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Order-1 differential algebraicity of Hilbert series for Koszul symmetric operads generated by one binary operation (Main Conjecture)

Establish that for every Koszul symmetric operad P generated by a single binary operation, the Hilbert series f_P(t) is differential algebraic of order 1 over Z[t] (equivalently, f_P(t) and its derivative f_P'(t) are algebraically dependent over Z[t]).

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Background

Motivated by a growing collection of examples where Hilbert series calculations are available, the authors observe that in all known cases of Koszul symmetric operads generated by a single binary operation, the Hilbert series satisfies a first-order differential algebraic relation. They therefore formulate a stronger conjecture than the general KP conjecture, restricted to this class, and later verify it for all such set-operads classified in the paper.

References

In all the examples we know, the Hilbert series of these operads are differential algebraic of order $1$ over $\mathbb{Q}[t]$. We conjecture that this is always the case: \begin{cnj*}\ref{cnj:main} Let $P$ a Koszul symmetric operad generated by one element in arity $2$, then the Hilbert series of $P$ is differential algebraic of order $1$ over $\mathbb{Z}[t]$. \end{cnj*}

— On Hilbert series of Koszul operads and a classification result for set-operads (2509.14419 - Laubie, 17 Sep 2025) in Introduction; also restated in Section 3 as Conjecture 3.1 (labelled Conjecture \ref{cnj:main})