Failure of disjoint additivity under magnetic Gauss constraints

Abstract

Under local magnetic Gauss constraints, vertex-disjoint regions can have commuting physical observable algebras that do not generate all observables on their union. We construct such counterexamples for every finite non-Abelian group on regular square tori, retaining the full magnetic physical space and original supports inside a fixed contractible disk. A peripheral projector selects inputs on which a joint transition preserves the constraints, while its loop factor alone fails the universal test over physical exterior completions. For a specified support family around a fixed eight-edge square loop, additivity fails if and only if the group is non-Abelian and each inner face is covered by at least one of its two prescribed peripheral edges. Leaving a face uncovered restores full additivity. For the same loop, additivity also holds with any region at vertex-set graph distance at least two. These results refute a recent group-double additivity proposition.

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BibTeX RIS

BibTeX

@misc{liang2026failure,
  title = {{Failure of disjoint additivity under magnetic Gauss constraints}},
  author = {Yi Liang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25329},
  eprint = {2610.25329},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25329},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Failure of disjoint additivity under magnetic Gauss constraints
AU  - Yi Liang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25329
UR  - https://arxiv.org/abs/2610.25329
AB  - Under local magnetic Gauss constraints, vertex-disjoint regions can have commuting physical observable algebras that do not generate all observables on their union. We construct such counterexamples for every finite non-Abelian group on regular square tori, retaining the full magnetic physical space and original supports inside a fixed contractible disk. A peripheral projector selects inputs on which a joint transition preserves the constraints, while its loop factor alone fails the universal test over physical exterior completions. For a specified support family around a fixed eight-edge square loop, additivity fails if and only if the group is non-Abelian and each inner face is covered by at least one of its two prescribed peripheral edges. Leaving a face uncovered restores full additivity. For the same loop, additivity also holds with any region at vertex-set graph distance at least two. These results refute a recent group-double additivity proposition.
ER  -

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