Kernel orientation, rescaling, and modular arithmetic

For $N\in K^{s\times L}$,

Their dimensions are $s-\mathrm{rank}N$ and $L-\mathrm{rank}N$. A right-nullspace routine must be applied to $N^T$ to obtain left separators.

If row denominator clearing gives

then a separator $\tilde{z}$ of the cleared data corresponds to $z=\tilde{z}D$. The row rescaling must be retained in the certificate.

For a rational matrix and a prime avoiding entry denominators,

A nonzero modular minor has a nonzero rational lift. But a modular augmented-rank jump need not prove rational nonmembership:

has a modular separator, while over $\mathbb{Q}$ it has the fit $1/p$. Discovery by modular arithmetic must therefore be followed by an exact characteristic-zero certificate.