No finite stopping certificate without a rank bound

Exact statement

After any prefix through degree $N-1$ , the distinct rational series $F(q)$ and $F(q)+cq^N/(1-\lambda q)$ , with $c\ne0$ , agree on every observed coefficient. Therefore unrestricted exact-prefix learning cannot certify from finite data that its current rational hypothesis is final.

StatusProved by an explicit rational tail

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Learning in the limit allows early hypotheses to change. Finite certification makes the stronger demand that the learner know when no later coefficient can force a change.

Proof or evidence

The added rational tail starts at degree N, so it changes no observed coefficient but produces a different rational target.

Rational-tail indistinguishability →

Verification notes

The construction was checked for arbitrary finite prefixes and nonzero rational c; no probabilistic or asymptotic assumption is used.

Limitations

The result does not prevent identification in the limit.
A known rank bound or active validation can restore finite certification.

Open work

Identify a usable rank bound, equivalence query, or finality certificate in the intended application.