Karatsuba carry handling

The blocks in the Karatsuba three-product identity are integers, not single digits: $p_0$ can need $2m$ positions and $h$ can exceed $q-1$. Concatenating the displayed blocks therefore fails. Accumulate their shifted values with ordinary addition, normalizing each digit position by Euclidean division: a coefficient $v$ becomes the digit $v\mathrm{mod}B$, with carry $\lfloor v/B\rfloor$ passed to the next position. Subtraction likewise propagates borrows. Although $h\ge 0$ for these nonnegative integers, intermediate digitwise subtraction can still borrow.

In the four-digit Karatsuba worked multiplication, grouping by base $q=100$ gives unnormalized coefficients $[672,2840,2652]$, written from high to low. Normalize from the right: $2652=26\cdot 100+52$; $2840+26=2866=28\cdot 100+66$; $672+28=700=7\cdot 100+0$. The final groups are $[7,00,66,52]$, yielding $7,006,652$. The zero group must retain its two positions.

Since $a,b,c,d<B^m$, their sums are less than $2B^m\le B^{m+1}$ and can need $m+1$ digits. Allocate that extra digit before the third recursive product; discarding it can change the answer. The GMP manual describes this sum growth and uses a difference-based form to avoid it.^[ref]