A Dyadic Numerical Method for the Prime Counting Function: Logarithmic Compression, Reversible Encoding, and Hereditary Automaton Structure

This paper presents a discrete numerical method for the prime counting functionbased on dyadic sampling at the nodesx^n = 2^n − 1, P (n) = π(2^n − 1).The method has three main components.First, dyadic sampling yields exact cumulative prime counts at dyadic nodes andexact prime counts in each dyadic interval, while requiring only N + 1 samples to coverthe range up to 2^N − 1.Second, the liftingY (n) = 2^P (n) − 1is exactly reversible throughP (n) = log2(Y (n) + 1),so no information is lost when passing from the dyadic prime-counting sequence to theexponential scale.Third, after normalization of the inter-dyadic counts, a sign-oscillation layer isobtained that is captured exactly, on the computed range up to 2^25 −1, by a deterministicfinite automaton with hereditary block-to-block state transfer. The observed automatonremains deterministic on the full tested range, the positive-sign regime persists fromn = 9 onward in the computed data, and the extension from n = 24 to n = 25 enlargesthe state space by only one new state while preserving determinism.The result is not presented as a universal closed formula for primes. Rather, it is afinite, exact, and highly compressed dyadic state architecture for prime-counting data,together with a hereditary propagation mechanism across computed dyadic blocks.