Three open connections for extending transformation-law theory

Establish a nonlinear analogue of the mixing ratio that uses a local four-block Jacobian decomposition to predict local realisation defects; construct neural architectures that explicitly carry transformation laws and compose defect budgets across depth; and disentangle the selection and reorganisation effects of learning through a controlled intervention.

Background

The paper identifies three unresolved directions for extending its framework. First, its exact mixing-ratio defect law applies to linear encoders, while rectifier encoders are treated through a qualitative crossing decomposition and empirical correlations. The authors propose using the local intertwining condition between encoder Jacobians and transformation-action Jacobians to define a nonlinear, local mixing ratio that could yield a quantitative defect law.

Second, the paper asks whether transformation laws can be made explicit in neural architectures rather than merely measured in pretrained representations. Such architectures would need to carry transformation laws across layers while composing their associated defect budgets, allowing researchers to test whether the observed operability floor is intrinsic to learned representations or instead caused by the way those representations are read out.

Third, the paper reports that transformation structure is partly inherited from input statistics and architecture and partly reorganised by training, but does not separate learning’s selection and reorganisation effects through a controlled intervention. Resolving these connections would extend the theory from descriptive measurement and post hoc construction toward mechanistic explanation and architectural design.

References

Three connections are open, and they are ordered by how much they would change the picture. The first is a nonlinear analogue of the mixing ratio: where the site is differentiable, the exact intertwining condition implies the local condition $J\psi(\tau s)\,D\tau_s=D\rho_\tau(\psi(s))\,J\psi(s)$ on two Jacobians, so a locally defined $\kappa$, built from the local four-block decomposition, should predict the local defect --- which would turn the crossing decomposition of \S{}3.5 into a quantitative local law rather than a decomposition plus a correlation. The second is a carrier architecture: layers that carry transformation laws explicitly, with defect budgets composed across depth the way \S{}5.4 composes them across a chain, would test whether the operability floor of \S{}6.1 is a property of learned representations or of how they are read. The third is the mechanism-level separation of learning's selection from its reorganisation (\S{}4.5) --- currently the inheritance result shows training moves the organisation only modestly while the depth profile moves it a great deal, and the two effects are not yet separated by a controlled intervention.

— Transformation Laws in Neural Representations: Structure, Realisability, and Construction  (2609.18190 - Sun, 16 Sep 2026) in Section 8.6, “The next explanatory step”