  Given a bigraded exact couple of modules over some ring, we determine the
meaning of the $E^{\infty}$-terms of its associated spectral sequence: Let
$L^{\ast}$ and $L_{\ast}$ denote the limit and colimit abutting objects of the
exact couple, filtered by the kernel and image objects to the associated cone
and cocone diagrams. Then the unstable E-infinite extension theorem states how
adjacent filtration quotients of the colimit filtration are extended by
$E^{\infty}$ objects over corresponding adjacent filtration quotients of the
kernel filtration.
  The stable E-infinity extension theorem is based on the fact that the
derivation process of the exact couple admits a transfinite recursion which is
beyond the scope of the traditional spectral sequence perspective. The
transfinite recursion always stabilizes at some ordinal. The resulting stable
$E$-objects are (a) always subobjects of $E^{\infty}$ and (b) extend adjacent
filtration quotients of the colimit filtration over corresponding adjacent
filtration quotients of the kernel filtration {\em without} the need for lim-1
corrective terms.
  The E-infinity extension theorems enable conclusions about the filtered
limit/colimit abutments even in cases where the spectral sequence is far from
converging in any traditional sense. We develop such results in the context of
'comparing' the spectral sequences via the morphism that is induced by a
morphism of underlying exact couples.
  We also contribute to 'reverse comparison' in a spectral sequence; that is
using information about the universal abutment(s) of the underlying exact
couple to extract information about one or more pages of the spectral sequence.
These results overlap with Zeeman's comparison theorems in a generalizing
fashion.
