Question 20248: A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail

Original question and givens

Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=24, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).

• G001 — Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=24, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).

Definition and scope

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.

• mko-markov-inequality

Step-by-step derivation

source operation=markov-negative-variable-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary]

Markov complete probability mass := sum(1/2, 1/2)

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/2, -23)

Markov original atom0 weighted X = -23/2

Markov original atom1 weighted X := mul(1/2, 25)

Markov original atom1 weighted X = 25/2

Markov complete SAME-law expectation := sum(-23/2, 25/2)

Markov complete SAME-law expectation = 1

ORIGINAL complete nonempty masses=[1/2, 1/2];ALL aligned values X=[-23, 25];full mass=1;ALL weighted X terms=[-23/2, 25/2]

finite exact expectation=1;prototype EVERY-ENTRY nonnegative=No;actual positive-mass nonnegativity=No;null-negative atoms=[];positive-mass negative atoms=[0]

Markov original atom0 threshold-event mass := mul(1/2, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=-23;positive threshold=24;original mass=1/2

event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0

Markov original atom1 threshold-event mass := mul(1/2, 1)

Markov original atom1 threshold-event mass = 1/2

ORIGINAL atom=1;value X=25;positive threshold=24;original mass=1/2

event X>=threshold=Yes;EQUALITY INCLUDED;event mass contribution=1/2

Markov complete exact tail probability := sum(0, 1/2)

Markov complete exact tail probability = 1/2

ALL original threshold-event flags=[No, Yes];ALL law-weighted indicator terms=[0, 1/2]

exact complete-law tail probability=1/2;this event calculation ALONE does not assert Markov applicability

Markov RAW expectation over positive threshold := div(1, 24)

Markov RAW expectation over positive threshold = 1/24

ACTUAL negative X at positive mass exists;original values=[-23, 25];tail=1/2;finite expectation=1;positive threshold=24

SOURCE pointwise applicability=No;raw E[X]/threshold=1/24;actual numeric tail<=raw quotient happens=No;NO general Markov guarantee or reverse-inequality claim

Reverse checks

• 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 Markov RAW expectation over positive threshold 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 Markov complete exact tail probability 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 Markov original atom1 threshold-event mass 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 Markov original atom0 threshold-event mass 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 Markov complete SAME-law expectation with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Markov original atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Markov original atom0 weighted X with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Markov complete probability mass 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.

Result

([-23, 25], No, 1/2, 1/24, Markov_not_applicable_to_signed_variable_without_transformation)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.