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| Formato: | Recurso digital |
| Idioma: | inglês |
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Zenodo
2026
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| Acesso em linha: | https://doi.org/10.5281/zenodo.20173318 |
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| _version_ | 1866901795272392704 |
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| author | Carroll, Jeremy H. |
| author_facet | Carroll, Jeremy H. |
| contents | <p>This preprint classifies the obstruction geometry selected by SO(k)-invariant local gauges on finite graph cochains with vector-valued edge defects. Within the M-compatible gauge framework, SO(k) invariance and the standard norm axioms force each local gauge on R^k to be proportional to the Euclidean norm. When this local geometry is combined with replica-extensive, edge-decomposable aggregation, the resulting global diagnostic is the mixed norm ℓ¹(E;ℓ²), with edge weights retained unless edge-exchange symmetry or explicit normalization is imposed.</p> <p>The paper proves the associated primal-dual obstruction formula: the quotient obstruction Φ₂,₁ is the infimum of the ℓ¹(E;ℓ²) residual modulo exact repairs, and its dual witnesses are Euclidean-unit-bounded divergence-free vector circulations. It also presents a finite triangle example separating scalar-coordinate ℓ¹ observers from SO(k)-invariant vector observers: the same cycle class receives obstruction magnitude 2 under coordinate-separable ℓ¹ geometry and √2 under Euclidean vector geometry.</p> <p>The result is observer-relative and conditional. It does not claim universal Euclidean physics or universal ℓ¹ geometry. Rather, it identifies the precise two-stage selection mechanism: SO(k) symmetry determines the local ℓ² gauge, while replica-extensive edge aggregation determines the outer ℓ¹ sum.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20173318 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | SO(k)-Invariant Gauge Classification and the ℓ¹(E; ℓ²) Obstruction Geometry Carroll, Jeremy H. SO(k) invariance M-compatible gauges vector-valued cochains finite graph cochains ℓ¹(E;ℓ²) mixed norms Euclidean gauge obstruction geometry quotient obstruction divergence-free dual witnesses finite obstruction calculus cohomological obstruction theory replica extensivity edge-decomposable diagnostics convex duality Minkowski functional observer-relative geometry <p>This preprint classifies the obstruction geometry selected by SO(k)-invariant local gauges on finite graph cochains with vector-valued edge defects. Within the M-compatible gauge framework, SO(k) invariance and the standard norm axioms force each local gauge on R^k to be proportional to the Euclidean norm. When this local geometry is combined with replica-extensive, edge-decomposable aggregation, the resulting global diagnostic is the mixed norm ℓ¹(E;ℓ²), with edge weights retained unless edge-exchange symmetry or explicit normalization is imposed.</p> <p>The paper proves the associated primal-dual obstruction formula: the quotient obstruction Φ₂,₁ is the infimum of the ℓ¹(E;ℓ²) residual modulo exact repairs, and its dual witnesses are Euclidean-unit-bounded divergence-free vector circulations. It also presents a finite triangle example separating scalar-coordinate ℓ¹ observers from SO(k)-invariant vector observers: the same cycle class receives obstruction magnitude 2 under coordinate-separable ℓ¹ geometry and √2 under Euclidean vector geometry.</p> <p>The result is observer-relative and conditional. It does not claim universal Euclidean physics or universal ℓ¹ geometry. Rather, it identifies the precise two-stage selection mechanism: SO(k) symmetry determines the local ℓ² gauge, while replica-extensive edge aggregation determines the outer ℓ¹ sum.</p> |
| title | SO(k)-Invariant Gauge Classification and the ℓ¹(E; ℓ²) Obstruction Geometry |
| topic | SO(k) invariance M-compatible gauges vector-valued cochains finite graph cochains ℓ¹(E;ℓ²) mixed norms Euclidean gauge obstruction geometry quotient obstruction divergence-free dual witnesses finite obstruction calculus cohomological obstruction theory replica extensivity edge-decomposable diagnostics convex duality Minkowski functional observer-relative geometry |
| url | https://doi.org/10.5281/zenodo.20173318 |