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R0023-12219 · q-20261013-advance-20261026-r05-covariance-0069
Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[1, 6, 1, -6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).
G001 — Use the complete list of distinct integer pairs=[[6, -1], [7, 0], [8, 1], [9, 5]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[1, 6, 1, -6]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).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
joint law total mass := sum(1/8, 1/4, 1/8, 1/2)
After
joint law total mass = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001
Before
original distinct joint pairs=[[6, -1], [7, 0], [8, 1], [9, 5]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]
After
normalized aligned masses=[1/8, 1/4, 1/8, 1/2];nonnegative=Yes;total=1
Conditions: none
Use the complete original joint table on one common probability space,preserving order,pairing and zero masses.Covariance requires nonnegative masses with exact total1.
Rule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S001
Before
original atom0 weighted X := mul(1/8, 6)
After
original atom0 weighted X = 3/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
original atom0 weighted Y := mul(1/8, -1)
After
original atom0 weighted Y = -1/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original atom0 XY := mul(6, -1)
After
original atom0 XY = -6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original atom0 weighted XY := mul(1/8, -6)
After
original atom0 weighted XY = -3/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
original atom1 weighted X := mul(1/4, 7)
After
original atom1 weighted X = 7/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
original atom1 weighted Y := mul(1/4, 0)
After
original atom1 weighted Y = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
original atom1 XY := mul(7, 0)
After
original atom1 XY = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
original atom1 weighted XY := mul(1/4, 0)
After
original atom1 weighted XY = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
original atom2 weighted X := mul(1/8, 8)
After
original atom2 weighted X = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
original atom2 weighted Y := mul(1/8, 1)
After
original atom2 weighted Y = 1/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
original atom2 XY := mul(8, 1)
After
original atom2 XY = 8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
original atom2 weighted XY := mul(1/8, 8)
After
original atom2 weighted XY = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
original atom3 weighted X := mul(1/2, 9)
After
original atom3 weighted X = 9/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
original atom3 weighted Y := mul(1/2, 5)
After
original atom3 weighted Y = 5/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
original atom3 XY := mul(9, 5)
After
original atom3 XY = 45
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
original atom3 weighted XY := mul(1/2, 45)
After
original atom3 weighted XY = 45/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
original E_X := sum(3/4, 7/4, 1, 9/2)
After
original E_X = 8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
original E_Y := sum(-1/8, 0, 1/8, 5/2)
After
original E_Y = 5/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
original E_XY := sum(-3/4, 0, 1, 45/2)
After
original E_XY = 91/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
original product of means := mul(8, 5/2)
After
original product of means = 20
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
original covariance := sub(91/4, 20)
After
original covariance = 11/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
original:E_X=8;E_Y=5/2;E_XY=91/4
After
Cov=E_XY-E_X*E_Y=11/4
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S023
Before
affine atom0 scaled X := mul(1, 6)
After
affine atom0 scaled X = 6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
affine atom0 U := sum(6, 6)
After
affine atom0 U = 12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
affine atom0 scaled Y := mul(1, -1)
After
affine atom0 scaled Y = -1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
affine atom0 V := sum(-1, -6)
After
affine atom0 V = -7
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
affine atom1 scaled X := mul(1, 7)
After
affine atom1 scaled X = 7
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
affine atom1 U := sum(7, 6)
After
affine atom1 U = 13
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
affine atom1 scaled Y := mul(1, 0)
After
affine atom1 scaled Y = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
affine atom1 V := sum(0, -6)
After
affine atom1 V = -6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
affine atom2 scaled X := mul(1, 8)
After
affine atom2 scaled X = 8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
affine atom2 U := sum(8, 6)
After
affine atom2 U = 14
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
affine atom2 scaled Y := mul(1, 1)
After
affine atom2 scaled Y = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
affine atom2 V := sum(1, -6)
After
affine atom2 V = -5
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
affine atom3 scaled X := mul(1, 9)
After
affine atom3 scaled X = 9
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
affine atom3 U := sum(9, 6)
After
affine atom3 U = 15
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
affine atom3 scaled Y := mul(1, 5)
After
affine atom3 scaled Y = 5
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
affine atom3 V := sum(5, -6)
After
affine atom3 V = -1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
affine transformed atom0 weighted X := mul(1/8, 12)
After
affine transformed atom0 weighted X = 3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
affine transformed atom0 weighted Y := mul(1/8, -7)
After
affine transformed atom0 weighted Y = -7/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
affine transformed atom0 XY := mul(12, -7)
After
affine transformed atom0 XY = -84
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
affine transformed atom0 weighted XY := mul(1/8, -84)
After
affine transformed atom0 weighted XY = -21/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
affine transformed atom1 weighted X := mul(1/4, 13)
After
affine transformed atom1 weighted X = 13/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
affine transformed atom1 weighted Y := mul(1/4, -6)
After
affine transformed atom1 weighted Y = -3/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
affine transformed atom1 XY := mul(13, -6)
After
affine transformed atom1 XY = -78
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
affine transformed atom1 weighted XY := mul(1/4, -78)
After
affine transformed atom1 weighted XY = -39/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
affine transformed atom2 weighted X := mul(1/8, 14)
After
affine transformed atom2 weighted X = 7/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
affine transformed atom2 weighted Y := mul(1/8, -5)
After
affine transformed atom2 weighted Y = -5/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
affine transformed atom2 XY := mul(14, -5)
After
affine transformed atom2 XY = -70
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
affine transformed atom2 weighted XY := mul(1/8, -70)
After
affine transformed atom2 weighted XY = -35/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
affine transformed atom3 weighted X := mul(1/2, 15)
After
affine transformed atom3 weighted X = 15/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
affine transformed atom3 weighted Y := mul(1/2, -1)
After
affine transformed atom3 weighted Y = -1/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
affine transformed atom3 XY := mul(15, -1)
After
affine transformed atom3 XY = -15
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
affine transformed atom3 weighted XY := mul(1/2, -15)
After
affine transformed atom3 weighted XY = -15/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
affine transformed E_X := sum(3/2, 13/4, 7/4, 15/2)
After
affine transformed E_X = 14
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
affine transformed E_Y := sum(-7/8, -3/2, -5/8, -1/2)
After
affine transformed E_Y = -7/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
affine transformed E_XY := sum(-21/2, -39/2, -35/4, -15/2)
After
affine transformed E_XY = -185/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
affine transformed product of means := mul(14, -7/2)
After
affine transformed product of means = -49
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
affine transformed covariance := sub(-185/4, -49)
After
affine transformed covariance = 11/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
affine transformed:E_X=14;E_Y=-7/2;E_XY=-185/4
After
Cov=E_XY-E_X*E_Y=11/4
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S061
Before
affine two scale product := mul(1, 1)
After
affine two scale product = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
affine scaled original covariance := mul(1, 11/4)
After
affine scaled original covariance = 11/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
U=(1)*X+(6);V=(1)*Y+(-6);same indexed atoms and weights
After
E_UV-E_U*E_V=(1)*(1)*(E_XY-E_X*E_Y)=11/4
Conditions: none
Expand E[UV]=acE[XY]+adE[X]+bcE[Y]+bd and subtract(aE[X]+b)(cE[Y]+d);all shift terms cancel.Even zero scaling may duplicate image points:retain original probability atoms rather than treating images as a new distinct support.
Rule: affine_covariance_shift_cancellation_not_new_joint_support · Uses: G001, S064
S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S064 · premise_and_result_audit — Recompute affine scaled original covariance with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Recompute affine two scale product with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S061 · premise_and_result_audit — Recompute affine transformed covariance with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute affine transformed product of means with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute affine transformed E_XY with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute affine transformed E_Y with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute affine transformed E_X with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute affine transformed atom3 weighted XY with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute affine transformed atom3 XY with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute affine transformed atom3 weighted Y with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute affine transformed atom3 weighted X with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute affine transformed atom2 weighted XY with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute affine transformed atom2 XY with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute affine transformed atom2 weighted Y with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute affine transformed atom2 weighted X with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute affine transformed atom1 weighted XY with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute affine transformed atom1 XY with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute affine transformed atom1 weighted Y with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute affine transformed atom1 weighted X with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute affine transformed atom0 weighted XY with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute affine transformed atom0 XY with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute affine transformed atom0 weighted Y with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute affine transformed atom0 weighted X with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute affine atom3 V with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute affine atom3 scaled Y with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute affine atom3 U with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute affine atom3 scaled X with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute affine atom2 V with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute affine atom2 scaled Y with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute affine atom2 U with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute affine atom2 scaled X with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute affine atom1 V with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute affine atom1 scaled Y with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute affine atom1 U with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute affine atom1 scaled X with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute affine atom0 V with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute affine atom0 scaled Y with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute affine atom0 U with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute affine atom0 scaled X 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 original covariance with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute original product of means with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute original E_XY with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute original E_Y with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute original E_X with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute original atom3 weighted XY with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute original atom3 XY with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute original atom3 weighted Y with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute original atom3 weighted X with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute original atom2 XY with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute original atom1 XY with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute original atom0 XY with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute original atom0 weighted X with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Recompute joint law total mass with exact rational arithmetic, not rounded decimals.(1, 6, 1, -6, 11/4, 11/4, Cov(aX+b,cY+d)=ac Cov(X,Y))
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: 1976e619a26fcd89882baa8ce76d2280aeb5dd3dd9d6f895435d135aba4a79cccandidate record SHA-256: 9092822256576819d2c058d3e76af508156bcea5518740fd63da976db58ffbafsemantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED