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R0027-14080 · q-20261014-advance-20261028-r06-random-variable-independence-0080
Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.
G001 — Use the complete joint grid with original x_values=[-8, 17], y_values=[1, 24], and aligned row-by-column exact masses=[[12/49, 2/49], [30/49, 5/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (independent?,E_X,E_Y,E_XY,covariance,implication_pass,boundary=zero_covariance_converse_not_claimed). Despite the source field name, implication_pass is the conjunction of both verified independence and zero covariance in this protocol, not a truth value of material implication.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 grid model whole mass sum := sum(12/49, 2/49, 30/49, 5/49)
After
joint grid model whole mass sum = 1
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001
Before
joint grid model:raw aligned masses=[[12, 49], [2, 49], [30, 49], [5, 49]]
After
normalized masses=[12/49, 2/49, 30/49, 5/49];nonnegative=Yes;total=1
Conditions: none
First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model.
Rule: whole_finite_probability_law_not_uniform_assumption · Uses: G001, S001
Before
original x row labels=[-8, 17];original y column labels=[1, 24]
After
normalized aligned row-by-column joint grid=[[12/49, 2/49], [30/49, 5/49]]
Conditions: none
Each value list is distinct but need not be numerically sorted.Pair every mass with its original row/column,including zero cells;sorting values must not misalign the grid.
Rule: RV_joint_grid_original_order_not_value_sort · Uses: G001, S002
Before
X row0 marginal := sum(12/49, 2/49)
After
X row0 marginal = 2/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
X row1 marginal := sum(30/49, 5/49)
After
X row1 marginal = 5/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Y column0 marginal := sum(12/49, 30/49)
After
Y column0 marginal = 6/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Y column1 marginal := sum(2/49, 5/49)
After
Y column1 marginal = 1/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
joint cell row0 column0 marginal product := mul(2/7, 6/7)
After
joint cell row0 column0 marginal product = 12/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
original row=0;column=0;value labels=[-8, 1];joint mass=12/49;marginal product=12/49
After
cell factorizes=Yes
Conditions: none
Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order.
Rule: RV_independence_every_joint_cell · Uses: G001, S008
Before
joint cell row0 column1 marginal product := mul(2/7, 1/7)
After
joint cell row0 column1 marginal product = 2/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
original row=0;column=1;value labels=[-8, 24];joint mass=2/49;marginal product=2/49
After
cell factorizes=Yes
Conditions: none
Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order.
Rule: RV_independence_every_joint_cell · Uses: G001, S010
Before
joint cell row1 column0 marginal product := mul(5/7, 6/7)
After
joint cell row1 column0 marginal product = 30/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
original row=1;column=0;value labels=[17, 1];joint mass=30/49;marginal product=30/49
After
cell factorizes=Yes
Conditions: none
Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order.
Rule: RV_independence_every_joint_cell · Uses: G001, S012
Before
joint cell row1 column1 marginal product := mul(5/7, 1/7)
After
joint cell row1 column1 marginal product = 5/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
original row=1;column=1;value labels=[17, 24];joint mass=5/49;marginal product=5/49
After
cell factorizes=Yes
Conditions: none
Audit every cell's singleton-event product,including zeros.One failed cell gives a singleton-preimage counterexample.Retain original row-major failure order.
Rule: RV_independence_every_joint_cell · Uses: G001, S014
Before
all joint factorization failures=[]
After
random variables independent=Yes;all-cell factorization covers every value-event rectangle
Conditions: none
If every cell factorizes,any value-event rectangle sum splits into a product of marginal sums,covering every preimage-event choice.This implies no identical distribution,IID,Markov behavior or larger-family mutual independence.
Rule: RV_all_cells_to_all_event_preimages · Uses: G001, S015
Before
moment row0 col0 weighted X := mul(12/49, -8)
After
moment row0 col0 weighted X = -96/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
moment row0 col0 weighted Y := mul(12/49, 1)
After
moment row0 col0 weighted Y = 12/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
moment row0 col0 XY := mul(-8, 1)
After
moment row0 col0 XY = -8
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
moment row0 col0 weighted XY := mul(12/49, -8)
After
moment row0 col0 weighted XY = -96/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
moment row0 col1 weighted X := mul(2/49, -8)
After
moment row0 col1 weighted X = -16/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
moment row0 col1 weighted Y := mul(2/49, 24)
After
moment row0 col1 weighted Y = 48/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
moment row0 col1 XY := mul(-8, 24)
After
moment row0 col1 XY = -192
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
moment row0 col1 weighted XY := mul(2/49, -192)
After
moment row0 col1 weighted XY = -384/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
moment row1 col0 weighted X := mul(30/49, 17)
After
moment row1 col0 weighted X = 510/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
moment row1 col0 weighted Y := mul(30/49, 1)
After
moment row1 col0 weighted Y = 30/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
moment row1 col0 XY := mul(17, 1)
After
moment row1 col0 XY = 17
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
moment row1 col0 weighted XY := mul(30/49, 17)
After
moment row1 col0 weighted XY = 510/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
moment row1 col1 weighted X := mul(5/49, 17)
After
moment row1 col1 weighted X = 85/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
moment row1 col1 weighted Y := mul(5/49, 24)
After
moment row1 col1 weighted Y = 120/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
moment row1 col1 XY := mul(17, 24)
After
moment row1 col1 XY = 408
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
moment row1 col1 weighted XY := mul(5/49, 408)
After
moment row1 col1 weighted XY = 2040/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
grid E_X := sum(-96/49, -16/49, 510/49, 85/49)
After
grid E_X = 69/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
grid E_Y := sum(12/49, 48/49, 30/49, 120/49)
After
grid E_Y = 30/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
grid E_XY := sum(-96/49, -384/49, 510/49, 2040/49)
After
grid E_XY = 2070/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
grid product of means := mul(69/7, 30/7)
After
grid product of means = 2070/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
grid covariance := sub(2070/49, 2070/49)
After
grid covariance = 0
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
verified independence=Yes;E_X=69/7;E_Y=30/7;E_XY=2070/49;covariance=0
After
source implication_pass is conjunction(independent,covariance0)=Yes,not material implication
Conditions: none
Verified all-cell independence distributes the finite double sum to E[XY]=E[X]E[Y],hence Cov=0.Despite the field name implication_pass,the protocol asks both facts to hold.A dependent zero-covariance model still returns No;this neither refutes the theorem nor proves the converse.
Rule: RV_zero_covariance_conjunction_not_converse · Uses: G001, S037
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Recompute grid covariance with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute grid product of means with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute grid E_XY with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute grid E_Y with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute grid E_X with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute moment row1 col1 weighted XY with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute moment row1 col1 XY with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute moment row1 col1 weighted Y with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute moment row1 col1 weighted X with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute moment row1 col0 weighted XY with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute moment row1 col0 XY with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute moment row1 col0 weighted Y with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute moment row1 col0 weighted X with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute moment row0 col1 weighted XY with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute moment row0 col1 XY with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute moment row0 col1 weighted Y with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute moment row0 col1 weighted X with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute moment row0 col0 weighted XY with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute moment row0 col0 XY with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute moment row0 col0 weighted Y with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute moment row0 col0 weighted X with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute joint cell row1 column1 marginal product with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute joint cell row1 column0 marginal product with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute joint cell row0 column1 marginal product with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute joint cell row0 column0 marginal product with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Y column1 marginal with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute X row1 marginal with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute X row0 marginal 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 — 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 grid model whole mass sum with exact rational arithmetic, not rounded decimals.(Yes, 69/7, 30/7, 2070/49, 0, Yes, zero_covariance_converse_not_claimed)
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.
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