Dyadic Inheritance, Inflation, and Exact Coarse–Fine Reconstruction with Applications to Prime Counting and the Logarithmic Integral

We develop a unified dyadic framework based on binary scale, binary closure, Euclidean inheritance, inflation, and exact coarse–fine reconstruction. The structural part of the paper is purely exact: it establishes the dyadic inheritance law, proves that inflation is exactly grouped inheritance, and shows that fine and coarse descriptions belong to a single compatible architecture. We then apply this framework to dyadic increment chains for prime counting and the logarithmic integral, defined by Δπ(n) = π(2^n − 1) − π(2^(n−1) − 1) and ΔLi(n) = Li(2^n − 1) − Li(2^(n−1) − 1). For the inflated prime-counting chain we prove exact invertibility through the variables S_n = Δπ(2n−1) + Δπ(2n), D_n = Δπ(2n) − Δπ(2n−1), and q_n = D_n / S_n, whenever S_n ≠ 0. This shows that the pair of fine dyadic prime increments is recovered exactly from the inflated block together with its normalized asymmetry. Finally, we formulate the dyadic automaton problem in exact symbolic terms, replacing empirical threshold quantization by canonical dyadic codings. In this way the paper isolates a theorem-level exact core, while the finite-state closure question is stated as a precise mathematical problem and a focused conjectural direction.