Universal Inclusion of Prescribed Primes in 3x3 Magic Squares

Abstract

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 \(\vt(n)=\log n\) on the primes and \(0\) off the primes, while \(Λ\) 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.