Tier-2 proximity scaling for Yang-Mills-Higgs minimisers

Let $M$ be a closed oriented Riemannian manifold with a non-zero relevant topological invariant, and let $D=\{(x_+,+1),(x_-,-1)\}$ consist of two oppositely labelled points at geodesic distance $d$. Let $(A^*,\phi^*)$ be the YMH minimiser. Tier-2 proximity scaling. As $d\to0$, the matter-sector energy

$$\int_M\|D_{A^*}\phi^*\|^2$$

diverges at a rate set by the regularised Green kernel, and this is the dominant contribution to the YMH energy in every dimension. At leading order with the connection fixed at the BPS connection, the exponent is $\min(2\nu-n,2)$; the conjectural assertion is that this divergence persists under full back-reaction, with $A^*=A^*(d)$. In dimension $4$ for an $SU(2)$ instanton, the curvature is conjectured to concentrate in a single midpoint core with scale $\lambda\sim d$ and peak $\|F_{A^*}\|_{\max}\sim1/d^2$, while the Yang--Mills energy remains $8\pi^2|c_2|$. In dimension $2$ for a $U(1)$ monopole, the leading gauge field remains the constant-curvature connection, its deformation is bounded, and no $1/d^2$ curvature peak occurs. This extends proximity scaling from the matter-sector approximation to the fully coupled non-abelian YMH problem; the fixed-connection leading-order result is established, whereas persistence under back-reaction remains unproved.