The first-order part of the Ordered Variable Word theorem

An ordered $\omega$-variable sequence over a finite alphabet~$A$ is an infinite sequence over the extended alphabet $A\bigsqcup \{x_0,x_1,\dots \}$, where each variable kind appears at least once, and the last occurrence of $x_i$ appears before the first occurrence of $x_{i+1}$. The Ordered Variable Word theorem states that for every finite coloring of the words over a finite alphabet~$A$, there is an ordered $\omega$-variable sequence~$W$ over~$A$ such that every word induced by substituting variables with letters, and cutting before the first undefined variable, has the same color. In this article, we prove that the Ordered Variable Word theorem is ${\mathrm{\Pi }}_1^1$-conservative over ${\mathsf{R}\mathsf{C}\mathsf{A}}_0+\mathsf{B}{\mathrm{\Sigma }}_2^0$. It follows that the tree theorem for singletons by Chubb, Hirst, and McNicholl is ${\mathrm{\Pi }}_1^1$-conservative over ${\mathsf{R}\mathsf{C}\mathsf{A}}_0+\mathsf{B}{\mathrm{\Sigma }}_2^0$, answering a long-standing open question. Lemma 3.13 was found by Claude Opus 5.5 (Anthropic).