Strong Approximate Innerness for Diagonal Graph Actions without Aperiodicity

Every finite abelian action on the C*-algebra of a finite strongly connected noncycle graph that fixes vertices and scales edges by characters is strongly approximately inner. Neither graph aperiodicity nor action outerness is assumed. The proof uses Barlak and Szabo's aperiodic outer-action theorem as its analytic input, removes the graph period through a return-path corner and fixed-frame lifting, and removes the inner kernel through vertex coboundaries and extension of subgroup characters. A block-diagonal polar correction restores fixed implementers for the original action. The result answers the finite abelian diagonal quasi-free action question in Oberwolfach Report 42/2017, p. 2621, for simple purely infinite Cuntz-Krieger algebras. This is an AI-assisted, self-audited, unrefereed preprint; no formal verification, independent human review or absolute priority is claimed.