Non-rationality conjecture for Mordell–Weil and Néron–Severi height specializations

Let $k=\mathbb{C}$ and $K=\mathbb{C}(z)$. For the motivic height series $\mathcal{Z}(u,v;t)$, define

$$Z_{\operatorname{MW}}(v;t)\coloneqq \mathcal{Z}(1,v;t), \qquad Z_{\operatorname{NS}}(w;t)\coloneqq \mathcal{Z}(w,w;t).$$

Non-rationality conjecture. The specializations are not rational in $t$ with coefficients in $K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][v]$ and $K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][w]$, respectively; equivalently,

$$Z_{\operatorname{MW}}(v;t)\notin K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][v](t), \qquad Z_{\operatorname{NS}}(w;t)\notin K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][w](t).$$

The conjecture reflects the global nature of Mordell–Weil rank and Néron–Severi rank jumps, which are invisible to the finite local Kodaira factor stratification. No resolution is stated here; the preceding discussion motivates the expected failure of a finite Euler-product rationality mechanism for these specializations.