Consider for example a non archimedean field $K$, then one can take: $$ R = K\langle T^{\pm \frac{1}{p^\infty}}\rangle $$ so in practice if I'm not wrong we' re taking: $$ R \cong \varinjlim_{n}(K[ T^{ \frac{1}{p^n}}, T^{ -\frac{1}{p^n}}])^\wedge $$ Now, not thinking so much to the formal definition (in which i'm not sure) in practice we're taking all the $p$-poter root of symbols $T$ and $T^{-1}$.
Clearly in the theory of perfectoids appears usually the tilting functor, so: $$ K^\flat := \varprojlim_{x \mapsto x^p}K/pK $$ My question is: if I consider in the ring $K\langle T^{\pm \frac{1}{p^\infty}}\rangle$ and the series of Frobenius maps. The tilt of $R$ is: $$ R^\flat \cong K^\flat\langle T^{\pm \frac{1}{p^\infty}}\rangle \hspace{0,5 cm}?? $$