Pure-swirl loss of boundedness under \(L^1_tL^2_x\) forcing: exact integrability of an explicit family

We give an explicit pure-swirl family of forced Navier-Stokes solutions in a bounded circular cylinder with no slip on the lateral wall and free slip on the horizontal faces. For each \(T\gt 0\) and each finite pair \(p,q\geq 1\) with \(1/p+1/q\gt 1\), we construct a corresponding force in \(L^q(0,T;L^p(D))\cap L^1(0,T;L^2(D))\) whose velocity loses \(L^\infty\) boundedness as \(t \uparrow T\). The construction refines the \(k=1\) case of our previous weighted ansatz: we correct the mixed-norm calculation and the convection identity, and impose an annular cancellation to ensure smoothness at the axis. For this family we prove exact mixed-norm membership criteria, including the spatial force endpoint \(p=1\), and two-sided rates for the force, velocity supremum norm and enstrophy. The velocity has a strong \(L^2\) trace at \(T\) and, with this trace, is the unique Leray-Hopf solution on \([0,T]\) and satisfies the energy equality. The same velocity and force solve Stokes with zero pressure.