For split characteristic polynomial, elementary divisors and Jordan strings give the same Jordan blocks

On the cyclic summand $F[x]\mathrm{/}((x-\lambda {)}^e)$, multiplication by $(x-\lambda {)}^k$ has kernel dimension $\min (k,e)$ for $k\ge 0$: it kills exactly the final $\min (k,e)$ translated-power basis vectors.

Kernel dimensions add over direct sums, so for each $\lambda$ the module elementary divisors give $\dim \ker (T-\lambda I{)}^k={\sum }_e\min (k,e)$, counted with multiplicity.

By [L2], successive differences of the sequence in step 2.1 recover the number of blocks of size at least $k$ and exactly $k$. These are the same shifted-power invariants used in [L1], so the module and Jordan-string constructions have identical block multisets. For $k=0$ the kernel is zero; after the largest exponent the sequence stabilizes, and on the zero space every sequence and list is empty.