Traverse one child at a time, retaining one product over groups and one sum over beta. Keep only the current interpolation index, tuple counters, graph, new descriptions and the polynomially many local coefficients needed for recovery. Precision, field coordinates and rational common denominators have polynomial bit length. Immutable older descriptions are referenced rather than recursively copied. There are O_C(log lambda) stack levels, so their combined storage remains polynomial in s+description_length. Original entry routines must obey the input polynomial-space contract, and the new routines have the same property. Exponentially many calls do not require storing the recursion tree.