Scalarized Einstein-Euler-Heisenberg black holes at the approach to extremality

We investigate extremal black holes with scalar hair in the generic Einstein-Euler-Heisenberg (EEH)-scalar theory with two scalar couplings to the Maxwell term. One is an exponential coupling with coupling constant $α$ and the other is its polynomial coupling. We first construct scalarized black holes on the cold (C)-horizon of EEH black holes described by mass M and magnetic charge P both at the linear level, through the existence curves αₙ(q) with $q=P/M$, and as fully backreaction solutions. As extremality (q=qₑ) is approached, all existence curves accumulate, following αₙ-αc~ 1/ln²(qₑ-q), at the critical branch αc fixed by the Breitenlohner-Freedman bound for the near-horizon (AdS₂× S²) throat. We obtain scalarized extremal black hole (SEBH) with constant secondary hair and it is recovered exactly from the entropy function approach working on the near-horizon throat. Its entropy is an attractor invariant, being independent of the asymptotic modulus, so the scalar hair remains secondary. Exploiting this constant scalar, we seek further SEBHs with scalar hair keeping its charge Qₛ for the zero asymptotic scalar (ϕ_∞=0). In the $(q,α)$ plane, we observe that these extremal solutions occupy q≥ qₑ. Hence, the existence curves αₙ(q) existing for q≤ qₑ and the extremal branches bound the scalarized domain from opposite sides and they meet only at q=qₑ.