Circular-Minor Positivity Barrier

Integrate the regularizer after an attention-logit matrix or graph message-passing matrix has been formed, before softmax or before applying the propagation operator. For an $n\times n$ matrix $M$, choose a fixed set $\mathcal C$ of circular pairs $(P,Q)$: all $1\times1$ minors, adjacent $2\times2$ minors, and selected larger contiguous minors with cyclic row and column order. If conservation is required, project with $M\leftarrow M-\operatorname{diag}(M\mathbf1)$, or parameterize $M$ as a Laplacian-like matrix. Normalize using $\widetilde M=M/(\|M\|_\infty+\epsilon)$. For every $(P,Q)\in\mathcal C$, extract $M_P^Q$, compute its determinant, multiply by $(-1)^{|P|}$, and add $\lambda\operatorname{softplus}(-\widehat s_{P,Q}/\tau)^2$ to the task loss. Pseudocode: construct $M$; normalize; initialize penalty to zero; loop over circular pairs; calculate $\widehat s_{P,Q}$; accumulate the barrier; backpropagate the task loss plus penalty. The determinant identity comes from the paper, while $\mathcal C$, $\lambda$, and $\tau$ are selected by validation. Start with a 4-layer Transformer on a synthetic noncrossing-sequence task and CIFAR-10 ViT-Tiny. Compare no regularizer, entrywise nonnegative regularization, and circular-minor regularization at matched FLOPs. Success is fewer unstable attention heads, lower sensitivity to depth and learning rate, and improved validation accuracy or calibration at equal training loss.