Exact 2ⁿ-Block Arithmetic and Prime-Enable Filtering: A Binary-Logical and Hardware-Native Framework

We develop a formal framework for exact arithmetic on block representations of size 2ⁿ, where nonnegative integers are decomposed into finite sequences of n-bit windows. The in-tended primitive viewpoint is binary-logical and blockwise rather than modular: 2ⁿ-blocks servesimultaneously as arithmetic units and as hardware-natural local units. Within this setting we define exact addition, subtraction, multiplication, and division on 2ⁿ-blocks. Division is organized through inherited-remainder block steps, in which each local partialremainder is combined with the next incoming block to form the next extended partial dividend. Divisibility is then reformulated not as a primitive modular notion, but as exact blockwiseclosure under division. This leads naturally to a prime-enable signal and to a conditioned-passarchitecture that outputs N if and only if N satisfies the primality condition in the tested divisorrange. The framework is simultaneously arithmetic, logical, and architectural: it is exact as integerarithmetic, compatible with binary hardware realization, and naturally parallelizable because theunderlying 2ⁿ-blocks form scalable local units. We also explain how the same construction can bereverted to decimal language through explicit integer-valued formulas. The contribution is not anew elementary formula for the nth prime, but a hardware-native and binary-logical organizationof primality that may serve as a primitive basis for further arithmetic reformulations.