A Dyadic Framework for Arithmetic Operations and a Binary Criterion for Primality via the Power-of-Two Transform

We present a dyadic framework for reinterpreting elementary arithmetic operationson nonnegative integers through the transformP2 (n) = 2ⁿ. Under this reading, integers are represented as positions of an active bit, addition andsubtraction become exponent shifts, multiplication becomes an iteration of shifts, anddivision becomes a block or grid-based reading. As an application, we study the wordRn = 2ⁿ − 1, consisting of n consecutive ones, and obtain a binary criterion for primality: n is primeif and only if Rn admits no nontrivial reconstruction from smaller blocks by meansof shifts and OR. Equivalently, n is prime if and only if Rn is irreducible under exactrepetition of smaller blocks Rd = 2ᵈ − 1 with 1 < d < n. The aim is not to claim historical novelty for the underlying isomorphism, but toestablish an operational and geometric reading in which divisibility, compositeness, and primality appear as properties of binary blocks, periodic masks, and exact wordreconstruction.