Use only the explicitly supplied finite list p_n=2/(n+3) for n=0 through prefix_length-1, with prefix_length=2. 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+3) for n=0 through prefix_length-1, with prefix_length=2. 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.
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.
only indices0 through 1 supplied;finite probability entries=[2/3, 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.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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, 3, 2, [2/3, 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.