Show live background task counts

app/androidApp/src/main/kotlin/com/example/aiapp/SessionScreen.kt

@@ -256,6 +256,8 @@ fun SessionScreen(
     val topEdgeHeld = remember { TopEdgeHold() }
     var items by remember { mutableStateOf(listOf<TranscriptItem>()) }
     var status by remember { mutableStateOf(subagent?.status ?: summary.status) }
+    var backgroundTasks by
+        remember(address) { mutableIntStateOf(if (isSubagent) 0 else summary.backgroundTasks) }
     // Seeded from the row this screen was opened from, so a conversation already under way says how
     // much it is holding before any turn happens here. Null is "nobody has measured it", which is a
     // different answer from an empty context and is drawn differently.
@@ -538,6 +540,9 @@ fun SessionScreen(
                         }
                     status = event.state
                 }
+                if (event is SessionEvent.BackgroundTasks && !isSubagent) {
+                    backgroundTasks = event.count
+                }
                 if (!isSubagent) loginOpen = authenticationPromptAfter(loginOpen, event)
                 // In order, always: one late event recorded ahead of the backlog would fold a
                 // streamed delta into whatever row happened to be last by then.
@@ -1842,6 +1847,7 @@ fun SessionScreen(
                 status = status,
                 compactingFor = compactingFor,
                 contextTokens = contextTokens,
+                backgroundTasks = backgroundTasks,
                 subagent = isSubagent,
             )
 
@@ -2362,6 +2368,8 @@ private fun SessionStatusRow(
     compactingFor: Long?,
     /** Context the session is holding, or null where nothing has measured it. */
     contextTokens: Long?,
+    /** Provider-reported live background work; zero is deliberately not drawn. */
+    backgroundTasks: Int,
     modifier: Modifier = Modifier,
     /**
      * Whether this row is for a subagent rather than a session, which changes only one word:
@@ -2395,6 +2403,14 @@ private fun SessionStatusRow(
                     // its own contrast, since the surface under it will not change to rescue it.
                     color = commandColor,
                 )
+                if (backgroundTasks > 0) {
+                    Text(
+                        backgroundTaskLabel(backgroundTasks),
+                        style = MaterialTheme.typography.labelSmall,
+                        color = MaterialTheme.colorScheme.onSurfaceVariant,
+                        modifier = Modifier.padding(start = 8.dp),
+                    )
+                }
                 LinearProgressIndicator(
                     color = commandColor,
                     trackColor = MaterialTheme.colorScheme.surfaceContainerHigh,
@@ -2415,6 +2431,14 @@ private fun SessionStatusRow(
                     color = MaterialTheme.colorScheme.onSurfaceVariant,
                     modifier = Modifier.padding(start = 8.dp),
                 )
+                if (backgroundTasks > 0) {
+                    Text(
+                        backgroundTaskLabel(backgroundTasks),
+                        style = MaterialTheme.typography.labelSmall,
+                        color = MaterialTheme.colorScheme.onSurfaceVariant,
+                        modifier = Modifier.padding(start = 8.dp),
+                    )
+                }
                 Spacer(Modifier.weight(1f))
             }
             // Every remaining state says which one it is, including the quiet one. The row used to
@@ -2423,13 +2447,22 @@ private fun SessionStatusRow(
             // showed nothing at all. The words and the colour are `sessionStatusWord`'s and
             // `sessionStatusColour`'s, shared with the session list so one state is not called two
             // things -- or drawn two colours -- depending which screen you are on.
-            else ->
+            else -> {
                 Text(
                     sessionStatusWord(status, subagent),
                     style = MaterialTheme.typography.labelSmall,
                     color = sessionStatusColour(status),
-                    modifier = Modifier.weight(1f),
                 )
+                if (backgroundTasks > 0) {
+                    Text(
+                        backgroundTaskLabel(backgroundTasks),
+                        style = MaterialTheme.typography.labelSmall,
+                        color = MaterialTheme.colorScheme.onSurfaceVariant,
+                        modifier = Modifier.padding(start = 8.dp),
+                    )
+                }
+                Spacer(Modifier.weight(1f))
+            }
         }
         // How full the session is, which is the number a reader is asking about -- how much room is
         // left before the next compaction -- rather than what has been spent getting here.