Let K=Q*J and pad the unknown polynomial coefficient list to degrees 0,...,K-1. The local coefficient [z^j]T^u at T=t+z is binom(u,j)*t^(u-j) for u>=j, otherwise zero. These integers form a K-by-K matrix V, one row per (t,j). The previously proved Hermite uniqueness makes V nonsingular. Its entries have O(K*log2(Q+2)) bits: binom(u,j)<=2^u and t<=Q. If the computed local data y share denominator Omega, then b=Omega*y is integral. Solving V*(Omega*h)=b recovers the polynomial coefficients h without floating-point interpolation.