A Four-Point Obstruction to Completeness for Three Atomic Inner Functions

For the atomic singular inner functions \(u_{\xi,\alpha}(z)=\exp[-\alpha(\xi+z)/(\xi-z)]\), we construct an explicit four-point functional annihilating all individual nonnegative powers at singularities \(1,-1,i\) with mass \(3\pi\), while detecting \(z^2\). Their linear span is therefore not weak-star dense in \(H^\infty\), not dense in \(H^p\) for any \(0<p<\infty\), and not dense for locally uniform convergence. The certificate also works with the fourth atom at \(-i\) and independent positive integer multiples of the masses. A real smooth Poisson-kernel density annihilates all bilateral boundary powers in \(L^\infty\). Four is the minimal possible support size for an evaluation annihilator associated with three distinct atomic singularities. Disk automorphisms give bad positive masses at every prescribed triple of distinct boundary locations. These complete explicit results disprove universal completeness but do not classify the complete parameter configurations or close the underspecified AIM source question. This is an unrefereed preprint with no absolute priority claim.