Universal Inclusion of Prescribed Primes in 3x3 Magic Squares

We present an integrated version of the global program proving that every prescribed prime $q_0\ge 5$ occurs in some $3\times 3$ magic square whose nine entries are distinct positive primes. The manuscript explicitly corrects the four points that had prevented the previous version from being regarded as closed: (i) the notation for the fixed prime $q_0$ is now kept uniformly distinct from the notation for the sieve moduli $d$; (ii) the weight convention is unified by working with the function $\text{vt}(n)=\log n$ on the primes and $0$ off the primes, while $\mathrm{\Lambda }$ is used only inside the analytic estimates where it is the natural variable; (iii) the full residual notation $(W,a_W,b_W,S_1,A_d,g(d))$ has been incorporated throughout the manuscript; and (iv) the final closure is replaced by a residual-completion theorem on the \emph{common support of the core}, thereby eliminating the logical gap produced by intersecting two independent theorems.