feat(gui): WatchdogState pure-logic helper for first-run modal

Convert gui/src/firstrun.rs to a module dir with watchdog.rs, the
pure-logic WatchdogState helper that the GTK widget code in Task 2
will poll. Three unit tests cover fresh/expired/poke transitions.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

gui/src/firstrun/mod.rs

@@ -1,6 +1,10 @@
 //! First-login modal. Phase 10 implements the body of `run`.
 
-#![allow(dead_code)] // body lands in Phase 10.
+#![allow(dead_code)] // body lands in Phase 10 Task 2.
+
+mod watchdog;
+#[allow(unused_imports)] // wired into run() in Phase 10 Task 2.
+pub(crate) use watchdog::WatchdogState;
 
 use crate::app_context::AppContext;
 

gui/src/firstrun/watchdog.rs

@@ -0,0 +1,55 @@
+//! Tracks the last-activity timestamp; the GTK timer queries this each tick
+//! to decide whether 60s of idle has elapsed.
+
+use std::time::{Duration, Instant};
+
+pub(crate) struct WatchdogState {
+    last_activity: Instant,
+    timeout: Duration,
+}
+
+impl WatchdogState {
+    pub fn new(timeout: Duration) -> Self {
+        Self {
+            last_activity: Instant::now(),
+            timeout,
+        }
+    }
+
+    pub fn poke(&mut self) {
+        self.last_activity = Instant::now();
+    }
+
+    /// True iff `timeout` has passed since the last poke. Once expired,
+    /// stays expired until the caller drops + recreates.
+    pub fn expired(&self) -> bool {
+        self.last_activity.elapsed() >= self.timeout
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn fresh_state_is_not_expired() {
+        let w = WatchdogState::new(Duration::from_secs(60));
+        assert!(!w.expired());
+    }
+
+    #[test]
+    fn zero_timeout_is_immediately_expired() {
+        let w = WatchdogState::new(Duration::from_millis(0));
+        std::thread::sleep(Duration::from_millis(2));
+        assert!(w.expired());
+    }
+
+    #[test]
+    fn poke_resets_the_clock() {
+        let mut w = WatchdogState::new(Duration::from_millis(100));
+        std::thread::sleep(Duration::from_millis(150));
+        assert!(w.expired());
+        w.poke();
+        assert!(!w.expired());
+    }
+}