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Sign / Phase Symmetry: Sometimes the Feature Is a Ray, Not a Vector

A feature can be physically the same mechanism even when its coordinates flip sign or rotate phase, provided the observables only depend on its projective class.

The intuition is simple: if downstream behavior only sees the line spanned by a feature, then \(v\) and \(-v\) are not two mechanisms. They are the same mechanism in opposite coordinates.

Theorem Reference

Lean anchors. SignPhaseGaugeOn, MergeEquivalenceModSign

Math statement.

\rho(\tau x)=\rho(x) \;\Longrightarrow\; x \sim_{\rho^\ast \mathcal O} \tau x.

In English. If a local twist \(\tau\) leaves the feature-class map \(\rho\) unchanged, then any observable that only depends on \(\rho(x)\) must treat \(x\) and \(\tau x\) as the same feature.

Physical intuition. If your readout only depends on projective feature class, then sign/phase partners should be merged before you start telling a mechanistic story.

physics: same mode, different chart engineering: merge/prune modulo sign interpretability: avoid overcounting features

Vector view vs ray view

Many interpretability stories implicitly use the left picture. This theorem family says some should use the right picture instead.

Why This Is Useful

Merging

Cluster neurons or features modulo sign/phase before claiming distinct mechanisms.

Pruning

Projective duplicates can often be treated as redundant realizations of one direction.

Representation geometry

The natural quotient is not Euclidean parameter space but a projective space of rays.

Physics reading: the sign is sometimes gauge, while the ray is the actual physical mode. The theorem tells you when that statement is mathematically justified by the observable family.