Length profiles of graphs in expanding squares: an exact realization theorem

Which functions describe the Euclidean length of a continuous graph inside every square centered at the origin? We characterize the finite continuous profiles that vanish at zero and are realized by continuous graphs on the real line through the origin: subtracting the square's side length must leave a nondecreasing function. Every such profile has an even, nonnegative realization locally of bounded variation, below any prescribed positive continuous height bound on (0,∞) while remaining below |x|; singular continuous profiles are included. If the target is locally absolutely continuous, a locally absolutely continuous graph can be chosen. A sign-balancing construction proves sufficiency and separates height from growth. We also prove that finite even convex graphs through the origin are determined by their profiles, except for reciprocal V-shaped pairs; differentiability at the origin gives uniqueness. Finally, a jump formula exhibits an obstruction absent from continuous targets, and the tangent at the origin yields a necessary condition for exact C1 realization. 2020 Mathematics Subject Classification: 26A45; 28A75; 26A51. The accompanying archive contains the LaTeX source, figure data, and scripts for checking the worked examples.