Exact Projections of the Real Part of ζ(s) Arithmetic Grids, Identities, Bounds, and a Mean-Square Formula in ℜ(s) > 1

We present a deterministic framework for decomposing the real part of the Riemann zeta function Re(zeta(s)) in the region Re(s) > 1 by means of _arithmetic grids_ (structured subsets of N). We deduce exact identities for multiplicative, additive, and double grids, along with elementary bounds on their contributions and a treatment of the primorial mollifier as a finite Dirichlet polynomial. As a _concrete_ application, we establish a **mean-square formula** for series of the form b(n) = sum_{d|n} a(d): for sigma > 1,lim_{T->inf} (1/(2T)) * int_{-T}^{T} |sum_{n>=1} b(n) / n^(sigma + it)|^2 dt = zeta(2*sigma) * sum_{d,e>=1} (a(d) * overline{a(e)}) / lcm(d,e)^(2*sigma).We provide explicit corollaries for the _additive grid_ (multiples of a finite set of primes) and for the _prime power grid_ with weights. These tools allow for the quantification, isolation, and bounding of layers of Re(zeta(s)) without resorting to unproven hypotheses.