Expected finite k-adic complexity in general integer bases

The manuscript claims $E_N^{k\text{-adic}}<N/2+0.241933621228105$ for every integer $k\ge2$ and $N\ge1$. It gives $E_N^{k\text{-adic}}\le N/2$ for prime-power bases and a strict inequality below $N/2$ for bases with exactly two distinct prime factors. The example $k=30,N=2$ disproves a universal zero-correction bound. Section 1 identifies the standard rational observable with the source question. Written arguments combine classical imports, finite certificates and infinite-tail estimates. The optimal additive constant and finite-length attainment remain open. Constrained FCSR connection integers and rounded register lengths are not identified with this observable. The earlier binary theorem is credited and imported.

The manuscript claims $E_N^{k\text{-adic}}<N/2+0.241933621228105$ for every integer $k\ge2$ and $N\ge1$. It gives $E_N^{k\text{-adic}}\le N/2$ for prime-power bases and a strict inequality below $N/2$ for bases with exactly two distinct prime factors. The example $k=30,N=2$ disproves a universal zero-correction bound. Section 1 identifies the standard rational observable with the source question. Written arguments combine classical imports, finite certificates and infinite-tail estimates. The optimal additive constant and finite-length attainment remain open. Constrained FCSR connection integers and rounded register lengths are not identified with this observable. The earlier binary theorem is credited and imported.