The Bieri-Neumann-Strebel invariant, usually called the BNS invariant, divides the possible real-valued directions on a group into regions with different finiteness behaviour. An AIM problem asks whether free-by-cyclic groups can have arbitrarily many of these regions even after symmetries from the outer automorphism group are taken into account. This anonymous candidate proposes an explicit family indexed by an integer m. Each group comes from a linearly growing automorphism of a free group of rank two m plus one. Its first Betti number is m plus two. The construction produces m plus one deleted hyperplanes, and these divide the BNS invariant into exactly two m plus two connected components. Counting components is not enough, because an outer automorphism could permute several of them. The proof therefore assigns each component a coordinate-free number: the smallest rank of the kernel of a primitive integral character in that component. A direct Bass-Serre calculation gives one value for the outer pair and m strictly separated values for the interior pairs. Since automorphisms preserve kernel rank, components with different minima cannot belong to the same orbit. This proves a lower bound of m plus one orbits, which grows without bound. It does not prove that this is the exact orbit count or compute the full outer automorphism group. The BNS identification imports a theorem of Cashen and Levitt, while finite Python and JavaScript checks only corroborate the universal written argument. This is an anonymous, AI-assisted, unrefereed candidate. This is release 0.2.0-candidate, dated 31 August 2026; the paper and evidence are linked at evidencepress.org. This AI-generated voice is a communication aid, not additional mathematical evidence.