A Layered Dyadic Architecture for the Inter-Dyadic Prime Counting Function

We study the inter-dyadic prime count Delta(n) = pi(2^n - 1) - pi(2^(n-1) - 1) through a layered discrete architecture. The first layer is the dyadic prime-number-theorem backbone F(n) = 2^(n-1)/(n ln 2). The second layer is the normalized ratio R(n) = Delta(n)/F(n), which is well approximated on the computed range by R(n) ≈ 1 + (1 - ln 2)/(n ln 2). The third layer is the scaled residual K(n) = -n^2(R(n) - 1 - (1 - ln 2)/(n ln 2)). Rather than fitting K(n) on arbitrary windows, we organize the analysis in natural dyadic blocks of the index n, namely [2^k, 2^(k+1) - 1]. On the block k = 4 (n = 16,...,31), K(n) exhibits a structured transitional regime, while on the block k = 5 (n = 32,...,50 in the available data) it enters a numerically stabilized tail near -0.28. In each block, K(n) is modeled empirically by a class-dependent modulation K(n) ≈ A_k(r) + B_k(r)(-1)^n/n, where r = n mod 4. After subtraction of this blockwise analytic correction, the remaining residual, quantized into bands, is governed by a deterministic finite-state transducer: on the first block it requires two steps of memory, whereas on the second block one-step memory suffices and the residual is almost entirely absorbed into the central band. The result is presented as a computationally observed dyadic state architecture, not as a theorem.