# Question 15064: Eventual Equality Supplies a Conditional Universal-Tail 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.

R0029-15064 · q-20261014-advance-20261030-r07-convergence-in-probability-0089



## Original question and givens

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=14; inspect n=17. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).


- G001 — On the declared common probability space assume X_n=X almost surely for every n≥settling_index=14; inspect n=17. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_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

declared common probability space;X_k=X almost surely for EVERY k>=14;inspected n=17

After

inspected deviation status=0;for EVERY positive epsilon,all k>=14 have deviation probability0

Conditions: none

Use the explicitly declared all-later almost-sure equality,not a pattern guessed from this index.Earlier not_yet_zero only means the certificate has not supplied0,not a known positive probability.The all-epsilon zero tail gives the conditional convergence-in-probability certificate.

Step identifiers and dependenciesRule: eventual_equality_full_tail_not_one_index · Uses: G001









## Reverse checks


- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(14, 17, 0, Yes, eventual_equality_universal_tail_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: 508f70f77edfa774cbf0bd83d954b6190075e0ac62a26830f2967aec1ad46376
- candidate record SHA-256: 0d473aa409592679b58e181e34d19e44d4d9158348461fc77e00b56ca3c769f1
- 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-0089.json)