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Theorem · external result

FPRD-MN-T01

Nonnegative powers form an effective ultimately periodic set

Exact statement

For every rational square matrix MM, the exponent set S(M)={n∈N:Mn≥0}S(M)=\{n\in\mathbb N:M^n\ge0\} is effectively computable and ultimately periodic.

StatusPublished theorem; FPRD exposition reconstructed from the source
External reviewPublished in the peer-reviewed STACS 2024 proceedings. This does not constitute external review of the FPRD exposition.

Context

The theorem computes the complete set of good exponents, not only a yes/no answer for an eventual tail.

Hypotheses and scope

  • M is a finite square matrix with rational entries.
  • Nonnegativity is entrywise.

Proof or evidence

The published proof splits exponents into finitely many nondegenerate residue classes. Dominated entry sequences have computable eventual signs; if one entry sequence in a residue is not dominated, no whole nonnegative matrix power occurs in that residue.

The theorem and proof architecture →

Verification notes

The source theorem number, residue-class argument, and unconditional scope were rechecked. A separate exact audit checked 4,282 signed permutation matrices and the signed-cycle hostile family.

Limitations

  • No polynomial-time bound is asserted.
  • The theorem concerns a single generated power sequence, not arbitrary noncommuting matrix semigroups.

Open work

Extract explicit certificate and complexity bounds from the effective spectral construction.