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R0023-12284 · q-20261013-advance-20261026-r05-covariance-0084
Take X uniformly on [−scale,0,scale] for nonzero positive scale=9, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
G001 — Take X uniformly on [−scale,0,scale] for nonzero positive scale=9, and Y=X² on that same finite space. Give (scale,covariance,joint_zero=P(X=0,Y=0),product_zero=P(X=0)P(Y=0),zero_covariance?,dependent?,zero-covariance-but-dependent). Use exact reduced rational outputs with positive denominators; omit /1 for integers.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
zero covariance scale squared := mul(9, 9)
After
zero covariance scale squared = 81
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
joint law total mass := sum(1/3, 1/3, 1/3)
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, S001
Before
original distinct joint pairs=[[-9, 81], [0, 0], [9, 81]];raw aligned masses=[[1, 3], [1, 3], [1, 3]]
After
normalized aligned masses=[1/3, 1/3, 1/3];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, S002
Before
original atom0 weighted X := mul(1/3, -9)
After
original atom0 weighted X = -3
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 weighted Y := mul(1/3, 81)
After
original atom0 weighted Y = 27
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 XY := mul(-9, 81)
After
original atom0 XY = -729
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 atom0 weighted XY := mul(1/3, -729)
After
original atom0 weighted XY = -243
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 X := mul(1/3, 0)
After
original atom1 weighted X = 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 weighted Y := mul(1/3, 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, S008
Before
original atom1 XY := mul(0, 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, S009
Before
original atom1 weighted XY := mul(1/3, 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, S010
Before
original atom2 weighted X := mul(1/3, 9)
After
original atom2 weighted X = 3
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 weighted Y := mul(1/3, 81)
After
original atom2 weighted Y = 27
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 XY := mul(9, 81)
After
original atom2 XY = 729
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 atom2 weighted XY := mul(1/3, 729)
After
original atom2 weighted XY = 243
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 E_X := sum(-3, 0, 3)
After
original E_X = 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, S015
Before
original E_Y := sum(27, 0, 27)
After
original E_Y = 54
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 E_XY := sum(-243, 0, 243)
After
original E_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, S017
Before
original product of means := mul(0, 54)
After
original product of means = 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, S018
Before
original covariance := sub(0, 0)
After
original covariance = 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, S019
Before
original:E_X=0;E_Y=54;E_XY=0
After
Cov=E_XY-E_X*E_Y=0
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, S020
Before
joint zero event mass := sum(1/3)
After
joint zero event mass = 1/3
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
X zero marginal mass := sum(1/3)
After
X zero marginal mass = 1/3
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
Y zero marginal mass := sum(1/3)
After
Y zero marginal mass = 1/3
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, S023
Before
zero event product of marginal masses := mul(1/3, 1/3)
After
zero event product of marginal masses = 1/9
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, S024
Before
X uniform on [-9, 0, 9];Y=X^2;Cov=0;P(X=0,Y=0)=1/3
After
P(X=0)*P(Y=0)=1/9;different=Yes
Conditions: none
Symmetry makes E[X] and E[XY] zero,hence zero covariance.Independence requires factorization for every event pair;the zero events with1/3≠1/9 give one exact refutation.
Rule: zero_covariance_explicit_independence_counterevent · Uses: G001, S025
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Recompute zero event product of marginal masses with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute Y zero marginal mass with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute X zero marginal mass with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute joint zero event mass with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Recompute original covariance with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute original product of means with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute original E_XY with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute original E_Y with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute original E_X with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute original atom2 XY with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute original atom1 XY with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute original atom0 XY with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute original atom0 weighted X with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute joint law total mass with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute zero covariance scale squared with exact rational arithmetic, not rounded decimals.(9, 0, 1/3, 1/9, Yes, Yes, zero-covariance-but-dependent)
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: 8ef3790e3a5893336cead7fae66d01fcfc74763e2a7814738eb77fea7a4f863bcandidate record SHA-256: 03dcccdaa4e007a65d48f1238812b498b8a1e4dc75fd319908706072c7fd6a0asemantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED