Inversion height two in the free field of the free group: strict hierarchies and a non-membership criterion

Let D be the free field of A= C[F₃] and E₂ the ring obtained from A by inverting twice. Two chains of subrings of E₂ are strict at every step: the rings generated over E₁ by inverses of sums of products of at most r inverse binomials (with coefficients in A), whose union is a proper subring of E₂, and the rings generated by A and the entries of the inverses of full matrices over A of size at most d, intersected with E₂. We also locate infinitely many stopping steps of the Dubrovin–Gräter–Hanke complexity series inside E₂, with explicit witnesses; that there are infinitely many stopping steps follows from the work of Herbera and Sánchez, but no infinite family had been located within a fixed stage. The tool is a non-membership criterion that compares one atomic factor with the allowed denominators through invariants that do not grow under passage to factors. MSC 2020: Primary 16K40; Secondary 16S85, 16S10, 16S34, 16E60. Companion site (blueprint): https://karlesmarin.github.io/blueprints/inversion-height-two/