The_McCleary_Lock

We prove that the entire content of McCleary's A User's Guide to Spectral Sequences — exact couples, derived couples, filtered differential graded modules, convergence theorems, the Serre spectral sequence, the Eilenberg–Moore spectral sequence, the Adams spectral sequence, and all attendant algebraic machinery — reduces, without remainder, to iterated applications of the rank-nullity theorem on filtered chain complexes over a field. Every "spectral sequence" is a sequence of quotient vector spaces Er = ker dr / im dr−1 where each dr is a linear map with dr2 = 0, and the passage Er ⇝ Er+1 is the single computation dim Er+1p,q = dim ker dr|Erp,q − dim im dr|Erp−r,q+r−1. This is rank-nullity. Everything else is notation. The reduction is organized as a chapter-by-chapter translation of McCleary's text into the language of the Lattice Spectral Classification Theorem (Paper 10 of this series). The lattice in question is the subspace lattice of the filtered chain complex; the meet operator is intersection of subspaces; the Möbius inversion is the inclusion-exclusion on filtration layers. The spectral sequence is the Krylov iteration of this meet operator. All results are finite linear algebra in ZFC without the Axiom of Choice.