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R0024-12518 · q-20261013-advance-20261026-r04-finite-joint-distribution-0043
Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.
G001 — Use the complete list of distinct integer pairs=[[5, -1], [6, 0], [7, 1], [8, 4]] 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. The joint table must qualify first. Give (x_marginal,y_marginal), each as [label,mass] rows sorted numerically by label, retaining listed labels even when their aggregated mass is zero.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 exact total := sum(1/8, 1/4, 1/8, 1/2)
After
joint law exact total = 1
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001
Before
original ordered support=[[5, -1], [6, 0], [7, 1], [8, 4]];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;qualified=Yes
Conditions: none
Keep the complete joint table's original pair-mass alignment with distinct support pairs.Zero masses are allowed.Check nonnegativity and exact unit total separately;do not substitute marginals or a uniform law.
Rule: whole_joint_table_not_marginal_substitution · Uses: G001, S001
Before
marginal coordinate0 label5 := sum(1/8)
After
marginal coordinate0 label5 = 1/8
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
coordinate=0;label=5;original support indices=[0]
After
aggregated marginal mass=1/8;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S003
Before
marginal coordinate0 label6 := sum(1/4)
After
marginal coordinate0 label6 = 1/4
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
coordinate=0;label=6;original support indices=[1]
After
aggregated marginal mass=1/4;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S005
Before
marginal coordinate0 label7 := sum(1/8)
After
marginal coordinate0 label7 = 1/8
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
coordinate=0;label=7;original support indices=[2]
After
aggregated marginal mass=1/8;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S007
Before
marginal coordinate0 label8 := sum(1/2)
After
marginal coordinate0 label8 = 1/2
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
coordinate=0;label=8;original support indices=[3]
After
aggregated marginal mass=1/2;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S009
Before
marginal coordinate1 label-1 := sum(1/8)
After
marginal coordinate1 label-1 = 1/8
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
coordinate=1;label=-1;original support indices=[0]
After
aggregated marginal mass=1/8;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S011
Before
marginal coordinate1 label0 := sum(1/4)
After
marginal coordinate1 label0 = 1/4
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
coordinate=1;label=0;original support indices=[1]
After
aggregated marginal mass=1/4;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S013
Before
marginal coordinate1 label1 := sum(1/8)
After
marginal coordinate1 label1 = 1/8
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
coordinate=1;label=1;original support indices=[2]
After
aggregated marginal mass=1/8;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S015
Before
marginal coordinate1 label4 := sum(1/2)
After
marginal coordinate1 label4 = 1/2
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
coordinate=1;label=4;original support indices=[3]
After
aggregated marginal mass=1/2;listed label retained even at mass0
Conditions: none
Aggregate equal-coordinate original atoms,then sort labels numerically.Retain every listed label even at zero aggregated mass;do not multiply marginals to invent a joint law.
Rule: joint_marginal_all_original_atoms · Uses: G001, S017
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute marginal coordinate1 label4 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 — Recompute marginal coordinate1 label1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute marginal coordinate1 label0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute marginal coordinate1 label-1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute marginal coordinate0 label8 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute marginal coordinate0 label7 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute marginal coordinate0 label6 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 marginal coordinate0 label5 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 exact total with exact rational arithmetic, not rounded decimals.([[5, 1/8], [6, 1/4], [7, 1/8], [8, 1/2]], [[-1, 1/8], [0, 1/4], [1, 1/8], [4, 1/2]])
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: 517e4fa6e9366790f3992462370dbe73c4a52d7c21b508192b6cb712f5124d58candidate record SHA-256: e6c3034776d306b1550e583a9ecbe57f94729ae76f5f338476c7d4e3a6ec4b36semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8Lean: NOT_RUN · human review: NOT_PERFORMED