Boundary Layers and Inverse Reconstruction in Canonical Multi-Gap PLRS Numeration

Abstract

We study carry propagation in a family of canonical positive linear recurrence sequence (PLRS) numeration systems whose coefficient vectors consist of alternating blocks of k ones and
prescribed zero runs. We first derive the exact combinatorial structure of maximal legal words directly from the recursive legality definition. This yields an explicit periodic maximal word, an
exact description of the exceptional carry fibres, exact fibre offsets, and the universal identity that the mean carry length is identically two for every member of the family.
Using the associated generating functions, we develop a complete asymptotic expansion for the variance of the carry length. The expansion naturally decomposes into primitive travelling boundary layers determined by the coefficient signature and exponentially smaller corrections
arising from perturbations of the dominant characteristic root. The first moment is shown to be universal, while the higher-order asymptotic boundary layers contain sufficient information to reconstruct the defining coefficient signature, as proved in Section 10.
Finally, we prove an inverse reconstruction theorem. After separating the universal root contribution from the primitive boundary layers, the coefficient signature is recovered recursively from the asymptotic expansion of the variance. This establishes that the asymptotic carry statistics uniquely determine the underlying multi-gap PLRS family.