Finite nontrivial tensor-resolution sets for infinite spectra

Construct a natural infinite spectrum whose set of positive two-factor tensor resolutions is finite and nontrivial, without imposing additional restrictions; in particular, investigate whether an incommensurate d-dimensional rectangular well has exactly 2^{d-1} such resolutions.

Background

The paper defines the two-factor resolution set Σ2(X)\Sigma_2(X) of a spectral alphabet as the positive factorizations X=ABX=AB, modulo interchange of the factors. It establishes three examples: the one-dimensional box has only the trivial resolution, finite equally spaced internal spectra have finitely many resolutions, and the harmonic oscillator has infinitely many resolutions arising from binary expansions.

The authors explicitly report that they did not find a natural infinite spectrum exhibiting the intermediate behavior of a finite but nontrivial resolution set. They propose an incommensurate rectangular well as a candidate example, with the conjectural count ∣Σ2∣=2d−1|\Sigma_2|=2^{d-1}.

References

We did not find a natural infinite spectrum with a finite but nontrivial resolution set without imposing additional restrictions. An example would be a $d$-dimensional incommensurate rectangular well, i.e. the energy scales in each direction are linearly independent over $\mathbb Q$, with $|\Sigma_2|=2{d-1}$.

— The Arithmetic of Spectra: Factorization, Statistics, and Symmetric Functions  (2609.10391 - Chaudhary, 9 Sep 2026) in Section 6, Tensor resolutions of a spectral alphabet