Why it is a question
A34 §6G: “A raw parameter gradient, gradient angle, or norm value can depend on the chosen coordinates.” “An architecture-sensitive modular norm is more informative than an arbitrary Euclidean parameter norm, but it is still a specified geometry; it is not automatically invariant under every function-preserving reparameterization.” A34: “Without it, some supposed differences in organizational strain may reflect coordinate choices rather than different learning dynamics.” The owner, at U35: “interesting question, double click on this.”
A35 §2 sorted five diagnostics:
| Diagnostic | A35’s expected invariance |
|---|---|
| Pairwise Euclidean gradient angles | Not invariant under general reparameterization |
| Common strict first-order descent, all tangent directions allowed | Invariant under smooth invertible coordinate changes |
| Budgeted feasibility, update geometry transported | Locally invariant under the same changes |
| Actual finite-step trajectories | Not necessarily invariant unless the update rule and its state transform too |
| Feasibility of specified changes to the represented function | Invariant under function-preserving parameterization, with the function class and allowed changes fixed |
CLAIM-366 proves the first three rows. It does not answer the question, for two reasons.
- Function-preserving symmetries are not charts. Permuting hidden units, rescaling across a ReLU, or the orthogonal map on a transformer’s residual stream (THEORY-112) relate distinct points that compute the same function. Quotienting by them (CLAIM-263’s protocol) is not a diffeomorphism of one parameter space, and at singular points the Fisher metric that would supply a principled geometry is degenerate (THEORY-082; CLAIM-305: a blow-up adds no observation). Whether a frustration verdict is constant on symmetry orbits is open.
- Trajectories carry the optimizer. A35: “two parameterizations can compute the same current function while yielding different future trajectories under SGD, Adam, weight decay, or another fixed optimizer.” And: “If we are evaluating an actual training system, optimizer dependence is part of what we’re trying to explain—not necessarily a nuisance to quotient away.”
A39 §5 restated it as an open question: “Which frustration diagnoses survive reparameterization or quotienting by function-preserving symmetries?” A40 §3.5 split it into three invariances: coordinates, function-preserving symmetries, and future learning dynamics.
What would count as an answer
For each diagnosis in use (the compatibility complex of CLAIM-347, the residual Φ_r of CLAIM-365, governance regret, modular localization), a statement of which group it is invariant under, with a proof or a counterexample, and so whether it is a property of the represented function, of a chosen update geometry, or of one training organization with its optimizer. A40 §3.5 named the target: “Determine which frustration diagnoses are properties of the represented function, which belong to an update geometry, and which belong to a particular training organization.”
Note of 2026-10-27: the coordinate part split off
The status review of this date split this question under ADR-049. Changes of chart are QUESTION-286, Answered by CLAIM-366. This question keeps what is still open: function-preserving symmetries, which are not charts of one space, and trajectories, which carry the optimizer.