Asymptotic equivalence of length complexity and Turing step count in thickened Turing-machine bordisms

Prove that for the dynamical bordism M_G obtained by thickening a given Turing machine graph G and equipped with the flat tube metric g_0, the length complexity LenC_{(M_G,g_0)}(n) of computing on input n is asymptotically equal to the number of Turing steps T(n), i.e., establish that lim_{n→∞} LenC_{(M_G,g_0)}(n) / T(n) = 1.

Background

In the Topological Kleene Field Theory framework, computable functions are represented by dynamical bordisms whose reaching maps realize partial recursive functions. The authors define a geometric “length complexity” LenC based on integrating the norm of the computing flow along its trajectory under a natural piecewise-flat metric g_0 on clean bordisms.

They conjecture that for bordisms constructed by thickening a Turing machine’s finite-state graph, the geometric length complexity matches, asymptotically, the Turing machine’s step count. Establishing this would bridge standard time complexity and intrinsic geometric measures, opening a path to a geometrically grounded complexity theory.