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 Deltaₚi (n) = pi (2ⁿ - 1) - pi (2^ (n-1) - 1) and DeltaLi (n) = Li (2ⁿ - 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) = -n² (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 Sₙ = Deltaₚi (2n - 1) + Deltaₚi (2n) and qₙ = (Deltaₚi (2n) - Deltaₚi (2n - 1) ) / (Deltaₚi (2n) + Deltaₚi (2n - 1) ). The imbalance variable qₙ admits a simple empirical law qₙ 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.
Ricardo Adonis Caraccioli Abrego (Sat,) studied this question.