Residual Norm Homology Is the Augmentation Ideal Modulo Its Socle
Epistemic-graph event: 1362.
For a collapsed k-bit kernel algebra A_B with augmentation ideal I_B
and top socle line F_2 epsilon_B, the mod-two norm homology is
(I_B/F_2 epsilon_B) tensor A_(B^c).
For k>=2 its collapsed-kernel Loewy length is k-1, with successive layer
ranks
C(k,1), C(k,2), ..., C(k,k-1)
times the untouched deck module. The one-bit module vanishes; the full
five-bit layers have dimensions 5,10,10,5.
Thus the residual bad-prime object is the proper Boolean augmentation module, not merely a numerical rank defect. This remains formal deck algebra and does not authorize physical branch intersections or relative-chain maps.
Research note:
research/grothendieck/mod2-norm-homology-loewy-module.md.