Carry–Run Theorem and Sakib Index for the Exact Distribution of ν_p( (2n choose n) ) over n mod p^k

Kummer’s theorem identifies the p-adic valuation of a binomial coefficient with the number of carries in base-p addition. Specializing to the central binomial co- efficient 2n n
converts the valuation νp 2n n
into the carry count of doubling n in base p. This manuscript develops an exact finite-state carry-chain representation for doubling, and packages the resulting distribution over n mod pk into explicit closed forms and rational generating functions. We further propose a normalized carry-density statistic, the Sakib Index, to compare valuation intensity across (p, k) at scale. All reported numerical plots are based on deterministic enumeration / ex- act recurrences (no Monte Carlo) and are included as dataset-based figures, along with a small set of conceptual diagrams that summarize the pipeline.