feat(web): done dot on sessions that finished while unviewed

A session that stops running while it is not the selected session arms a
green 'done' reminder dot on its sidebar row, so the operator notices a
finished background session and returns to it; opening the session clears
the dot, and a re-run re-arms it on completion.

SessionManager owns the reminder set (a sibling of the waiting-approval
bit): a running->idle edge of a non-selected session arms it, select()
consumes it, removal prunes it, and it survives connection generations.
The bit rides SessionListEntry/SessionSummary into the workspace browser
rows, which render the existing StateDot done state (running keeps the
spinner) and label the hover card '已完成/Completed'.

packages/client/runtime/src/client/sessions/lineage.ts

@@ -29,6 +29,8 @@ export interface SessionListEntry {
   projectionValues?: Readonly<Partial<SessionProjectionMap>>
   /** User interaction currently blocking this session, derived from live mux frames. */
   pendingInteraction?: PendingInteractionStatus
+  /** Finished running while not selected and not yet opened — the sidebar's green "done" reminder (clears on select or the next run). */
+  completed: boolean
   /** Lineage indent depth: root = 0; the UI just multiplies by the indent width. */
   depth: number
 }
@@ -39,11 +41,13 @@ export interface SessionListEntry {
  * hydrated list from mutable timestamps.
  * @param summaries - the host's session.list items.
  * @param pendingInteractions - current manager-owned interaction status by session.
+ * @param completed - sessions with a pending completion reminder (manager-owned live fact; absent = false).
  * @returns display rows in render order.
  */
 export function flattenLineage(
   summaries: readonly TitledSessionSummary[],
   pendingInteractions?: ReadonlyMap<SessionId, PendingInteractionStatus>,
+  completed?: ReadonlySet<SessionId>,
 ): SessionListEntry[] {
   const byId = new Map<SessionId, TitledSessionSummary>()
   for (const s of summaries) byId.set(s.sessionId, s)
@@ -72,6 +76,7 @@ export function flattenLineage(
     out.push({
       ...s,
       ...(pendingInteraction === undefined ? {} : { pendingInteraction }),
+      completed: completed?.has(s.sessionId) ?? false,
       depth,
     })
     const kids = children.get(s.sessionId)

packages/client/runtime/src/client/sessions/manager.ts

@@ -109,6 +109,14 @@ export class SessionManager {
    *  sessions never instantiated. Cleared per connection generation — the reopen replay re-adds
    *  still-pending requests — and on session-removed. */
   private readonly pendingInteractions = new Map<SessionId, Map<string, PendingInteractionStatus>>()
+  /**
+   * Sessions that finished running while not selected — the sidebar's green
+   * "done" reminder (manager-owned, survives connection generations; cleared
+   * on select and session-removed, re-armed by the next completion).
+   */
+  private readonly completedNotifications = new Set<SessionId>()
+  /** Last-observed running bits per session; the true→false edge here arms {@link completedNotifications}. */
+  private readonly prevRunning = new Map<SessionId, boolean>()
   /** Per-session projection value stores, retained independently of instance arrival (the
    *  title-snapshot precedent, generalized): push frames land here whether or not the Session
    *  is instantiated (list rows read the 'title' key), and an instantiated Session adopts the
@@ -175,6 +183,8 @@ export class SessionManager {
         : this.catalogs.get(address.parentSessionId)?.parentAvailable ?? false,
     )
     this.selected = sessionId
+    // Looking at the session consumes its completion reminder (dot clears).
+    this.completedNotifications.delete(sessionId)
     void this.refreshSubagents(sessionId)
     this.notifier.notifyNow()
   }
@@ -192,6 +202,7 @@ export class SessionManager {
     this.addresses.set(address.childSessionId, address)
     this.sessions.get(address.childSessionId)?.configureSubagent(address, catalog?.parentAvailable ?? false)
     this.selected = address.childSessionId
+    this.completedNotifications.delete(address.childSessionId)
     void this.refreshSubagents(address.childSessionId)
     this.notifier.notifyNow()
   }
@@ -414,13 +425,28 @@ export class SessionManager {
       try {
         const { result } = await this.api.sessions.list({})
         if (result.ok) {
-          let summaries = this.listPhase === 'pending'
+          const baseline = this.listPhase === 'pending'
             ? result.value.items
             : mergeOrderedBaseline(established, result.value.items, summary => summary.sessionId)
-          for (const mutation of mutations) summaries = applyMutation(summaries, mutation)
+          // Seed first observations from the pull-time baseline BEFORE replaying
+          // in-flight mutations, then reconcile the reminders after EVERY
+          // replayed mutation: an edge that happens entirely between mutations
+          // (baseline idle → running → idle) must still arm, which a single
+          // sync on the folded result would collapse away.
+          for (const s of baseline) {
+            if (!this.prevRunning.has(s.sessionId)) this.prevRunning.set(s.sessionId, s.running)
+          }
+          let summaries = baseline
+          for (const mutation of mutations) {
+            summaries = applyMutation(summaries, mutation)
+            this.summaries = summaries
+            this.syncCompletedNotifications()
+          }
           this.summaries = summaries
           this.listState = 'idle'
           this.listPhase = 'ready'
+          // Covers the empty-mutations pull (a plain baseline carries no edge).
+          this.syncCompletedNotifications()
           // Push running/blank bits down to instantiated Sessions (the list is the authoritative summary source).
           for (const s of this.summaries) {
             const session = this.sessions.get(s.sessionId)
@@ -566,6 +592,8 @@ export class SessionManager {
   private recordMutation(mutation: SessionListMutation): void {
     this.listMutations?.push(mutation)
     this.summaries = applyMutation(this.summaries, mutation)
+    // Eager edge reconciliation — a snapshot-build-time pass would miss consecutive status frames.
+    this.syncCompletedNotifications()
     this.notifier.markDirty()
   }
 
@@ -893,6 +921,38 @@ export class SessionManager {
     })
   }
 
+  /**
+   * Reconcile completion reminders against the latest summaries, eagerly after
+   * every mutation and pull (a snapshot-build-time pass would collapse
+   * consecutive status frames into one observation). A running→idle edge of a
+   * non-selected session arms its reminder; running disarms it; removal drops
+   * it. First observation only records the running bit — sessions already
+   * idle at load get no reminder.
+   */
+  private syncCompletedNotifications(): void {
+    const seen = new Set<SessionId>()
+    for (const s of this.summaries) {
+      seen.add(s.sessionId)
+      const prev = this.prevRunning.get(s.sessionId)
+      if (prev === undefined) {
+        this.prevRunning.set(s.sessionId, s.running)
+        continue
+      }
+      if (prev && !s.running) {
+        if (s.sessionId !== this.selected) this.completedNotifications.add(s.sessionId)
+      } else if (s.running) {
+        this.completedNotifications.delete(s.sessionId)
+      }
+      this.prevRunning.set(s.sessionId, s.running)
+    }
+    for (const id of this.prevRunning.keys()) {
+      if (!seen.has(id)) this.prevRunning.delete(id)
+    }
+    for (const id of this.completedNotifications) {
+      if (!seen.has(id)) this.completedNotifications.delete(id)
+    }
+  }
+
   private buildListSnapshot(): SessionListSnapshot {
     const merged: TitledSessionSummary[] = this.summaries.map((summary) => {
       // List rows read the generic 'title' projection key (host-computed unit
@@ -914,7 +974,7 @@ export class SessionManager {
       const status = statuses.find(candidate => candidate !== 'approval') ?? statuses[0]
       if (status !== undefined) pendingInteractions.set(sessionId, status)
     }
-    const fresh = flattenLineage(merged, pendingInteractions)
+    const fresh = flattenLineage(merged, pendingInteractions, this.completedNotifications)
     const items = fresh.map((entry) => {
       const prev = this.entryCache.get(entry.sessionId)
       if (
@@ -924,6 +984,7 @@ export class SessionManager {
         && prev.origin === entry.origin && prev.title === entry.title && prev.depth === entry.depth
         && prev.pendingInteraction === entry.pendingInteraction
         && prev.projectionValues === entry.projectionValues
+        && prev.completed === entry.completed
       ) return prev
       this.entryCache.set(entry.sessionId, entry)
       return entry

packages/client/runtime/src/client/sessions/service.ts

@@ -51,6 +51,8 @@ export interface SessionSummary {
   running: boolean
   /** User interaction currently blocking this session (sidebar amber-dot state). */
   pendingInteraction?: PendingInteractionStatus
+  /** Finished while not selected and not yet opened — the sidebar's green "done" reminder. Absent = false. */
+  completed?: boolean
   /**
    * Empty-log bit (host summary derivation mirror). New Session reuses a blank
    * one targeting the same workspace. Filtering stays with the consumer: the
@@ -614,6 +616,7 @@ export class SessionsService implements ISessions {
         id: entry.sessionId,
         displayTitle: displayTitleOf(entry.title, entry.cwd, entry.sessionId),
         running: entry.running,
+        ...(entry.completed ? { completed: true } : {}),
         blank: entry.blank,
         updatedAt: entry.updatedAt,
         ...(entry.pendingInteraction === undefined