Natural Dyadic Closures, Prime Profiles, and Exact Hereditary Divisibility

We introduce a dyadic-hereditary perspective on divisibility by odd primes. For each oddprime p, we define its natural dyadic closure lengthLₚ = ordₚ (2), and interpret it as the minimal binary block size at which divisibility by p becomes exactlyhereditary under block decomposition. This leads to the notion of a prime profile (p, Lp). The main result is an exact hereditary divisibility theorem: if an integer is decomposed intobinary blocks of length Lp, then divisibility by p is determined exactly by hereditary accumulationof the block values. We also prove a simple structural law for Mersenne primes: if p = 2m − 1 isprime, then Lₚ = m. These notions induce a classification of primes into dyadically heavy and dyadically favor-able families according to the ratio Lₚ/ (p − 1). The heavy case Lₚ = p − 1 is exactly theclassical condition that 2 be a primitive root modulo p, connecting the framework to standardmultiplicative-order theory and Artin-type questions. We provide a numerical example, a tableof prime profiles, and a brief computational summary showing that early dyadic closure is notrare.