Generalized spectral conjecture

Definitions and mathematical statement

Let $S\subset{\mathbb{R}}$ be a unital subring and let $A$ be an $n\times n$ matrix over $S$. Its nonzero spectrum means the multiset of nonzero complex eigenvalues, with algebraic multiplicity. A matrix $B$ over $S$ is primitive in the nonnegative-matrix sense if all its entries are nonnegative real numbers and $B^k$ has all entries strictly positive for some integer $k\geq1$.

Square matrices $A_0,A_1$ over $S$, possibly of different sizes, are elementary strong shift equivalent over $S$ if rectangular matrices $R,T$ over $S$ satisfy $A_0=RT$ and $A_1=TR$. Strong shift equivalence over $S$ is the equivalence relation generated by these elementary steps: there must be a finite chain from $A$ to $B$. Entries of the intermediate matrices and rectangular factors are in $S$; the source does not require them to be nonnegative.

The proposed generalized spectral statement is: whenever the nonzero spectrum of $A$ satisfies the source's "spectral-conjecture conditions", there is a primitive matrix $B$ over $S$ that is strong shift equivalent over $S$ to $A$. No size for $B$ is prescribed.

The record does not reproduce those spectral conditions or their dependence on $S$, so the hypothesis is not self-contained. In particular, a list of Perron, trace, or arithmetic conditions cannot be supplied from this record alone. The supplied research summary also distinguishes results for dense subrings and ${\mathbb{Z}},{\mathbb{Q}},{\mathbb{R}}$ from obstructions over other rings, and discusses shift equivalence versus strong shift equivalence. These relations cannot be silently identified: the requested conclusion is specifically strong shift equivalence. The scope of an unrestricted assertion for every unital subring must be checked rather than inferred from the registry's solved label.

Short English statement

Can a matrix satisfying the appropriate nonzero-spectral conditions be transformed by strong shift equivalence over its real subring into a primitive nonnegative matrix? The source omits the spectral conditions and qualifies which ring cases are covered by the reported solutions.

Sources

• Local UnsolvedMath registry, permanent ID 4600006, exact source code catalogue entry, "Generalized spectral conjecture"; proposed year 2008, no proposer specified. Registry status solved, classification SOLVED-IN-LITERATURE.
• Mike Boyle, Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 6.2, https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf. Supplied pdf-text extraction accessed July 29, 2026. The source presents the item as open; its provenance records no later authoritative resolution check. No PDF was accessed here.
• The supplied research summary reports solutions for the principal dense-subring and ${\mathbb{Z}}/{\mathbb{Q}}/{\mathbb{R}}$ cases, building on Kim--Ormes--Roush and Boyle--Schmieding. It warns that, for general other subrings, shift equivalence need not imply strong shift equivalence because of an $\operatorname{NK}_1$ obstruction. No precise theorem, spectral hypothesis list, or full publication reference is supplied. This qualification is preserved separately from the solved registry label; it is not asserted as a complete classification of the primitive-realization problem.
• Provenance flags status and rights NEEDS_REVIEW: the historical source is open, no later authoritative check is recorded there, and public access did not establish redistribution permission.