Multi-Bit Mod-Two Branch Norms Have Residual Square-Zero Homology
Epistemic-graph event: 1358.
On an even fiber of size d, the mod-two source norm is the all-ones matrix
J_d. It is square-zero with rank one, so
dim(ker J_d / im J_d)=d-2.
Therefore a one-bit fiber (d=2) is norm-acyclic, while every larger branch
fiber carries residual square-zero homology. For the five-site k-bit
collapses, the total dimensions for k=1,...,5 are
0, 16, 24, 28, 30.
This refines the existing mod-two augmentation filtration: nonzero square-zero norm classes exist at every branch degree, but only the one-bit norm has image equal to kernel. The result remains formal deck-module algebra and does not construct the missing physical relative specialization.
Research note:
research/grothendieck/mod2-fiber-norm-homology-rank.md.