MATH.9:11 - SoTA-Echoing

Question: when does a symmetry-compatible deterministic choice exist, how can it be constructed, and which changed requirement resolves an obstruction?

Milne, Group Theory v4.01, November 2025, chapter 4 and exercise 4-1 with its solution, characterizes maps between transitive group actions through stabilizers. Adopt that mathematical relationship. The argument in :4.2-:4.4 develops it as a construction constrained by the permitted-answer set. Mathlib’s fixed-point action results, including MulActionHom.map_mem_fixedPoints, supply the corresponding necessary preservation property.

Ma and colleagues, A Canonicalization Perspective on Invariant and Equivariant Learning, 2024, §2.1, distinguishes canonicalized inputs from a single equivariant group-valued canonicalizer, which can be obstructed at inputs with nontrivial stabilizers. Adapt this distinction in :4.3/:5.3/:10. It gives a concrete alternative to assuming normalization has selected a unique return transformation.

Dym, Lawrence and Siegel, Equivariant Frames and the Impossibility of Continuous Canonicalization, 2024, §2.2, separates orbit and group canonicalization and examines continuity; §6 retains stabilizer-related conditions when extending weighted constructions to equivariant outputs. Adopt the separate regularity question in :4.4. Their weighted-frame methods address additional geometric-learning requirements; they are not replaced by the finite construction here.

An arbitrary tie-break is cheaper but can fail the requested transformation law. A supplied direct rule with that law is preferable when it already works. The orbit construction earns its cost by revealing impossible outputs or generating a consistent rule. The cases demonstrate those differences; they do not compare runtimes of geometric-learning implementations.

Reopen when the input attributes, action, permitted answer, demanded regularity or available construction changes.