The shuffle‑word‑embedding conjecture, which asks whether certain patterns of operator products can be reproduced by simpler diagonal models, now holds for many key cases. For occurrence numbers m = 1, 2 and 4 any finite set of shuffle traces can be approximated as closely as desired by commuting diagonal self‑adjoint contractions. For every odd m ≥ 3 the conjecture is exact: there exist diagonal self‑adjoint matrices X and Y, with ∥X∥≤1, that reproduce the shuffle traces exactly for all degrees n (and for all n if the original operators have finite spectrum).

The proof works by constructing specific real numbers in [−1, 1] and building the diagonal matrices from them. The resulting matrices match the normalized shuffle coefficients for any prescribed finite set of degrees n, and give a perfect match when the original operators have a finite spectrum.

A key insight is that for even occurrence numbers the whole conjecture is equivalent to a positivity condition on a linear functional Λ(m)_{A,B}. If this functional maps every real polynomial that is non‑negative on [−1, 1] to a non‑negative number, then a positive measure can be found that produces the required diagonal model.

For m = 6 the situation is more subtle. Attempts to prove positivity of the associated kernel by splitting six‑step cyclic paths into two three‑step paths fail in general—the kernel is not positive semidefinite. This suggests that the contraction model property may not hold for m = 6, leaving that case open.

These results sharpen our understanding of when complex operator products can be faithfully represented by simple commuting diagonal models.