[cmpadden]

Summary

• Move Pi pane aggregation and lease checks out of synchronous tmux status-format expansion into a background helper.
• Use tmux's built-in #() job cache to retain the last badge while a refresh runs, without cache files or a daemon.
• Keep the existing pane-local Pi status protocol and priority ordering unchanged.
• Add control-client regression coverage for blocked refreshes, expired badges, and preserved window labels.

Context

Investigated intermittent window-label flickering. tmux 3.7c shares a 100 ms format-expansion budget across nested window rendering; artificially exceeding that budget reproduced missing labels, though the live flicker was not directly captured. This change removes repeated pane scans from that rendering path.

Validation

• All seven tmux-status tests pass, including the real-tmux integration test.
• make format and the helper's shfmt check pass.
• make lint and commit hooks pass with ShellCheck supplied through an ephemeral uvx --from shellcheck-py environment.

Rollout

Reload tmux with prefix+r. Existing Pi processes do not need to reload; badge updates normally appear within about a second.