Mersennian Resonance in Dyadic Additive Blocks Canonical Coordinates, Divisor Control, and a Connection with Fermat Numbers

This paper introduces a dyadic block framework for studying divisibility patterns near numbers of the form 2^s - 1. For each integer s >= 1, the interval from 2^s - 1 to 2^(s+1) - 2 is written as N = (2^s - 1) + r. The main result proves that the common divisor between N and r is exactly the same as the common divisor between 2^s - 1 and r. Therefore, all resonance values inside the block are determined only by the divisors of 2^s - 1. This gives an exact divisor-residue structure for every dyadic block. The paper derives formulas for the number and density of resonant composite residues, describes the special collapse that occurs when 2^s - 1 is prime, and connects the block density with Euler's totient function. A further result shows that, when s is a power of two, the condition that the block density equals one half is equivalent to the primality of the corresponding initial Fermat numbers. The contribution is not presented as a new factorization algorithm, but as an exact structural decomposition showing how the arithmetic of 2^s - 1 controls local divisibility phenomena in its dyadic block.