NSVZ-Compatible Three-Loop Gauge $β$-Functions and Regulator-Driven Scheme Structure in Supersymmetric Theories with Exponential Higher Covariant Derivative Regularization

We study the three-loop gauge $β$-functions in general N=1 supersymmetric gauge theories regularized by higher covariant derivatives (HCD) supplemented with Pauli--Villars subtraction. The all-structure three-loop form is known in the HCD framework (e.g.~{Kazantsev2020,Haneychuk2022,Haneychuk2025}) and involves regulator-dependent parameters. Here we evaluate these parameters explicitly for the exponential regulators R(x)=exⁿ and F(x)=exᵐ. We obtain the constants $A(n)$ and $B(m)$ in closed form, together with their large-$n,m$ asymptotics, and substitute them into the general three-loop expressions. This yields fully explicit, regulator-parameterized $β$-functions and a systematic expansion in $1/n$ and $1/m$ that cleanly organizes finite, scheme-dependent terms. We then exhibit finite coupling redefinitions that map the renormalized DR̄ result to an NSVZ-compatible scheme. Our analysis clarifies how exponential higher-derivative regulators preserve the NSVZ relation at the bare level and illustrates the regulator-driven structure of supersymmetric RG flows.