Question 15082: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Original question and givens

Use only the explicitly supplied finite list p_n=2/(n+8) for n=0 through prefix_length-1, with prefix_length=7. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

• G001 — Use only the explicitly supplied finite list p_n=2/(n+8) for n=0 through prefix_length-1, with prefix_length=7. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_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.

• mko-convergence-in-probability

Step-by-step derivation

positive profile parameters c=2;s=8;raw fraction=c/(n+s)

nonnegative valid probability indices start at=0;only there0<profile<=1

finite prefix index0 denominator := sum(0, 8)

finite prefix index0 denominator = 8

finite prefix index0 probability := div(2, 8)

finite prefix index0 probability = 1/4

finite prefix index1 denominator := sum(1, 8)

finite prefix index1 denominator = 9

finite prefix index1 probability := div(2, 9)

finite prefix index1 probability = 2/9

finite prefix index2 denominator := sum(2, 8)

finite prefix index2 denominator = 10

finite prefix index2 probability := div(2, 10)

finite prefix index2 probability = 1/5

finite prefix index3 denominator := sum(3, 8)

finite prefix index3 denominator = 11

finite prefix index3 probability := div(2, 11)

finite prefix index3 probability = 2/11

finite prefix index4 denominator := sum(4, 8)

finite prefix index4 denominator = 12

finite prefix index4 probability := div(2, 12)

finite prefix index4 probability = 1/6

finite prefix index5 denominator := sum(5, 8)

finite prefix index5 denominator = 13

finite prefix index5 probability := div(2, 13)

finite prefix index5 probability = 2/13

finite prefix index6 denominator := sum(6, 8)

finite prefix index6 denominator = 14

finite prefix index6 probability := div(2, 14)

finite prefix index6 probability = 1/7

only indices0 through 6 supplied;finite probability entries=[1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7]

general convergence established=No;no infinite continuation or every-epsilon tail supplied

Reverse checks

• 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 finite prefix index6 probability with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute finite prefix index6 denominator with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite prefix index5 probability with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite prefix index5 denominator with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute finite prefix index4 probability with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute finite prefix index4 denominator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute finite prefix index3 probability with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite prefix index3 denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite prefix index2 probability with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute finite prefix index2 denominator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute finite prefix index1 probability with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute finite prefix index1 denominator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite prefix index0 probability with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite prefix index0 denominator 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

(2, 8, 7, [1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7], No, finite_prefix_not_convergence_certificate)

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.