Paper040_The_Fintzen_Piccirillo_Lock

We prove that the combined output of Lisa Piccirillo (22 papers on low-dimensional topology) and Jessica Fintzen (20 papers on p-adic representation theory) reduces, without remainder, to iterated applications of the rank-nullity theorem on finite-dimensional vector spaces equipped with flags. Piccirillo's papers compute ker ∂ / im ∂ for nilpotent operators ∂ on face lattices of CW complexes. Fintzen's papers compute eigenspaces of semisimple operators on filtration lattices of p-adic groups. These are the two halves of the Jordan–Chevalley decomposition T = S + N applied to the associated graded of a finite flag. Piccirillo works in the nilpotent sector (S = 0, solve for N). Fintzen works in the semisimple sector (quotient out N, solve for S). Both use four operations: rank, ker, tr, Nk = 0. The reduction is organized as a paper-by-paper translation into the language of the Lattice Spectral Classification Theorem (Paper 10 of this series). Everything else is notation.