A Closed Curve with Finite Critical Menger Curvature and No Tangent at One Point

We give an explicit embedded planar loop of length one with finite integral Menger curvature M_3 whose arclength parametrization is not differentiable at one point. The curve is smooth everywhere else. Its local graph has slope a sin(log log(1/|x|)) plus a smooth term. A direct three-point estimate bounds the critical Menger energy of a Lipschitz graph by the W^(1/3,3) seminorm of its slope. The oscillating slope has finite seminorm, while its secants have two distinct limiting directions. A smooth circle-arc replacement closes the example. This gives a complete negative answer to the everywhere-differentiability subquestion in OWR 22/2013, p. 1353, and Problem 5.3 of Strzelecki and von der Mosel. It does not close the bundled circle-minimization or stationary-regularity questions. Earlier Mobius-energy counterexamples and fractional-Sobolev methods are credited. This is an unrefereed, AI-assisted preprint; priority is not certified.