Reconstruction of Ordered Prefix Languages from Coarsened Carry Traces

This paper develops an exact inverse theory for ordered carry traces on zero-prolongable prefix languages. It asks how much of an underlying legality language can be reconstructed when the reset depth between consecutive legal words is observed only through a possibly noninjective numerical decoder. The paper proves that every strict jump of the decoder reveals a complete ordered cylinder partition. Across all widths, the carry traces determine exactly the language’s -decimated ancestry system, where The complete ordered prefix tree is therefore universally reconstructible precisely when . Under gap-free branching, the traces also reconstruct the labelled language. A visibility graph associated with the decoder gives the exact finite observation horizon required for reconstruction: Matching lower bounds show that this horizon is optimal, even for binary gap-free languages. In the important first-jump case , one has , so traces through width recover the language through depth with no additional look-ahead. The paper also examines hidden state-dependent carry costs, explains why the depth-only assumption is substantive, and identifies uniform cross-state separation as a sufficient condition for extending part of the reconstruction theory.