# Question 15096: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0029-15096 · q-20261014-advance-20261030-r07-convergence-in-probability-0071



## Original question and givens

Use only the explicitly supplied finite list p_n=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. 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=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. 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](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/)





## Step-by-step derivation

S001 · impliesBefore

positive profile parameters c=1;s=8;raw fraction=c/(n+s)

After

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

Conditions: none

A positive denominator and value≤1 require n+s≥c.Keep the original index;do not clip values above1 or silently shift them.This checks arithmetic domain only.The infinite-process/common-space interpretation remains declared,and the prefix-only family gains no infinite continuation premise.

Step identifiers and dependenciesRule: deviation_profile_probability_domain_not_clipping · Uses: G001





S002 · calculationBefore

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

After

finite prefix index0 denominator = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

finite prefix index0 probability := div(1, 8)

After

finite prefix index0 probability = 1/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

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

After

finite prefix index1 denominator = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

finite prefix index1 probability := div(1, 9)

After

finite prefix index1 probability = 1/9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

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

After

finite prefix index2 denominator = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

finite prefix index2 probability := div(1, 10)

After

finite prefix index2 probability = 1/10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

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

After

finite prefix index3 denominator = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

finite prefix index3 probability := div(1, 11)

After

finite prefix index3 probability = 1/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

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

After

finite prefix index4 denominator = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

finite prefix index4 probability := div(1, 12)

After

finite prefix index4 probability = 1/12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

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

After

finite prefix index5 denominator = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

finite prefix index5 probability := div(1, 13)

After

finite prefix index5 probability = 1/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

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

After

finite prefix index6 denominator = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

finite prefix index6 probability := div(1, 14)

After

finite prefix index6 probability = 1/14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

finite prefix index7 denominator := sum(7, 8)

After

finite prefix index7 denominator = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

finite prefix index7 probability := div(1, 15)

After

finite prefix index7 probability = 1/15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

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

After

finite prefix index8 denominator = 16

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

finite prefix index8 probability := div(1, 16)

After

finite prefix index8 probability = 1/16

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

finite prefix index9 denominator := sum(9, 8)

After

finite prefix index9 denominator = 17

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

finite prefix index9 probability := div(1, 17)

After

finite prefix index9 probability = 1/17

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

finite prefix index10 denominator := sum(10, 8)

After

finite prefix index10 denominator = 18

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

finite prefix index10 probability := div(1, 18)

After

finite prefix index10 probability = 1/18

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

finite prefix index11 denominator := sum(11, 8)

After

finite prefix index11 denominator = 19

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

finite prefix index11 probability := div(1, 19)

After

finite prefix index11 probability = 1/19

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

finite prefix index12 denominator := sum(12, 8)

After

finite prefix index12 denominator = 20

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · calculationBefore

finite prefix index12 probability := div(1, 20)

After

finite prefix index12 probability = 1/20

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

finite prefix index13 denominator := sum(13, 8)

After

finite prefix index13 denominator = 21

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

finite prefix index13 probability := div(1, 21)

After

finite prefix index13 probability = 1/21

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

finite prefix index14 denominator := sum(14, 8)

After

finite prefix index14 denominator = 22

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

finite prefix index14 probability := div(1, 22)

After

finite prefix index14 probability = 1/22

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

finite prefix index15 denominator := sum(15, 8)

After

finite prefix index15 denominator = 23

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

finite prefix index15 probability := div(1, 23)

After

finite prefix index15 probability = 1/23

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · calculationBefore

finite prefix index16 denominator := sum(16, 8)

After

finite prefix index16 denominator = 24

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

finite prefix index16 probability := div(1, 24)

After

finite prefix index16 probability = 1/24

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

finite prefix index17 denominator := sum(17, 8)

After

finite prefix index17 denominator = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

finite prefix index17 probability := div(1, 25)

After

finite prefix index17 probability = 1/25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · calculationBefore

finite prefix index18 denominator := sum(18, 8)

After

finite prefix index18 denominator = 26

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037





S039 · calculationBefore

finite prefix index18 probability := div(1, 26)

After

finite prefix index18 probability = 1/26

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

finite prefix index19 denominator := sum(19, 8)

After

finite prefix index19 denominator = 27

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

finite prefix index19 probability := div(1, 27)

After

finite prefix index19 probability = 1/27

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · calculationBefore

finite prefix index20 denominator := sum(20, 8)

After

finite prefix index20 denominator = 28

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · calculationBefore

finite prefix index20 probability := div(1, 28)

After

finite prefix index20 probability = 1/28

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

only indices0 through 20 supplied;finite probability entries=[1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28]

After

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

Conditions: none

List the original zero-indexed prefix without dropping an invalid first entry.Even correct decreasing entries supply no every-epsilon/all-later obligation;an empty prefix is not a certificate either.

Step identifiers and dependenciesRule: finite_probability_prefix_not_infinite_process_certificate · Uses: G001, S043









## Reverse checks


- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Recompute finite prefix index20 probability with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Recompute finite prefix index20 denominator with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Recompute finite prefix index19 probability with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute finite prefix index19 denominator with exact rational arithmetic, not rounded decimals.
- 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

(1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 8817de706d3ca95904acb8f2ab1c09e4f4e92b70ce961fa79367f03637cb31a9
- candidate record SHA-256: 167c8854734634b6f0bedd049d9603e4e18d04bdf8f47fad5c65888a19663ede
- semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
- question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
- Lean: NOT_RUN · human review: NOT_PERFORMED

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