McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds

McMullen’s compact Kobayashi-geodesic curve $V\subset X_L$, arising from the hyperbolic triangle group $\mathrm{\Delta }(14,21,42)$ via a modular embedding into the Hilbert modular sixfold $X_L={\mathbb{H}}^6\mathrm{/}{\mathrm{S}\mathrm{L}}_2({\mathcal{O}}_L)$ attached to the totally real cyclic field $L=\mathbb{Q}(\cos \frac{\pi }{21})$, is not contained in any proper Shimura subvariety of $X_L$, and the generic fiber $A_v$ satisfies $\mathrm{M}\mathrm{T}(A_v)={\mathrm{R}\mathrm{e}\mathrm{s}}_{L\mathrm{/}\mathbb{Q}}{\mathrm{S}\mathrm{L}}_2$, hence carries no exceptional Hodge tensors. The Weil locus ${\mathcal{W}}_K\subset X_L$ parametrizing abelian sixfolds of Weil type for $K=\mathbb{Q}(\sqrt{-d})$ has codimension $3$ and $20$ irreducible components; the expected dimension $1+3-6=-2$ makes any non-empty $V\cap {\mathcal{W}}_K$ super-atypical in the sense of Zilber-Pink. We prove that $V\cap {\mathcal{W}}_K$ is finite, possibly empty: every intersection point is a CM point with ${\mathrm{E}\mathrm{n}\mathrm{d}}^0(A_v)=M=KL$, a degree-$12$ CM field with $\mathrm{G}\mathrm{a}\mathrm{l}(M\mathrm{/}\mathbb{Q})\cong \mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}3$, established by two independent methods: the André-Oort theorem for ${\mathcal{A}}_6$ and the Ax-Schanuel theorem for period maps. The Hodge-Weil classes in $H^{3,3}$ at intersection points are absolute Hodge yet inaccessible to all existing algebraicity theorems, due to three independent obstructions: CM isolation, absence of a $K$-secant structure, and uncontrolled discriminant. For $d\in \{3,7\}$, so that $M=\mathbb{Q}({\zeta }_{42})$, we reduce non-emptiness of $V\cap {\mathcal{W}}_K$ to $44\times 64=2816$ explicit algebraic equations for the prime $\mathrm{\ell }=43$ via Hecke correspondences on $X_L$, and isolate the remaining open steps toward a new case of the Hodge conjecture for abelian sixfolds.