Surface unread counts on sidebar launcher rows

The design's Slack/Linear/Gmail rows show numeric badges (12, 3) that
real apps publish in their tab title prefix — "(12) Slack | …",
"(3) Inbox — Linear", "(99+) Gmail". Add a pure parser
ely_domain::parse_title_unread_count that recognises the leading
"(N)" pattern and surfaces it through BrowserTab::unread_count(); the
sidebar launcher row renders a glass pill badge whenever the count
is non-zero, capped at "99+" for very high counts.

No domain field, no fabrication: the badge appears only when a real
website publishes its own unread count via the title. Tests cover
canonical formats, non-prefixed titles, leading whitespace, and
non-numeric / overflow inputs.

crates/ely_app/src/shell/chrome/sidebar.rs

@@ -220,6 +220,7 @@ impl ElyShell {
         let host = tab.url().host().map(|host| host.to_string());
         let title = tab.title().to_string();
         let initial = title.chars().next().unwrap_or('?').to_string();
+        let unread = tab.unread_count();
         let group_name = SharedString::from(format!("launcher-{}", tab.id().as_str()));
         let close_id = SharedString::from(format!("launcher-close-{}", tab.id().as_str()));
 
@@ -252,6 +253,7 @@ impl ElyShell {
                     .text_color(rgb(text_color))
                     .child(title),
             )
+            .when(unread > 0, |el| el.child(render_unread_badge(unread)))
             .child(
                 div()
                     .id(close_id)
@@ -354,6 +356,24 @@ fn profile_initial(name: &str) -> String {
         .to_string()
 }
 
+fn render_unread_badge(count: u32) -> impl IntoElement {
+    let label = if count > 99 {
+        "99+".to_string()
+    } else {
+        count.to_string()
+    };
+
+    div()
+        .px(px(6.0))
+        .py(px(1.0))
+        .rounded(px(999.0))
+        .bg(rgba(UNREAD_BADGE_BG))
+        .text_size(px(10.0))
+        .font_weight(gpui::FontWeight(500.0))
+        .text_color(rgb(colors::INK_3))
+        .child(label)
+}
+
 fn section_tabs_label(count: usize) -> impl IntoElement {
     div()
         .pt(px(12.0))
@@ -389,6 +409,7 @@ fn section_label(label: &'static str) -> impl IntoElement {
 
 pub(crate) const ACTIVE_NAV_BG: u32 = 0xffffffd9;
 pub(crate) const PANEL_BG: u32 = 0xffffffe0;
+const UNREAD_BADGE_BG: u32 = 0x281e140f;
 
 pub(crate) fn panel_shadow() -> Vec<BoxShadow> {
     vec![

crates/ely_domain/src/lib.rs

@@ -64,7 +64,7 @@ pub use sync::{
 };
 pub use tab::{
     BrowserTab, DEFAULT_ZOOM_PERCENT, MAX_ZOOM_PERCENT, MIN_ZOOM_PERCENT, TabFlags, TabState,
-    ZOOM_PERCENT_STEP, validate_zoom_percent,
+    ZOOM_PERCENT_STEP, parse_title_unread_count, validate_zoom_percent,
 };
 pub use tab_group::TabGroup;
 pub use update::UpdatePolicy;

crates/ely_domain/src/tab.rs

@@ -106,6 +106,11 @@ impl BrowserTab {
         &self.title
     }
 
+    #[must_use]
+    pub fn unread_count(&self) -> u32 {
+        parse_title_unread_count(&self.title)
+    }
+
     #[must_use]
     pub fn url(&self) -> &UrlText {
         &self.url
@@ -274,3 +279,46 @@ pub fn validate_zoom_percent(value: u16) -> Result<u16, DomainError> {
 
     Err(DomainError::InvalidZoomPercent { value, min: MIN_ZOOM_PERCENT, max: MAX_ZOOM_PERCENT })
 }
+
+#[must_use]
+pub fn parse_title_unread_count(title: &str) -> u32 {
+    let trimmed = title.trim_start();
+    let Some(rest) = trimmed.strip_prefix('(') else {
+        return 0;
+    };
+    let Some(close) = rest.find(')') else {
+        return 0;
+    };
+    rest[..close].parse::<u32>().unwrap_or(0)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::parse_title_unread_count;
+
+    #[test]
+    fn parses_leading_parenthesised_count() {
+        assert_eq!(parse_title_unread_count("(12) Slack"), 12);
+        assert_eq!(parse_title_unread_count("(3) Inbox — Linear"), 3);
+        assert_eq!(parse_title_unread_count("(0) Quiet"), 0);
+    }
+
+    #[test]
+    fn ignores_non_prefixed_titles() {
+        assert_eq!(parse_title_unread_count("Slack"), 0);
+        assert_eq!(parse_title_unread_count("Inbox (12)"), 0);
+        assert_eq!(parse_title_unread_count(""), 0);
+    }
+
+    #[test]
+    fn rejects_non_numeric_or_overflow_counts() {
+        assert_eq!(parse_title_unread_count("(99+) Gmail"), 0);
+        assert_eq!(parse_title_unread_count("(abc) Mail"), 0);
+        assert_eq!(parse_title_unread_count("(99999999999999999999) huge"), 0);
+    }
+
+    #[test]
+    fn tolerates_leading_whitespace() {
+        assert_eq!(parse_title_unread_count("   (7) Slack"), 7);
+    }
+}