Gauge-Equivariant Attention for Rotation-Stable 360^circ Scene Understanding

Panoramic 360^circ scene understanding increasingly relies on icosphere transformers, but a state-of-the-art spherical model loses more than half of its segmentation accuracy when the camera rotates by 90^circ, and controlled ablations identify gauge dependence in its relative-position bias as a major contributor. We propose gauge-equivariant relative position encoding (GE-RPE): a parameter-free Reynolds average of the bias over a finite cyclic subgroup Cn!subset!SO(2) of gauge rotations. Plugged into a SphereUFormer backbone the change is invisible at deployment---zero added parameters and 1.5--4.4% forward latency---and the matched three-seed GE-RPE model records a 1.3% drop; the published SphereUFormer checkpoint records 53% under the same stress protocol but a different training recipe. Once the gauge defect is removed and a teacher-token permutation πR aligns the SSL views to the rotated student frame, iBOT+MAE pretraining stops being a liability and becomes a clean low-label lever: the full framework EquiSSL (GE-RPE + πR + iBOT+MAE) tightens the drop to 0.8% at 68.30% val mIoU and lifts 1%-label fine-tuning by +2.39 mIoU on the N=373 test split (and by +4.10 on the smaller N=40 val split); the same fix carries over to monocular depth and to zero-shot Structured3D segmentation. The construction is provably Cn-invariant and O(n-2)-close to the continuous SO(2) average, making the resulting model a usable 360^circ visual-computing primitive across panoramic relighting, immersive video, and cross-dataset transfer. Code is available at https://github.com/Jaywalk18/equissl-release.