Calogero-Moser Polynomial Approximations and Dwork Congruences for Complete Flags

We construct $p$-adic polynomial approximations to the zero-framing cohomological quasimap vertex of cotangent bundles to complete flag varieties from the trigonometric Calogero–Moser system. Negative integer couplings give regular staircase Jack polynomials with nonnegative integer coefficients. We compare them with complete-flag master polynomials extending the construction of Smirnov–Varchenko. Both families converge coefficientwise to the vertex and satisfy the full commuting equations modulo $p^s$. For $n\geq3$, we prove sharp degree-dependent congruences and compute leading residues at every $p$-power scale. Their discrepancy has coefficient norm $p^{-1}$ and an exact weighted Gauss norm giving uniform decay on smaller closed polydiscs. For the master family, multivariable Dwork congruences reconstruct the vertex by an infinite product. The results hold in every rank for $0<r/d\leq1/2$ and odd prime $p$.