Eventual matrix nonnegativity interleaves into one LRS

Exact statement

For a $d\times d$ rational matrix $M$ , all $d^2$ entry sequences of $M^n$ can be interleaved into one effectively constructed rational LRS of order at most $d^3$ , whose ultimate nonnegativity is equivalent to eventual entrywise nonnegativity of $M^n$ .

StatusReconstructed and proved; useful upper-bound reduction but not the decision argument

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

Context

Each matrix entry is an LRS. Interleaving preserves a common tail condition across the finitely many entries.

Proof or evidence

With $s=d^2$ , enumerate the entries as $u_r$ and define $v_{sn+r}=u_r(n)$ . If $U_r(x)=N_r(x)/D(x)$ , then $V(x)=\sum_r x^rU_r(x^s)$ , with denominator $D(x^s)$ of degree at most $d^3$ .

Generating-function interleaving →

Verification notes

The order bound and both directions of the universal-tail equivalence were rechecked; 3,564 Cayley--Hamilton entry identities and 4,212 interleaved values passed independently.

Limitations

The reduction lands in the open general Ultimate Positivity problem and therefore is not itself a decision procedure.
No reverse reduction or hardness consequence is claimed.

Open work

Keep this reduction separate from the stronger structured matrix theorem and from any unproved reverse reduction.