Analogue of the countable-setting lemma for simultaneous flatness

Prove an analogue of Lemma 3.1 of \cite{KL26} for the local \(S\)-invariant setting that guarantees simultaneous flatness, for all \(r\), of the quotient families \(\big(\mathfrak{a}_{r}(\nu)+\mathfrak{a}_{r}(\nu_0)\big)/\mathfrak{a}_{r}(\nu_0)\).

Background

The paper proves the minimizer theorem using a generic-limit construction over a countable subfield. The authors note that an alternative argument from \cite{KL26} would require a corresponding flatness result for the families of valuation-ideal quotients. They explicitly state that they do not know how to establish this result simultaneously for all real parameters rr, leaving a technical unresolved problem in the theory.

References

A difficulty in importing the argument of to extend Theorem \ref{누영이} is that we do not know how to prove an analog of Lemma 3.1 in our setting. A main problem is that we do not know how to ensure the flatness of $\frac{\mathfrak{a}_{r}(\nu)+\mathfrak{a}_r(\nu_0)}{\mathfrak{a}_r(\nu_0)}$ simultaneously for $r$.

— On the existence of minimizer on a log Fano cone singularity  (2608.23019 - Kim, 24 Aug 2026) in Section 3, subsection “Generic limit argument and proof of the main theorem,” final Remark