Question 15094: 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=4/(n+6) for n=0 through prefix_length-1, with prefix_length=19. 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=4/(n+6) for n=0 through prefix_length-1, with prefix_length=19. 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=4;s=6;raw fraction=c/(n+s)

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

finite prefix index0 denominator := sum(0, 6)

finite prefix index0 denominator = 6

finite prefix index0 probability := div(4, 6)

finite prefix index0 probability = 2/3

finite prefix index1 denominator := sum(1, 6)

finite prefix index1 denominator = 7

finite prefix index1 probability := div(4, 7)

finite prefix index1 probability = 4/7

finite prefix index2 denominator := sum(2, 6)

finite prefix index2 denominator = 8

finite prefix index2 probability := div(4, 8)

finite prefix index2 probability = 1/2

finite prefix index3 denominator := sum(3, 6)

finite prefix index3 denominator = 9

finite prefix index3 probability := div(4, 9)

finite prefix index3 probability = 4/9

finite prefix index4 denominator := sum(4, 6)

finite prefix index4 denominator = 10

finite prefix index4 probability := div(4, 10)

finite prefix index4 probability = 2/5

finite prefix index5 denominator := sum(5, 6)

finite prefix index5 denominator = 11

finite prefix index5 probability := div(4, 11)

finite prefix index5 probability = 4/11

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

finite prefix index6 denominator = 12

finite prefix index6 probability := div(4, 12)

finite prefix index6 probability = 1/3

finite prefix index7 denominator := sum(7, 6)

finite prefix index7 denominator = 13

finite prefix index7 probability := div(4, 13)

finite prefix index7 probability = 4/13

finite prefix index8 denominator := sum(8, 6)

finite prefix index8 denominator = 14

finite prefix index8 probability := div(4, 14)

finite prefix index8 probability = 2/7

finite prefix index9 denominator := sum(9, 6)

finite prefix index9 denominator = 15

finite prefix index9 probability := div(4, 15)

finite prefix index9 probability = 4/15

finite prefix index10 denominator := sum(10, 6)

finite prefix index10 denominator = 16

finite prefix index10 probability := div(4, 16)

finite prefix index10 probability = 1/4

finite prefix index11 denominator := sum(11, 6)

finite prefix index11 denominator = 17

finite prefix index11 probability := div(4, 17)

finite prefix index11 probability = 4/17

finite prefix index12 denominator := sum(12, 6)

finite prefix index12 denominator = 18

finite prefix index12 probability := div(4, 18)

finite prefix index12 probability = 2/9

finite prefix index13 denominator := sum(13, 6)

finite prefix index13 denominator = 19

finite prefix index13 probability := div(4, 19)

finite prefix index13 probability = 4/19

finite prefix index14 denominator := sum(14, 6)

finite prefix index14 denominator = 20

finite prefix index14 probability := div(4, 20)

finite prefix index14 probability = 1/5

finite prefix index15 denominator := sum(15, 6)

finite prefix index15 denominator = 21

finite prefix index15 probability := div(4, 21)

finite prefix index15 probability = 4/21

finite prefix index16 denominator := sum(16, 6)

finite prefix index16 denominator = 22

finite prefix index16 probability := div(4, 22)

finite prefix index16 probability = 2/11

finite prefix index17 denominator := sum(17, 6)

finite prefix index17 denominator = 23

finite prefix index17 probability := div(4, 23)

finite prefix index17 probability = 4/23

finite prefix index18 denominator := sum(18, 6)

finite prefix index18 denominator = 24

finite prefix index18 probability := div(4, 24)

finite prefix index18 probability = 1/6

only indices0 through 18 supplied;finite probability entries=[2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6]

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

Reverse checks

• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute finite prefix index18 probability with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix index18 denominator with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix index17 probability with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute finite prefix index17 denominator with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute finite prefix index16 probability with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute finite prefix index16 denominator with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute finite prefix index15 probability with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute finite prefix index15 denominator with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute finite prefix index14 probability with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute finite prefix index14 denominator with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix index13 probability with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix index13 denominator with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix index12 probability with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute finite prefix index12 denominator with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute finite prefix index11 probability with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute finite prefix index11 denominator with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite prefix index10 probability with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite prefix index10 denominator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute finite prefix index9 probability with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute finite prefix index9 denominator with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute finite prefix index8 probability with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite prefix index8 denominator with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite prefix index7 probability with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute finite prefix index7 denominator with exact rational arithmetic, not rounded decimals.
• 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

(4, 6, 19, [2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6], 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.