Asymptotics of Erd\H{o}s's \(L^2\) Lagrange Interpolation Problem: Arcsine Distribution and Airy Endpoint Universality

Let \(x_1,\dots,x_n\in[-1,1]\) be distinct nodes and let\[l_k(x)=\prod_{i\neq k}\frac{x-x_i}{x_k-x_i}\]denote the associated Lagrange interpolation polynomials. Erd\H{o}s posed the problem of minimizing the functional\[I(x_1,\dots,x_n)=\int_{-1}^1 \sum_{k=1}^n |l_k(x)|^2\,dx\]and determining its asymptotic behavior as \(n\to\infty\). It was known that\[2-O\!\left(\frac{(\log n)^2}{n}\right)\le \inf I \le 2-\frac{2}{2n-1},\]with the upper bound attained by nodes related to Legendre polynomials. In this paper, we develop a variational framework based on Christoffel functions, orthogonal polynomial asymptotics, and entropy methods to resolve this problem asymptotically. Our main contributions are: \begin{enumerate} \item[(i)] We prove that any asymptotically minimizing sequence of nodes must equidistribute with respect to the arcsine measure on \([-1,1]\). \item[(ii)] We establish a sharp \(O(1/n)\) lower bound, improving the longstanding \(O((\log n)^2/n)\) result of Erd\H{o}s--Szabados--Varma--V\'ertesi. \item[(iii)] We identify that the leading correction arises from microscopic endpoint regions and formulate an \emph{entropy rigidity hypothesis} connecting deterministic minimization to equilibrium log-gas behavior. \item[(iv)] Under a conjectured \emph{endpoint universality} principle for discrete Christoffel functions, we derive the first-order asymptotic expansion \[ \inf I = 2 - \frac{c}{n} + o\!\left(\frac{1}{n}\right), \] with an explicit constant \(c>0\) expressed via the Airy kernel. \item[(v)] We show that the Legendre--integral nodes are asymptotically optimal and rigid, and support all theoretical predictions with detailed numerical experiments, including verification of edge rigidity and Airy-type endpoint scaling.\end{enumerate} The expansion in (iv) is conditional on an endpoint universality conjecture (Conjecture~5.1), whose rigorous proof remains an open problem. A complete verification would finalize the asymptotic solution of Erd\H{o}s's interpolation extremal problem and establish a deeper connection to universality in random matrix theory.