Existence of an FPS-generating type-II family for the Δ(S_{d-2}×S_2) case with m=2

Determine whether a type-II critical-point family with isotropy Δ(S_{d-2}×S_2), two excess neurons (m=2), and a fractional power series representation exists and originates from the known m=1 type-II solution.

Background

The paper presents numerical results for type-II critical points with isotropy Δ(S_{d-2}×S_2) for m=0 and m=1, where m=k-d is the number of excess neurons. The authors do not provide the analogous m=2 family.

The unresolved issue is whether the m=1 construction extends to an m=2 type-II solution possessing a fractional power series representation.

References

Only results are given for $m = 0,1$ as at the time of writing, we are unaware of a type II solution with an FPS which originates from the type II solution with $m = 1$ described below.

— Population loss in shallow ReLU networks: Bias & families of critical points  (2609.30661 - Field, 25 Sep 2026) in Section 14, subsection “Results for isotropy Δ(S_{d-2}×S_2), type II, m=0,1”

There is no difficulty in constructing a path from $d=10$ to $d=9$. However, $d = 8$ provides a challenge and even with a piecewise linear path with several linear segments, we were unable to reach $d = 8$.

— Population loss in shallow ReLU networks: Bias & families of critical points  (2609.30661 - Field, 25 Sep 2026) in Section 15, subsection “Asymmetric targets and fractional power series,” example “Partial asymmetry with a 6×6 asymmetric block”