Both natural decision readings are decidable

Exact statement

Power nonnegativity and eventual nonnegativity of a rational matrix are decidable by testing, respectively, whether the effective ultimately periodic set $S(M)$ contains a positive exponent and whether it is cofinite.

StatusProved as an immediate consequence of the published theorem

External reviewThe underlying ultimate-periodicity theorem is published; no external review of this FPRD consequence note is recorded.

Context

The published theorem provides enough information for both decision tests.

Proof or evidence

Emptiness above zero and cofiniteness are decidable properties of an effectively presented ultimately periodic subset of the natural numbers.

Two decision consequences →

Verification notes

The n=0 case was excluded from the existential reading and the universal-tail reading was kept distinct. Eighty-one small integer matrices supplied 41 bounded witnesses separating the quantifiers.

Limitations

This is not an FPRD closure of an open problem; it identifies and applies an existing published answer.

Open work

Extract explicit certificates and complexity bounds for both decision tests from the effective periodic representation.