<p>Social Announcement Logic (SAL) is a framework for reasoning about belief diffusion in social networks, where information propagates locally based on an underlying follower structure. A key challenge in this area is to logically characterize what agents can achieve through communication, a problem often addressed using powerful but complex arbitrary announcement operators. Previous work on SAL with such operators resulted in infinitary axiomatizations, leaving the question of a finitary system open. This paper solves this open problem by presenting the first finitary, sound, and weakly complete axiomatization for a Social Announcement Logic with an arbitrary sincere announcement operator. Our central technical innovation is a novel, two-stage model transformation technique used to prove the soundness of the crucial discharge rule for arbitrary announcements. This method avoids the complex necessity-form techniques common in the literature on Arbitrary Public Announcement Logic, resulting in a more direct, Henkin-style completeness proof. Overall, the work establishes a robust and computable proof-theoretic foundation for reasoning about belief dynamics in social networks.</p>