Convergence range of the type-A fractional power series

Establish whether the fractional power series for the type-A family of spurious critical points converges for every dimension in which the type-A solutions exist, rather than only for sufficiently large dimension.

Background

The paper reviews a type-A family of critical points for the bias-free ReLU student–teacher model with isotropy ΔS_d. The family is represented by convergent fractional power series in d{-1/2}, and numerical evidence suggests that the series converges throughout the full range of dimensions for which the type-A solutions exist.

The cited implicit-function-theorem argument establishes convergence only for sufficiently large d. Thus, the convergence of the displayed fractional power series at smaller admissible dimensions remains unresolved.

References

The series appears to converge to the critical point for all $d$ for which type A solutions exist but the existence proof\S 8.3, based as it is on the implicit function theorem, only gives convergence for ``sufficiently large'' $d$.

— Population loss in shallow ReLU networks: Bias & families of critical points  (2609.30661 - Field, 25 Sep 2026) in Section 13, subsection “A family of spurious minima with isotropy ΔS_d, k=d”