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 \(g\)-decimated ancestry system, where
The complete ordered prefix tree is therefore universally reconstructible precisely when \(g=1\). 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 \(D(1)>D(0)\), one has \(g=\rho_D=1\), so traces through width \(M\) recover the language through depth \(M\) 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.