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

This paper studies the prime counting function through dyadic sampling at the nodesxn = 2^n − 1, P (n) = π(2^n − 1).The resulting dyadic prime-counting sequence is shown to possess four structural features.First, the dyadic samples determine exactly the cumulative prime counts at dyadicnodes and the exact number of primes in each dyadic interval.Second, the liftingY (n) = 2^P (n) − 1is exactly reversible throughP (n) = log2(Y (n) + 1).Third, the inter-dyadic prime counts∆(n) = P (n) − P (n − 1)are normalized by the dyadic prime-number-theorem backboneF (n) = 2^n−1n ln 2 ,leading to the ratioR(n) = ∆(n)F (n) .On the computed range up to 225−1, the data show that R(n) > 1 for every n = 9, . . . , 25,and that the two-term approximationR(n) ≈ 1 + (1 − ln 2)/(n ln 2)is highly accurate on the tested tail.Fourth, after removing this two-term correction, the scaled residualK(n) = −n2*(R(n) − 1 − (1 − ln 2)/(n ln 2))still exhibits hereditary structure by dyadic blocks when quantized into finitely manybands. This yields a deterministic hereditary transducer for the fine correction layer.The resulting picture is not presented as a universal closed formula for primes. Rather,it is a finite, exact, and highly compressed dyadic state architecture for prime-countingdata, together with a hereditary propagation mechanism across computed dyadic blocks.