Use complete normalized finite law with exact masses=[1/15, 2/15, 12/15] and aligned total values X=[0, 10, 12]. 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. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
G001 — Use complete normalized finite law with exact masses=[1/15, 2/15, 12/15] and aligned total values X=[0, 10, 12]. 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. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
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.
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
The prototype flag tests every table entry,whereas almost-sure nonnegativity concerns positive-mass atoms only.The English source-review guard excludes null-only negative ambiguities from non-tail families rather than calling them a theorem-premise failure.The tail itself can still be computed for signed X.
SOURCE qualification repeats its pointwise flag=Yes;pointwise nonnegativity is SUFFICIENT,not NECESSARY in general for the theorem's a.s. premise
Conditions: none
Retain both occurrences of the same source prototype flag rather than changing the answer to a different a.s.test.Expectations in this finite model are finite.The tag names the audited condition;false does not certify that it holds,and null-only ambiguities remain guarded.
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.