Hereditary and Inflated Dyadic Residual Chains for Prime Counting and the Logarithmic Integral

We study inter-dyadic residual laws for both prime counting and the logarithmic integral through a framework based on finite-state heredity, local inflation, and scale inflation. At the standard dyadic scale we define Delta_pi(n) = pi(2^n - 1) - pi(2^(n-1) - 1) and Delta_Li(n) = Li(2^n - 1) - Li(2^(n-1) - 1), together with the normalized ratio and scaled residual R(n) = Delta(n)/(2^(n-1)/(n ln 2)) and K(n) = -n^2(R(n) - 1 - (1 - ln 2)/(n ln 2)). On the computed range n = 16,...,50, both series admit finite-state residual descriptions, but only partial determinism at the standard scale. The central observation is that dyadic inflation substantially simplifies the architecture. On the inflated scale alpha = 2, both PIDN and LIDN collapse to the same optimal residual form, namely raw K combined with a deterministic two-memory mod-8 transducer. We also study inflation chains of the form [a,b] -> [a,b+1] -> [a,b+2] -> ..., showing that the automata may remain stable for long stretches and then reorganize in a controlled way, without destroying determinism. For the prime-counting case, the inflated block is naturally invertible through S_n = Delta_pi(2n - 1) + Delta_pi(2n) and q_n = (Delta_pi(2n) - Delta_pi(2n - 1)) / (Delta_pi(2n) + Delta_pi(2n - 1)). The imbalance variable q_n admits a simple empirical law q_n \approx A + B/n, and its residual is exactly captured in range by the same deterministic two-memory mod-8 architecture. The paper is presented as a computational structural study of residual chains, not as a theorem claiming full asymptotic resolution.