Unconditional proof of minimal annihilator status
Prove unconditionally, using creative telescoping, that the irreducible third-order Picard–Fuchs operator M is the minimal annihilator of the hyperkagome lattice Green's function.
References
Every structural statement above is a rigorous consequence of $M$ together with Theorems~\ref{thm:modular}, \ref{thm:y0}, \ref{thm:noalg} and \ref{thm:phi}; that $M$ is the minimal annihilator of the Green's function is established to the guess-and-verify standard of the lattice-statistics literature, and an unconditional creative-telescoping proof of minimality remains the one open step.
The remaining task is to specialize such an evaluation to the curve above, or to integrate $D_+$ termwise---it is quadratic in each $\cos\theta_i$, so one angular integration yields an inverse square root and the next an elliptic integral, leaving a one-dimensional algebraic period that the orthogonal structure forces into the $\mathrm{Sym}2(V_2)$ form. We record the reduction here; matching such a classical evaluation, term by term, against the modular parametrization of Section~\ref{sec:modular} and the closed form eq:y0closed is left to future work.