A Binary-Logical Foundational Framework for Reconstructing Arithmetic: AND, OR, NOT, 2^n-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 2^n. 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 2^n-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.