Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=12, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=12, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
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.
declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)
Conditions: none
The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices.
selected function evaluation:original point=1;n=12;limit=12;signed coefficient=-13
After
f_n(x)=155/14;f_n(x)-limit=-13/14;this error field remains SIGNED
Conditions: none
Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.
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 selected function evaluation reverse signed error with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute selected function evaluation function value with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute selected function evaluation signed correction with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute selected function evaluation 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
(12, 1, 155/14, 12, -13/14)
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.