Loop special relativity: Kaluza-Klein area metric as a line element for stringy events

Let a physical event constitute a simple loop in spacetime. This in turn calls for a generalized loop line element (=distance² between two neighboring loops) capable of restoring, at the shrinking loop limit, the special relativistic line element (=distance² between the two neighboring centers-of-mass, respectively). Sticking at first stage to a flat Euclidean/Minkowski background, one is led to such a preliminary loop line element, where the role of coordinates is played by the oriented cross sections projected by the loop event. Such cross sections are generically center-of-mass independent, unless (owing to a topological term) the loop events are intrinsically wrapped around a Kaluza-Klein--like compact fifth dimension. Serendipitously, it is the Kaluza-Klein ingredient which, on top of its traditional assignments, is shown to govern the extension of the Pythagoras theorem to loop space. Associated with M₄S₁ is then a ten-dimensional loop spacetime metric, whose four-dimensional center-of-mass core term is supplemented by a six-dimensional Maxwell-style fine structure. The imperative inclusion of a positive (say Nambu-Goto) string tension within the framework of loop special relativity is fingerprinted by a low periodicity breathing mode. Nash global isometric embedding is conjectured to play a major role in the construction of loop general relativity.