Integrality of the limiting varifold family

Prove that the family of limiting varifolds associated with the Allen–Cahn energy measures is integral.

Background

The convergence argument identifies, for almost every time, a unique rectifiable varifold associated with the limiting energy measure. Rectifiability alone does not imply integrality, which would require integer-valued multiplicity in the varifold representation.

While proving measurability of the limiting varifolds, the authors explicitly acknowledge that integrality has not been established. Consequently, the measurability proof is formulated only for the portions that do not require integrality.

References

However, since the integrality of our family of varifolds ${\mu_t}_{t\geq 0}$ remains open, we prove only the parts where the arguments differ.

— Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow  (2608.26842 - Laux et al., 27 Aug 2026) in Proposition 3.?, proof of Proposition \ref{prop:measurability}, Section 3, Proof of Theorem \ref{thm:ACtoBrakke}