Norm Splitting and Defect Detection Have Three Arithmetic Regimes
Epistemic-graph event: 1337.
From S T=d id, splitting and detection impose different arithmetic
requirements:
- if
dis a unit, the norm splits and invariant defects are detected; - if
dis a nonunit non-zero-divisor, invariant defects are still detected although scalar splitting is unavailable; - with nonzero
d-torsion, even invariant defects may be annihilated.
The middle regime is essential integrally. For every five-site branch
degree 2^k, integral chain modules detect trace-submodule defects despite
the absence of division by 2^k. After inverting 2, splitting also
appears. In characteristic two, both properties can fail at every one-bit
stage.
This refines the arithmetic audit without constructing the unavailable physical relative-chain pushforward or trace.
Research note:
research/grothendieck/norm-splitting-versus-defect-detection-regimes.md.