Exact ranks for the regular 17-, 18-, 19-, and 20-gons

A computer-assisted exclusion of four-dimensional, eight-facet lifts proves
that the regular $n$-gon slack matrix has real nonnegative rank nine for
$17\le n\le20$; explicit nine-term factorizations give the matching upper
bound. The reduction uses both factors of a nonnegative factorization, a
complete finite cover of planar shadows, and exact determinant and
Plücker-relation calculations over finite fields. The finite-field
calculations certify ranks over the cyclotomic number field; they are not
searches for finite-field points and use no numerical infeasibility
tolerance. The exact checks succeed for all 6,610 incidence patterns, and
again in a second pass with different primes, coordinate gauges and
determinant subsets. A limiting argument includes nonsimple lifts and
collapsed fibers. The regular 16-gon's known rank-eight factorization is
retained as a positive control.