Failure of separatedness or completeness can destroy the claimed abutment

Put $C_0=k$ with zero differential and $F_p{C}_0=k$ for every integer $p$. Each graded quotient is $k\mathrm{/}k=0$, so $E^0=0$ and every later page is zero by successive homology. But $H_0(C)=k$ and every homology filtration term equals $k$, whose intersection is nonzero. The zero limiting page agrees with the zero associated graded of this target; it does not imply that the target is zero. Thus this is weak convergence without separatedness or strong convergence.

Next take $C_1=S$, $C_0=P$, with differential the inclusion. On either nonzero degree set $F_{-m}{C}_j=T_m{C}_j$ for $m\ge 0$ and $F_p{C}_j=C_j$ for $p>0$. The inclusion preserves every tail, so these are subcomplexes. The filtration is increasing and exhaustive because $F_{0}C=C$. Its intersection is zero in both degrees, but its degree-one completion map is the proper inclusion $S\to P$; hence the filtered complex is incomplete.

For $p=-m\le 0$, the successive quotient in either nonzero chain degree is $T_m\mathrm{/}{T}_{m+1}\cong k$, by the coordinate $m$ map with zero-extension inverse. The induced differential between these two graded terms is the identity of $k$. For $p>0$ the quotient is zero. Thus every fixed-$p$ graded complex is either $k\stackrel{1}{\to }k$ in degrees $1,0$ or the zero complex; its homology vanishes. Hence $E^1=0$, and all later pages vanish. In contrast $H_1(C)=0$ and $H_0(C)=P\mathrm{/}S$; the constant-one sequence gives a nonzero class because it is not finitely supported.

Every $x\in P$ differs from its tail obtained by deleting coordinates $0,\dots ,m-1$ by an element of $S$. Thus $T_{m}P\to P\mathrm{/}S$ is surjective for every $m$, and the induced homology filtration has $F_p{H}_0(C)=P\mathrm{/}S$ for every integer $p$. This explains the lost target: the limiting zero page agrees with a zero associated graded, while the homology filtration is nonseparated. The chain complex's incompleteness was already checked in step 1.2; no complete-convergence theorem applies to it.

Finally take the zero-differential complexes $S[0]$ and $P[0]$ with the same tail filtrations. Their associated-graded terms are $k$ at $(p,q)=(-m,m)$ for each $m\ge 0$, and zero elsewhere. Every spectral differential is zero because the chain differential is zero, so these graded identifications persist on every page. Their homology targets are respectively $S$ and $P$, which are not isomorphic even as sets by [F1]. The inclusion induces the page isomorphisms and is the completion map, but is not onto on homology. Thus equal graded pages cannot replace the missing completeness hypothesis. The initial level $m=0$, zero positive levels and empty deleted prefix all satisfy the displayed formulas. Every construction uses fixed coordinates or finite truncations, without AC.