Finite reduction of the full gauge-family row problem

Theorem thm:rank certifies this finite block system. Rational families may be treated over $\mathbb{K}(\xi )$ or after clearing declared denominators on a specified domain. Exceptional ranks and denominator zeros require separate checks: $\xi c=1$ is generically solvable and fails at zero.

The complete $M_{\xi },y_{\xi }$, original minor, and complete kernels for the changed-gauge 23-row problem remain absent. Full gauge-family row closure stays . D04 and the ghost equation are cross-checks, not replacement rows.