Collapsed vector-space spectral sequences split noncanonically

Convergence identifies each graded quotient with the corresponding stationary page term. Under AC, F2 makes each such quotient projective. Apply F1 to the finite filtration; a decreasing filtration is first read in reverse order. The resulting isomorphism is the asserted direct sum, with partial sums in the filtration order. Zero quotients use the zero section. If the whole target is zero, its empty nonzero sum is zero; if only one quotient is nonzero, the normalized endpoints identify it with the target canonically.

In the finite-dimensional branch, choose a finite basis of each quotient in the fixed degree and choose a lift of each basis vector to the preceding extension. Linear extension is a section: composing with the quotient sends every basis vector to itself. There are only finitely many such bases and lifts, so finite induction supplies them in ZF. The same successive direct-sum construction as F1 therefore works without AC in that degree. This does not assert a simultaneous choice of splittings across an arbitrary family of degrees or spectral sequences.

For nonnaturality take $H=k^2$ with $0\subset k{e}_1\subset H$. The quotient has basis ${\bar{e}}_2$ and any linear section sends it to $a{e}_1+e_2$ for some $a\in k$. The filtration-preserving automorphism $T(e_1)=e_1$, $T(e_2)=e_2+e_1$ induces the identity on both graded pieces but moves every such section, since $a+1\mathrm{\ne }a$. These filtered data occur as a collapsed first-quadrant cohomological sequence: put $H$ in cochain degree one, differential zero, with $F^{0}H=H,F^{1}H=k{e}_1,F^{2}H=0$. The two associated graded entries at $(0,1)$ and $(1,0)$ remain unchanged on every page. A natural splitting would have to commute with $T$, contradicting the calculation. Thus even finite-dimensional collapse gives no general natural splitting.