feat: sort both home-screen session lists by activity, not tab order

The phone overview and the desktop tab rail list the same sessions, so
they now share one order (CodemanSessionOrder in constants.js, pure and
unit-tested): blocked on you first (longest-blocked at the top), then
running longest-turn-first, then quiet most-recently-quiet first.

The tiebreak flips direction halfway down on purpose: for a state a
session is still in, longer is more urgent; for a state it has stopped
in, more recent is more relevant. The running group keys off the pane's
last Enter (lastSubmitAt), never lastActivityAt, because a working pane
repaints about once a second and would rank every turn as freshly
started. A 0 stamp means "unknown" and sorts last within its state.

The desktop rail was previously in raw tab order. Its number badge stays
the Alt+1..9 index, so on a sorted rail it deliberately no longer runs
1,2,3 downward: it names a shortcut, not a row position. Its second
stamp changes from "active 3m ago" to the state duration the order is
computed from ("created 1d ago . working 40m"), since both working rows
otherwise read "active just now" and the order looked arbitrary.

The tab strip itself is untouched: still user-ordered and drag-sortable.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

src/web/public/constants.js

@@ -437,6 +437,94 @@ function computeSseStale(input) {
   return now - lastMessageAt >= timeoutMs;
 }
 
+// Home-screen session order: one comparator for both overviews.
+//
+// The phone overview (mobile-overview.js) and the desktop tab rail
+// (home-sessions.js) list the same sessions, so they answer the same question
+// and must answer it the same way: "which of these wants me next?".
+//
+//   1. Anything blocked on a human first (red question, then error, then a
+//      yellow idle prompt), longest-blocked at the top: a session that has been
+//      sitting on a permission dialog for 20 minutes is starving, one that
+//      raised it 5 seconds ago is not.
+//   2. Then whatever is running, LONGEST-RUNNING first, since that is the turn most
+//      likely to be finished, or stuck, by the time you look.
+//   3. Then everything quiet, MOST RECENTLY quiet first: when nothing is
+//      running, the session that just finished is the one you came back for,
+//      and the one you abandoned yesterday sinks.
+//
+// So the tiebreak flips direction halfway down the list, and that is the point:
+// for a state something is still doing, longer = more urgent; for a state
+// something has stopped in, more recent = more relevant.
+//
+// Pure: no DOM, no clock (every input is an epoch-ms stamp already on the
+// session payload), no `this`. Unit-tested in test/session-overview-order.test.ts.
+const SESSION_ACTIVITY_RANK = {
+  needs: 0,
+  error: 1,
+  waiting: 2,
+  working: 3,
+  idle: 4,
+  done: 5,
+};
+
+/** States still in progress, where the OLDEST stamp sorts first. */
+const SESSION_ACTIVITY_OLDEST_FIRST = ['needs', 'error', 'waiting', 'working'];
+
+/**
+ * When the row entered the state it is in.
+ *
+ * For everything quiet that is `lastActivityAt`, the last byte the pane printed:
+ * a Claude pane sitting at its composer prints nothing, so the end of the last
+ * turn is exactly when it went quiet.
+ *
+ * A WORKING pane is the opposite: it repaints about once a second, so its
+ * last-activity stamp is always "now" and would rank every running turn as
+ * freshly started. Its real start is the pane's last Enter (`lastSubmitAt`),
+ * persisted server-side and therefore stable across a Codeman restart. A
+ * working pane that has never submitted (spawned with its prompt on the command
+ * line, or an external CLI) falls back to last activity, which puts it at the
+ * short end of the running group rather than falsely at the head of it.
+ */
+function sessionActivityAnchor(row) {
+  const activeAt = Number(row && row.lastActivityAt) || 0;
+  if (row && row.state === 'working') return Number(row.lastSubmitAt) || activeAt;
+  return activeAt;
+}
+
+/**
+ * Sort comparator for one overview row against another.
+ * @param {{state: string, lastActivityAt?: number, lastSubmitAt?: number, orderIndex?: number}} a
+ * @param {{state: string, lastActivityAt?: number, lastSubmitAt?: number, orderIndex?: number}} b
+ */
+function compareSessionActivity(a, b) {
+  const rankA = SESSION_ACTIVITY_RANK[a.state];
+  const rankB = SESSION_ACTIVITY_RANK[b.state];
+  const rank = (rankA === undefined ? 99 : rankA) - (rankB === undefined ? 99 : rankB);
+  if (rank !== 0) return rank;
+
+  const atA = sessionActivityAnchor(a);
+  const atB = sessionActivityAnchor(b);
+  if (atA !== atB) {
+    // A row with no stamp at all gets no opinion: it sorts last either way
+    // rather than claiming to be the oldest (0) thing on the screen.
+    if (!atA) return 1;
+    if (!atB) return -1;
+    return SESSION_ACTIVITY_OLDEST_FIRST.includes(a.state) ? atA - atB : atB - atA;
+  }
+
+  // Equal stamps (or two unstamped rows): fall back to the user's tab order so
+  // the list is deterministic and cannot shuffle between renders.
+  const orderA = Number.isFinite(a.orderIndex) ? a.orderIndex : Number.MAX_SAFE_INTEGER;
+  const orderB = Number.isFinite(b.orderIndex) ? b.orderIndex : Number.MAX_SAFE_INTEGER;
+  return orderA - orderB;
+}
+
+/** Copy of `rows`, in overview order. Never sorts in place, so callers keep their array. */
+function sortSessionsByActivity(rows) {
+  return (Array.isArray(rows) ? rows.slice() : []).sort(compareSessionActivity);
+}
+
 if (typeof window !== 'undefined') {
   window.WEBGL_FALLBACK = WEBGL_FALLBACK;
   window.evaluateWebGLLongTaskTrip = evaluateWebGLLongTaskTrip;
@@ -464,6 +552,12 @@ if (typeof window !== 'undefined') {
     compute: computeSseStale,
     TIMEOUT_MS: SSE_STALE_TIMEOUT_MS,
   };
+  window.CodemanSessionOrder = {
+    RANK: SESSION_ACTIVITY_RANK,
+    anchor: sessionActivityAnchor,
+    compare: compareSessionActivity,
+    sort: sortSessionsByActivity,
+  };
 }
 
 // Scheduler API — prioritize terminal writes over background UI updates.