Do scheduled briefs count toward my analysis limit?

No. Competitive Briefs that Monitoring sends on a schedule don't count toward your monthly analysis limit. They're separate.

Two different things, both shaped like "analysis":

• Analysis runs is the number of times you can run a full analysis on a project per 30 days. Selecting Reanalyze uses one when the run completes successfully.
• Monitoring is what Competely does automatically in the background to spot changes between your analyses. It doesn't draw from your analysis runs. That's why a weekly brief and a monthly analysis run can coexist.

For the difference between the two in more detail, see Analysis vs Monitoring. For how analysis runs work, see how many analyses can I run per month?.