Let d_F=M(2L-1). For every retained t, request F(t+alpha zeta) modulo 2^(p_t) at d_F+1 binary-coordinate lifts of distinct residue-field elements. Local integrality puts this polynomial over A_(p_t), and unit point differences permit exact Lagrange interpolation there. Recover the whole degree-bounded polynomial and keep its first q_t coefficients for tapered Hermite recovery. The field has at least s^4 elements, while the required number is less than s^4 for sufficiently large s. This step specifies the evaluation task; the recursive mode construction must still compute those evaluations efficiently.