# Question 15076: A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

[Return to catalogue page 76](https://ocme.evemisslab.com/en/questions/page/76/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r07-convergence-in-probability-foundations/#q-20261014-advance-20261030-r07-convergence-in-probability-0051)

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.

R0029-15076 · q-20261014-advance-20261030-r07-convergence-in-probability-0051



## Original question and givens

Use only the explicitly supplied finite list p_n=1/(n+2) for n=0 through prefix_length-1, with prefix_length=1. 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+2) for n=0 through prefix_length-1, with prefix_length=1. 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=2;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, 2)

After

finite prefix index0 denominator = 2

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, 2)

After

finite prefix index0 probability = 1/2

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 · impliesBefore

only indices0 through 0 supplied;finite probability entries=[1/2]

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, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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, 2, 1, [1/2], 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: cd2c6727ede0ee9bc93c03fb16ae6a4c5ccb644e91340fa2f203758513b2dc6b
- candidate record SHA-256: 931dc0fd57eb1408bb41aacb6e5c3cc119f7a601c80cb40e3d0096548ab1adea
- semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
- question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0051.json)