Genuine modular rank and permanent verification rules

For

the separate rational original minor is $1/36$. Modulo five,

and the corresponding determinant is one. Both ranks are two. Composite moduli and noninvertible denominators are rejected.

For Gaussian-rational coefficients,

Restriction of scalars proves the identity. Alternatively, a compatible finite-field representation can be declared. Field labels are not interchanged.

Finite-field reconstruction and IBP reduction are discovery methods ManteuffelSchabinger,Peraro. A candidate becomes an exact certificate only after checking the original arrays, characteristic-zero identities, original minor, complete kernels, and target residual. Finite kinematic samples need coverage or certified degree/denominator bounds for a broader identity claim.