The Source-Side Mackey Norm Is a Degree-Idempotent
Epistemic-graph event: 1357.
If S T=dI, the source operator N=TS satisfies
N^2=dN.
Only after inverting d does (1/d)N become the projector onto the trace
image along the pushforward kernel. Integrally the image may have finite
index in the trace submodule.
At primes dividing d, the norm becomes square-zero. For the one-bit deck
quotient, S=[1 1] and T=[1;1], so modulo two the nonzero matrix
[[1,1],[1,1]] squares to zero and has image equal to its invariant kernel
line.
Thus a five-site integral norm identity does not imply a chain splitting; after mod-two reduction it produces nilpotent rather than projective structure. Physical use remains conditional on source-derived relative maps.
Research note:
research/grothendieck/source-norm-quasi-idempotent.md.