Layerwise linking-number topology probe

Use this as a layerwise representation probe first, and only then as an auxiliary loss. Construct two oriented closed polygonal curves from minibatch embeddings: for a toy benchmark, sample known linked and unlinked circles in $\mathbb{R}^{3}$; for real data, fit each class or subgroup with a closed principal curve and resample it at $K$ evenly spaced points. At selected layers $j$, obtain $P=F_j(M)\in\mathbb{R}^{K\times d}$ and $Q=F_j(N)\in\mathbb{R}^{K\times d}$, project to three coordinates with a fixed random orthogonal projection if $d>3$, and form the $K\times K$ Gauss-map grid $G_{ab}=(P_a-Q_b)/(\|P_a-Q_b\|+\epsilon)$. Split each parameter-grid cell into two triangles. Compute the signed spherical area of a triangle with unit vertices $u,v,w$ using $2\operatorname{atan2}(u\cdot(v\times w),1+u\cdot v+v\cdot w+w\cdot u)$, then sum areas and divide by $4\pi$. Use the rounded value only for logging; gradients should use the smooth separation term in $\mathcal{L}_{\mathrm{top}}$, because the integer degree changes discontinuously at collisions. Set $k_{ab}$ to the known toy link number or to a teacher-network measurement, and apply the loss only to selected class pairs. Estimate numerical reliability by doubling $K$ and checking that the rounded linking number is unchanged. First compare a width-3 plain MLP, explicit-fold ResNet, and invertible flow on linked/unlinked curve classification. Success is higher class accuracy or fewer layers at fixed accuracy, while the probe should reveal intentional changes in $\widehat{\operatorname{lk}}_j$ without frequent spurious changes caused by near-collisions.