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R0049-25181 · q-20261015-advance-20261114-r05-estimator-mean-squared-error-0056
Use scalar theta=-7, complete listed estimator values=[-7, -6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
G001 — Use scalar theta=-7, complete listed estimator values=[-7, -6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).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.
Before
source roles = [mse-zero-iff-almost-sure]
After
derived = [[theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, 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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1)
After
MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1, 0)
After
MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [MSE ORIGINAL single theta, -7, [-7, -6], [1, 0]]
After
derived = [[[-7, 1], [-6, 0]], 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.
Rule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S003
Before
MSE ORIGINAL single theta exact mean product0 := mul(-7, 1)
After
MSE ORIGINAL single theta exact mean product0 = -7
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
MSE ORIGINAL single theta exact mean cumulative0 := sum(0, -7)
After
MSE ORIGINAL single theta exact mean cumulative0 = -7
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
MSE ORIGINAL single theta exact mean product1 := mul(-6, 0)
After
MSE ORIGINAL single theta exact mean product1 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
MSE ORIGINAL single theta exact mean cumulative1 := sum(-7, 0)
After
MSE ORIGINAL single theta exact mean cumulative1 = -7
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
MSE ORIGINAL single theta error row0 := sub(-7, -7)
After
MSE ORIGINAL single theta error row0 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
MSE ORIGINAL single theta squared error row0 := mul(0, 0)
After
MSE ORIGINAL single theta squared error row0 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
MSE ORIGINAL single theta weighted error row0 := mul(1, 0)
After
MSE ORIGINAL single theta weighted error row0 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
MSE ORIGINAL single theta MSE cumulative0 := sum(0, 0)
After
MSE ORIGINAL single theta MSE cumulative0 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
MSE ORIGINAL single theta error row1 := sub(-6, -7)
After
MSE ORIGINAL single theta error row1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
MSE ORIGINAL single theta squared error row1 := mul(1, 1)
After
MSE ORIGINAL single theta squared error row1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
MSE ORIGINAL single theta weighted error row1 := mul(0, 1)
After
MSE ORIGINAL single theta weighted error row1 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
MSE ORIGINAL single theta MSE cumulative1 := sum(0, 0)
After
MSE ORIGINAL single theta MSE cumulative1 = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [MSE ORIGINAL single theta, -7, [[-7, 1, 0, 0], [-6, 0, 1, 0]]]
After
derived = [-7, 0, Yes, Yes, 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.
Rule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S016
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row1 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row0 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row0 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute MSE ORIGINAL single theta 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.(-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: d74b860cebdcdafed75435137d1d2c9c5219fa6a3aeec06bc5b60427ae6b99ffcandidate record SHA-256: 593615a75ed9fb20890ac4c1266b539db4154c1621bd4f8207e5b2f83ea61dbesemantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED