Functorial compatibility of exponent structures

Prove that the diagram relating the maps \gamma and \gamma\{p\} commutes for all approximation stages, thereby obtaining a calculus-based proof that \Omega W^{2n-1}_k and W^{2n-1}(k) have exponent p and recovering the Cohen--Moore--Neisendorfer theorem.

Background

The paper proposes an induction on the Goodwillie approximation degree for proving the exponent-p statement for the fiber W of the double suspension. The induction requires compatibility between chosen exponent retractions and the map \gamma. The authors verify the strategy only in low stages and leave the general commutativity problem open.

References

Can one prove that this diagram commutes in general, and thereby obtain a calculus-based proof of the Cohen--Moore--Neisendorfer theorem?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 1, subsection “Further direction II: the Cohen--Moore--Neisendorfer exponent theorem”

We do not know how to extend this argument to $k\geq 3.

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 1, subsection “Further direction II: the Cohen--Moore--Neisendorfer exponent theorem”