Model for the Goodwillie approximations of S^0

Construct a model for the Goodwillie approximation S^0_k at the prime 2 and determine whether S^0_k\simeq S^0\times\Omega^{\infty+1}S^1(k).

Background

The paper proposes a splitting for S0_k at p=2 suggested by the calculus EHP sequence and the model for S1_k. This is related to Dwyers question about whether the Goodwillie tower provides a contracting homotopy for the mod-2 Whitehead complex. The paper proves convergence of the tower independently but does not establish the proposed model.

References

Is there a similar model for $S0_k$? Is it true that $S0_k \simeq S0 \times \Omega{\infty+1}S1(k)$ at $p=2$?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 2, subsection “The model”