Weight-one counting misses Heisenberg growth

For every integer $N\ge 1$ the explicit word $[x^N,y^N]=x^N{y}^N{x}^{-N}{y}^{-N}=z^{N^2}$ has length at most 4N. Its normal coordinate tuple is (0,0,N^2), since the ordered form is unique. Hence the central coordinate in B(4N) can equal N^2.

If a fixed linear bound $\mathrm{\mid }k\mathrm{\mid }\le Cn$ held in every ball B(n), step 1.1 would imply $N^2\le 4CN$ for every positive integer N. Taking any integer N>4C contradicts this. Thus the claimed uniform linear bound fails. Moreover H has three independent coordinates but ball growth degree four by F1, so the predicted degree three from weight-one counting is false; positive multiples of n^4 cannot be bounded above by a fixed multiple of n^3 as n grows.