Assume the smaller child contractions are correct at their specified p_t. The mode identity gives each F(t+alpha zeta) at that precision. Residue-distinct interpolation then recovers the local jets, and tapered Hermite recovery combines them to obtain F(0) modulo the parent precision p. Because entry lifts were fixed throughout the call, F(0) is precisely its required contraction. Child shrinkage ensures termination at direct-summation leaves. Induction from those leaves proves this reconstruction procedure correct under the stated arithmetic and input hypotheses; it does not by itself establish the runtime bound.