Depth Invariance of Centered Carry Counts in Fibonacci Lifts

This paper establishes an exact depth-invariance theorem for centered carry counts in Fibonacci digit lifts in base 32. Starting from a fixed digit depth, we consider complete groups of Fibonacci additions whose indices are separated by the corresponding Pisano period. While the lower digits remain unchanged under these period shifts, the higher digits evolve according to an exact recursive lifting structure. The main theorem proves that, for any number of additional digit positions, the number of carries differs from half the size of the corresponding lift group by exactly the same amount as in the first lifting step. This deviation is independent of the number of additional digit positions, the starting digit depth, and the starting Fibonacci index, and its absolute value never exceeds two. Consequently, the entire frequency distribution of these centered carry deviations remains exactly invariant across arbitrarily many additional lift depths. A stronger bound holds for three of the six index classes modulo 6, where the absolute deviation never exceeds one. The proof is entirely algebraic and relies on Fibonacci matrix recurrences, exact 2-adic valuations, modular residue-class enumeration, and arithmetic progression identities. An infinite family of higher-digit carry-count problems is thereby reduced to a depth-independent arithmetic structure. The paper also establishes exact balance properties of higher digit positions within complete lift groups. A Python verification program is included to provide reproducible checks of selected finite cases. The work builds on established research concerning Fibonacci residues, Pisano periods, and stability phenomena in modular recurrence sequences. Its specific contribution is the pointwise invariance of centered carry counts across arbitrary lift depths and the resulting invariance of their frequency distributions. All results concern complete modular lift groups. They do not establish equidistribution over short consecutive Fibonacci-index intervals or normality of Fibonacci digit concatenations.