Brick infinite algebras admit infinitely many non-$\tau$-rigid bricks

Let $A$ be a finite dimensional algebra over an algebraically closed field.

Motivated by some foundational interactions between bricks and $\tau$-rigid modules, we prove, in full generality, that if all but finitely many bricks of the algebra $A$ are $\tau$-rigid, then $A$ is brick-finite.

Equivalently, any brick-infinite algebra admits infinitely many bricks which are not $\tau$-rigid.

Because $\tau$-rigidity implies rigidity, our result verifies a weaker version of an open conjecture which states that if (almost) all bricks over $A$ are rigid, then $A$ should be brick-finite.

In retrospect, this work strengthens some previous results and contributes to the recent studies of a series of challenging problems, all tied to the $2$nd brick-Brauer-Thrall conjecture.

More specifically, without any tameness assumption, we settle a question that was previously known only for $E$-tame algebras.