Matter-sector proximity scaling in the general case

Let $(M,g)$ be a closed Riemannian manifold with $n=\dim M$, let $\xi^*\in H^1(M,\mathbb{Z}_2)$ be a flat bundle class selected by the variational principle, and let $D=\{(x_i,y_i)\}$ be a labelled dataset containing opposite-label points $x_+,x_-\in M$ at geodesic distance $d=d_g(x_+,x_-)$. Assume the remaining $N-2$ points lie at non-degenerate positions in a compact region independent of $d$, and fix Matérn parameters $\nu>n/2$ and $\kappa>0$. Matter-sector proximity scaling. As $d\to0^+$, the matter-sector minimum on the selected bundle satisfies

$$E_{\min}(\xi^*)\sim \frac{C(M,g,\nu,\kappa)}{d^{2\nu-n}},$$

modulo logarithmic corrections when $2\nu-n\in2\mathbb{Z}_{\geq0}$, where $C(M,g,\nu,\kappa)>0$ is independent of $d$. The two-point case $N=2$ is established, while the general $N$ case remains open. The examples on $S^1$ with $\nu=1$ and on $T^2$ with $\nu=2$ realise the exponents $1$ and $2$, respectively, illustrating divergence of the matter energy as opposite labels approach. The result separates the discrete bundle selection from the continuous geometric cost of classifying nearby opposite labels.