Relations among cycle invariants of a three-armed quiver

Join two vertices by three paths, each with at least two edges, and give each edge an arrow in each direction, with variables $d_{ij},u_{ij}$. Put $D_i=\prod_jd_{ij}$ and $U_i=\prod_ju_{ij}$, and impose $D_1-D_2+D_3=0$. Put $v_{ij}=d_{ij}u_{ij}$ and $A=\prod_jv_{1j}$, $B=\prod_jv_{2j}$, $C=\prod_jv_{3j}$. Under independent rescalings at the vertices, the ring of invariant polynomial functions over $\mathbb C$ has presentation \[ \mathbb C[v_{ij},w_1,w_2,w_3]\big/I_2\!\begin{pmatrix} w_2&w_3&B\\ A&w_3+C&w_1 \end{pmatrix}. \] Here $(w_1,w_2,w_3)=(D_1U_2,D_2U_1,-D_2U_3)$, and $I_2$ denotes the ideal of $2\times2$ minors. Makonzi proved that these elements generate the invariant ring. We prove that the minors generate all their relations, with the corrected all-edge indexing in the paper, establishing the kernel formulation of Makonzi's conjecture. The key step is a specialization to a rational function field that proves independence of normal forms. Lean checks the kernel equality over $\mathbb C$ for every triple of path lengths at least two. Checked in Lean from the definitions; not yet reviewed by experts. Produced by AI models under the author's direction.