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Question 25116: Each Theta Has Its Own Complete Sampling Law and Risk

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0049-25116 · q-20261015-advance-20261114-r05-estimator-mean-squared-error-0041

Original question and givens

Use complete laws=[[3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]], [4, [3, 4, 5], [[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=[[3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]], [4, [3, 4, 5], [[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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [MSE ORIGINAL law index0, 3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]]

After

derived = [[[2, 1/4], [3, 1/2], [4, 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 dependencies

Rule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S004

S006 · calculation

Before

MSE ORIGINAL law index0 exact mean product0 := mul(2, 1/4)

After

MSE ORIGINAL law index0 exact mean product0 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

MSE ORIGINAL law index0 exact mean cumulative0 := sum(0, 1/2)

After

MSE ORIGINAL law index0 exact mean cumulative0 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

MSE ORIGINAL law index0 exact mean product1 := mul(3, 1/2)

After

MSE ORIGINAL law index0 exact mean product1 = 3/2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

MSE ORIGINAL law index0 exact mean cumulative1 := sum(1/2, 3/2)

After

MSE ORIGINAL law index0 exact mean cumulative1 = 2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

MSE ORIGINAL law index0 exact mean product2 := mul(4, 1/4)

After

MSE ORIGINAL law index0 exact mean product2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

MSE ORIGINAL law index0 exact mean cumulative2 := sum(2, 1)

After

MSE ORIGINAL law index0 exact mean cumulative2 = 3

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

MSE ORIGINAL law index0 error row0 := sub(2, 3)

After

MSE ORIGINAL law index0 error row0 = -1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

MSE ORIGINAL law index0 squared error row0 := mul(-1, -1)

After

MSE ORIGINAL law index0 squared error row0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

MSE ORIGINAL law index0 weighted error row0 := mul(1/4, 1)

After

MSE ORIGINAL law index0 weighted error row0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

MSE ORIGINAL law index0 MSE cumulative0 := sum(0, 1/4)

After

MSE ORIGINAL law index0 MSE cumulative0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

MSE ORIGINAL law index0 error row1 := sub(3, 3)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

MSE ORIGINAL law index0 MSE cumulative1 := sum(1/4, 0)

After

MSE ORIGINAL law index0 MSE cumulative1 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

MSE ORIGINAL law index0 error row2 := sub(4, 3)

After

MSE ORIGINAL law index0 error row2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

MSE ORIGINAL law index0 squared error row2 := mul(1, 1)

After

MSE ORIGINAL law index0 squared error row2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

MSE ORIGINAL law index0 weighted error row2 := mul(1/4, 1)

After

MSE ORIGINAL law index0 weighted error row2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

MSE ORIGINAL law index0 MSE cumulative2 := sum(1/4, 1/4)

After

MSE ORIGINAL law index0 MSE cumulative2 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [MSE ORIGINAL law index0, 3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]]]

After

derived = [3, 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 dependencies

Rule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [MSE ORIGINAL law index1, 4, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]

After

derived = [[[3, 1/4], [4, 1/2], [5, 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 dependencies

Rule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S027

S029 · calculation

Before

MSE ORIGINAL law index1 exact mean product0 := mul(3, 1/4)

After

MSE ORIGINAL law index1 exact mean product0 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

MSE ORIGINAL law index1 exact mean cumulative0 := sum(0, 3/4)

After

MSE ORIGINAL law index1 exact mean cumulative0 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

MSE ORIGINAL law index1 exact mean product1 := mul(4, 1/2)

After

MSE ORIGINAL law index1 exact mean product1 = 2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

MSE ORIGINAL law index1 exact mean cumulative1 := sum(3/4, 2)

After

MSE ORIGINAL law index1 exact mean cumulative1 = 11/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

MSE ORIGINAL law index1 exact mean product2 := mul(5, 1/4)

After

MSE ORIGINAL law index1 exact mean product2 = 5/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

MSE ORIGINAL law index1 exact mean cumulative2 := sum(11/4, 5/4)

After

MSE ORIGINAL law index1 exact mean cumulative2 = 4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

MSE ORIGINAL law index1 error row0 := sub(3, 4)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

MSE ORIGINAL law index1 error row1 := sub(4, 4)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

MSE ORIGINAL law index1 error row2 := sub(5, 4)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

source roles = [MSE ORIGINAL law index1, 4, [[3, 1/4, 1, 1/4], [4, 1/2, 0, 0], [5, 1/4, 1, 1/4]]]

After

derived = [4, 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 dependencies

Rule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S046

S048 · implies

Before

source roles = [[[[3, 1], [[[2, 1], [1, 4]], [[3, 1], [1, 2]], [[4, 1], [1, 4]]]], [[4, 1], [[[3, 1], [1, 4]], [[4, 1], [1, 2]], [[5, 1], [1, 4]]]]]]

After

derived = [[0, 1], [[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 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 dependencies

Rule: 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

([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 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: 25a9758acc95a72bb2aed226553306f39975d152828733c119f0dcfa71820952
  • candidate record SHA-256: e2f5a11fc28682aadf014e80ab948eea4ee583aeae6313d6a133f7ea71e588be
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON