The paper states
Theorem 1.1
In A group without fixed price, 5 October 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Put K=e^{96}96^{99}/95^{95}, choose
0<\alpha<1/200,\qquad K\alpha^3<1/2,
\qquad \eta=\alpha/100.
For the group and actions just defined,
\operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright X})
\ge1+\eta,
\qquad
\operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright Y_M})
\le1+\frac{99}{M}\quad(M>100).
In particular, choosing an integer M>\max\{100,99/\eta\} gives two essentially free p.m.p. actions of the same finitely generated group with strictly different costs. Thus the general fixed-price assertion is false.