree Conditions: none It is the conditional sample-outcome law that may vary, not the law of the constant statistic itself. Step identifiers and dependenciesRule: constant_statistic_boundary · Uses: G001, S030 ## Reverse checks - S031 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S030 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S029 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima. - S028 · premise_and_result_audit — Recompute conditional total (b23,0) with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute reconstruct atom 233 under b23 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute P_b23(233|T=0) with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute reconstruct atom 232 under b23 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute P_b23(232|T=0) with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute reconstruct atom 231 under b23 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute P_b23(231|T=0) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute reconstruct atom 230 under b23 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute P_b23(230|T=0) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute P_b23(T=0) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute conditional total (a23,0) with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute reconstruct atom 233 under a23 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute P_a23(233|T=0) with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute reconstruct atom 232 under a23 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute P_a23(232|T=0) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute reconstruct atom 231 under a23 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute P_a23(231|T=0) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute reconstruct atom 230 under a23 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute P_a23(230|T=0) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute P_a23(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_b23(Ω) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute P_a23(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 (constant, [[0, [[a23, [[230, 2/5], [231, 1/10], [232, 2/5], [233, 1/10]]], [b23, [[230, 1/10], [231, 2/5], [232, 1/10], [233, 2/5]]]], No]], No, [], null, No, Yes, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim) 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: 18783ff4ea38911e8e2f058afb6ea7635cfebcfe7ed8940d58acdb05ac5d2ace - candidate record SHA-256: c33b652df396e1cf1a094ff4cf671f78158392715a369f1c751fccf20cb86eaa - 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-0099.json)