Polynomiality of Stretched Schubert Structure Constants and Key Coefficients

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. The proof combines Demazure operators with vector partition functions and, in the Schubert case, P.~Magyar's orthodontic formula. These coefficient results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial, proving a conjecture of I.~Pak and Z.~Slonim. For key polynomials, this resolves the polynomiality part of a conjecture of P.~Alexandersson and E.~Alhajjar.