A Binary-Logical Foundational Framework for Reconstructing Arithmetic: AND, OR, NOT, 2ⁿ-Blocks, and Finite Structural Support

We propose a methodological framework in which classical arithmetic is reconstructed as aderived layer of a more primitive binary-logical system. The intended primitive basis consistsof the logical operations AND, OR, and NOT, together with organization into exact blocks ofsize 2ⁿ. Within this setting, derived binary constructions such as XOR and two’s complementsupport the reconstruction of addition, subtraction, multiplication, and division. The resulting framework is not presented as a replacement for standard arithmetic, nor as aclaim of novelty about binary computation itself, which is already standard in digital hardware. Its contribution is conceptual and methodological: it explicitly frames binary logic and 2ⁿ-blockorganization as a possible primitive layer from which ordinary arithmetic may be rebuilt andstudied. In this viewpoint, many concrete computations become concentrated on a finite supportof active blocks and inherited states, while infinity remains as a background mathematical horizonrather than the direct carrier of the computation. The paper formalizes the block framework, proves the correctness of inherited-remainderblock division, states an equivalence principle with ordinary arithmetic, and discusses themethodological consequences of taking binary-logical structure as foundational. The goal is notto proclaim a final foundation for mathematics, but to articulate a coherent, rigorous, and fertilepoint of view from which arithmetic and related structures may be reorganized.