# Question 25103: Each Theta Has Its Own Complete Sampling Law and Risk [Return to catalogue page 126](https://ocme.evemisslab.com/en/questions/page/126/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations/#q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028) 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. R0049-25103 · q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028 ## Original question and givens Use complete laws=[[-10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]], [-9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate). - G001 — Use complete laws=[[-10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]], [-9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate). ## Definition and scope 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. - [mko-estimator-mean-squared-error](https://ocme.evemisslab.com/en/mko/mko-estimator-mean-squared-error/) ## Step-by-step derivation S001 · impliesBefore source roles = [mse-multi-theta-table] After derived = [[law_risk_rows, certificate]] Conditions: none List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore MSE ORIGINAL law index0 ORIGINAL probability cumulative0 := sum(0, 1/4) After MSE ORIGINAL law index0 ORIGINAL probability cumulative0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore MSE ORIGINAL law index0 ORIGINAL probability cumulative1 := sum(1/4, 1/2) After MSE ORIGINAL law index0 ORIGINAL probability cumulative1 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore MSE ORIGINAL law index0 ORIGINAL probability cumulative2 := sum(3/4, 1/4) After MSE ORIGINAL law index0 ORIGINAL probability cumulative2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [MSE ORIGINAL law index0, -10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]] After derived = [[[-13, 1/4], [-10, 1/2], [-7, 1/4]], 1] Conditions: none Fix this same P_theta.Check original-order nonnegative masses summing to1,then sort values numerically with their aligned masses.Keep null rows,without renormalizing or treating estimator values as parameter indices. Step identifiers and dependenciesRule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S004 S006 · calculationBefore MSE ORIGINAL law index0 exact mean product0 := mul(-13, 1/4) After MSE ORIGINAL law index0 exact mean product0 = -13/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore MSE ORIGINAL law index0 exact mean cumulative0 := sum(0, -13/4) After MSE ORIGINAL law index0 exact mean cumulative0 = -13/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore MSE ORIGINAL law index0 exact mean product1 := mul(-10, 1/2) After MSE ORIGINAL law index0 exact mean product1 = -5 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore MSE ORIGINAL law index0 exact mean cumulative1 := sum(-13/4, -5) After MSE ORIGINAL law index0 exact mean cumulative1 = -33/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore MSE ORIGINAL law index0 exact mean product2 := mul(-7, 1/4) After MSE ORIGINAL law index0 exact mean product2 = -7/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore MSE ORIGINAL law index0 exact mean cumulative2 := sum(-33/4, -7/4) After MSE ORIGINAL law index0 exact mean cumulative2 = -10 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore MSE ORIGINAL law index0 error row0 := sub(-13, -10) After MSE ORIGINAL law index0 error row0 = -3 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore MSE ORIGINAL law index0 squared error row0 := mul(-3, -3) After MSE ORIGINAL law index0 squared error row0 = 9 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore MSE ORIGINAL law index0 weighted error row0 := mul(1/4, 9) After MSE ORIGINAL law index0 weighted error row0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore MSE ORIGINAL law index0 MSE cumulative0 := sum(0, 9/4) After MSE ORIGINAL law index0 MSE cumulative0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore MSE ORIGINAL law index0 error row1 := sub(-10, -10) After MSE ORIGINAL law index0 error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore MSE ORIGINAL law index0 squared error row1 := mul(0, 0) After MSE ORIGINAL law index0 squared error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore MSE ORIGINAL law index0 weighted error row1 := mul(1/2, 0) After MSE ORIGINAL law index0 weighted error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore MSE ORIGINAL law index0 MSE cumulative1 := sum(9/4, 0) After MSE ORIGINAL law index0 MSE cumulative1 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore MSE ORIGINAL law index0 error row2 := sub(-7, -10) After MSE ORIGINAL law index0 error row2 = 3 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore MSE ORIGINAL law index0 squared error row2 := mul(3, 3) After MSE ORIGINAL law index0 squared error row2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore MSE ORIGINAL law index0 weighted error row2 := mul(1/4, 9) After MSE ORIGINAL law index0 weighted error row2 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore MSE ORIGINAL law index0 MSE cumulative2 := sum(9/4, 9/4) After MSE ORIGINAL law index0 MSE cumulative2 = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore source roles = [MSE ORIGINAL law index0, -10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]]] After derived = [-10, 9/2, No, No, Yes] Conditions: none MSE directly sums all weighted squared errors,without a bias-variance decomposition.Nonnegative terms sum to0 iff the error vanishes at every positive-mass value;null values may differ.Both sides False still give a True iff. Step identifiers and dependenciesRule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S023 S025 · calculationBefore MSE ORIGINAL law index1 ORIGINAL probability cumulative0 := sum(0, 1/4) After MSE ORIGINAL law index1 ORIGINAL probability cumulative0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore MSE ORIGINAL law index1 ORIGINAL probability cumulative1 := sum(1/4, 1/2) After MSE ORIGINAL law index1 ORIGINAL probability cumulative1 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore MSE ORIGINAL law index1 ORIGINAL probability cumulative2 := sum(3/4, 1/4) After MSE ORIGINAL law index1 ORIGINAL probability cumulative2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore source roles = [MSE ORIGINAL law index1, -9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]] After derived = [[[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1] Conditions: none Fix this same P_theta.Check original-order nonnegative masses summing to1,then sort values numerically with their aligned masses.Keep null rows,without renormalizing or treating estimator values as parameter indices. Step identifiers and dependenciesRule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S027 S029 · calculationBefore MSE ORIGINAL law index1 exact mean product0 := mul(-10, 1/4) After MSE ORIGINAL law index1 exact mean product0 = -5/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore MSE ORIGINAL law index1 exact mean cumulative0 := sum(0, -5/2) After MSE ORIGINAL law index1 exact mean cumulative0 = -5/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore MSE ORIGINAL law index1 exact mean product1 := mul(-9, 1/2) After MSE ORIGINAL law index1 exact mean product1 = -9/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore MSE ORIGINAL law index1 exact mean cumulative1 := sum(-5/2, -9/2) After MSE ORIGINAL law index1 exact mean cumulative1 = -7 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore MSE ORIGINAL law index1 exact mean product2 := mul(-8, 1/4) After MSE ORIGINAL law index1 exact mean product2 = -2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore MSE ORIGINAL law index1 exact mean cumulative2 := sum(-7, -2) After MSE ORIGINAL law index1 exact mean cumulative2 = -9 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore MSE ORIGINAL law index1 error row0 := sub(-10, -9) After MSE ORIGINAL law index1 error row0 = -1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore MSE ORIGINAL law index1 squared error row0 := mul(-1, -1) After MSE ORIGINAL law index1 squared error row0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore MSE ORIGINAL law index1 weighted error row0 := mul(1/4, 1) After MSE ORIGINAL law index1 weighted error row0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore MSE ORIGINAL law index1 MSE cumulative0 := sum(0, 1/4) After MSE ORIGINAL law index1 MSE cumulative0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore MSE ORIGINAL law index1 error row1 := sub(-9, -9) After MSE ORIGINAL law index1 error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore MSE ORIGINAL law index1 squared error row1 := mul(0, 0) After MSE ORIGINAL law index1 squared error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore MSE ORIGINAL law index1 weighted error row1 := mul(1/2, 0) After MSE ORIGINAL law index1 weighted error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore MSE ORIGINAL law index1 MSE cumulative1 := sum(1/4, 0) After MSE ORIGINAL law index1 MSE cumulative1 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore MSE ORIGINAL law index1 error row2 := sub(-8, -9) After MSE ORIGINAL law index1 error row2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore MSE ORIGINAL law index1 squared error row2 := mul(1, 1) After MSE ORIGINAL law index1 squared error row2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore MSE ORIGINAL law index1 weighted error row2 := mul(1/4, 1) After MSE ORIGINAL law index1 weighted error row2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore MSE ORIGINAL law index1 MSE cumulative2 := sum(1/4, 1/4) After MSE ORIGINAL law index1 MSE cumulative2 = 1/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore source roles = [MSE ORIGINAL law index1, -9, [[-10, 1/4, 1, 1/4], [-9, 1/2, 0, 0], [-8, 1/4, 1, 1/4]]] After derived = [-9, 1/2, No, No, Yes] Conditions: none MSE directly sums all weighted squared errors,without a bias-variance decomposition.Nonnegative terms sum to0 iff the error vanishes at every positive-mass value;null values may differ.Both sides False still give a True iff. Step identifiers and dependenciesRule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S046 S048 · impliesBefore source roles = [[[[-10, 1], [[[-13, 1], [1, 4]], [[-10, 1], [1, 2]], [[-7, 1], [1, 4]]]], [[-9, 1], [[[-10, 1], [1, 4]], [[-9, 1], [1, 2]], [[-8, 1], [1, 4]]]]]] After derived = [[0, 1], [[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]]] Conditions: none Normalize and compute MSE under each separate theta law.Family order follows the source English canonical key of normalized RAT[theta,rows],not numeric theta or display strings.Without a prior,do not pool or average risks. Step identifiers and dependenciesRule: MSE_multi_theta_RAW_RAT_English_key_NOT_numeric_theta_or_display_key · Uses: G001, S047 ## Reverse checks - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 MSE cumulative2 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 weighted error row2 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 squared error row2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 error row2 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 MSE cumulative1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 weighted error row1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 squared error row1 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 error row1 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 MSE cumulative0 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 weighted error row0 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 squared error row0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 error row0 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean cumulative2 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean product2 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean cumulative1 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean product1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean cumulative0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 exact mean product0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 ORIGINAL probability cumulative2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute MSE ORIGINAL law index1 ORIGINAL probability cumulative0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 MSE cumulative2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 weighted error row2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 squared error row2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 error row2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 MSE cumulative1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 weighted error row1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 squared error row1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 error row1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 MSE cumulative0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 weighted error row0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 squared error row0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 error row0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean cumulative2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean product2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean cumulative1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean product1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean cumulative0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 exact mean product0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 ORIGINAL probability cumulative2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute MSE ORIGINAL law index0 ORIGINAL probability cumulative0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## 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: e92534af1843f4fc9a091bcb86dc531042afbc2732d13776ea6f462c5ecda1e1 - candidate record SHA-256: f0a97f51e5d5c37eda76f9d90e249365f215fad6089b10726643885337bafc27 - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028.json)