feat(web): list background tasks in the session header

The task registry has run every background bash, pwsh, pty-send, and
one-shot subagent since it landed, but only the model could read it: a
human at the Web client could not see that a build was running, tell a
finished task from a stuck one, or find its outcome anywhere but the
`run_in_background` tool card that printed an id and never updated.

Task state now reaches the browser as one whole-snapshot `session/tasks`
mux frame per session, pushed at every registry commit that changes what
that session can see. `TaskService` gains `onTasksChanged`, which is
owner-granular because owner-disposal removal is a change no per-task
record can express. The carrier reads the exact owner the listener hands
it, so a push stays correct while that scope tears down, and reads the
baseline through the non-resuming `ctx.agents.get` so listing never
revives a cold session. The client keeps a last-wins mirror on
`SessionListState`, and a new `dsh-client-ui-task` package renders it
beside the subagent catalog — rendering nothing at all until the session
has a task, so an ordinary conversation grows no new chrome.

Streamed per-task output and human-initiated cancellation are separate
phases; the note records why neither has to undo this channel, and why
no Web path may call the consuming `ctx.tasks.read()`.

packages/client/runtime/src/client/index.ts

@@ -36,7 +36,7 @@ export type {
   SessionBinding, SessionListState, SessionProvideContribution, SessionProvideDescriptor, SessionSummary,
 } from './sessions/service.ts'
 export type { SessionListPhase, SessionSearchResultItem, SubagentCatalogSnapshot } from './sessions/manager.ts'
-export type { SubagentAddress } from '@deepseek-ai/dsh-client-connection/client'
+export type { SubagentAddress, TaskView } from '@deepseek-ai/dsh-client-connection/client'
 export type { WorkspaceListPhase } from './workspaces/manager.ts'
 export type { WorkspaceListState } from './workspaces/service.ts'
 export type {

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

@@ -4,7 +4,7 @@
 
 import type {
   IApiClient, HostFrame, MuxFrame, RpcError, RpcRequest, RpcResult, SessionId,
-  SessionSummary, SubagentAddress, SubagentCatalog, WorkspaceId,
+  SessionSummary, SubagentAddress, SubagentCatalog, TaskView, WorkspaceId,
 } from '@deepseek-ai/dsh-client-connection/client'
 // Value import from the inline-safe wire layer (not the connection plugin):
 // plugin-to-plugin value imports are a bundle purity error.
@@ -47,6 +47,8 @@ export interface SessionListSnapshot {
   phase: SessionListPhase
   error: RpcError | null
   subagentsByParent: Readonly<Record<SessionId, SubagentCatalogSnapshot>>
+  /** Background tasks per session; an absent key is an empty set. */
+  tasksBySession: Readonly<Record<SessionId, readonly TaskView[]>>
   currentAddress: SubagentAddress | undefined
 }
 
@@ -137,6 +139,11 @@ export class SessionManager {
   private readonly catalogStale = new Set<SessionId>()
   private readonly openCatalogs = new Set<SessionId>()
   private readonly catalogDebounce = new Map<SessionId, ReturnType<typeof setTimeout>>()
+  /**
+   * Background tasks per session, last-wins from `session/tasks`. An empty set
+   * is stored as an absent key, so absence and `[]` are one representation.
+   */
+  private readonly tasksBySession = new Map<SessionId, readonly TaskView[]>()
 
   private selected: SessionId | undefined
 
@@ -657,10 +664,23 @@ export class SessionManager {
       this.notifier.markDirty()
       return
     }
+    if (frame.type === 'session/tasks') {
+      // Whole-set snapshot, so last-wins with no reconciliation. The Host omits
+      // the baseline for an empty set, which is the same fact an emptying change
+      // reports as `[]` — both land as an absent key.
+      if (frame.tasks.length === 0) this.tasksBySession.delete(frame.sessionId)
+      else this.tasksBySession.set(frame.sessionId, frame.tasks)
+      this.notifier.markDirty()
+      return
+    }
     if (frame.type === 'session/subscribed') {
       // Rows past the host's durable baseline rode state a restart lost; drop
       // them so last-wins cannot pin a phantom value over recomputed truth.
       this.projectionStores.get(frame.sessionId)?.truncate(frame.lastSeq)
+      // Same re-baseline reasoning as the queue below: this generation sends a
+      // task baseline only when the set is non-empty, so a mirror kept from the
+      // previous generation would survive as a phantom list.
+      this.tasksBySession.delete(frame.sessionId)
       this.notifier.markDirty()
       // New mux-generation baseline: discard the previous queue snapshot.
       // The host omits session/queue when the live queue is empty, so retaining
@@ -770,6 +790,11 @@ export class SessionManager {
         }
         this.pendingBuffers.delete(frame.sessionId) // a removed session's buffered frames must not replay on a future instantiation
         this.pendingInteractions.delete(frame.sessionId) // a removed session cannot wait on anyone
+        // Owner disposal already dropped these registry-side, but that lands on
+        // the mux stream while this frame rides the host stream, so the two have
+        // no relative order. Clearing here makes a detached Activation's rows
+        // disappear whichever arrives first.
+        this.tasksBySession.delete(frame.sessionId)
         if (!durableSubagent) this.projectionStores.delete(frame.sessionId)
         // A pull already in flight was requested before this removal and can
         // carry the pre-removal parentAvailable:true, which would resurrect
@@ -1006,6 +1031,7 @@ export class SessionManager {
       phase: this.listPhase,
       error: this.listError,
       subagentsByParent: Object.fromEntries(this.catalogs),
+      tasksBySession: Object.fromEntries(this.tasksBySession),
       currentAddress: current === undefined ? undefined : this.addresses.get(current),
     }
   }

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

@@ -17,7 +17,7 @@
  */
 import type { Context, Fiber } from 'cordis'
 import type {
-  IApiClient, RpcError, RpcResult, SessionId, SubagentAddress, WorkspaceId,
+  IApiClient, RpcError, RpcResult, SessionId, SubagentAddress, TaskView, WorkspaceId,
 } from '@deepseek-ai/dsh-client-connection/client'
 // Value import from the inline-safe wire layer (not the connection plugin):
 // plugin-to-plugin value imports are a bundle purity error.
@@ -80,6 +80,12 @@ export interface SessionListState {
   phase: SessionListPhase
   /** Direct durable catalogs keyed by their selected parent address. */
   subagentsByParent: Readonly<Record<SessionId, SubagentCatalogSnapshot>>
+  /**
+   * Background tasks each session can see, mirrored last-wins from
+   * `session/tasks`. A missing key is an empty set — the Host sends no baseline
+   * for a session without tasks — so consumers read absence, never a sentinel.
+   */
+  tasksBySession: Readonly<Record<SessionId, readonly TaskView[]>>
   /** Current session's catalog-derived address, absent on ordinary navigation. */
   currentAddress: SubagentAddress | undefined
 }
@@ -271,7 +277,7 @@ export class SessionsService implements ISessions {
     this.manager = new SessionManager(api, restored.sessionId, restored.subagentAddress)
     this.list = createSnapshotStore<SessionListState>({
       ids: [], byId: {}, current: undefined, phase: 'pending',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })
     // The manager owns wire truth; the store is its projection. Manager
     // notifications are already microtask-batched.
@@ -606,7 +612,7 @@ export class SessionsService implements ISessions {
   /** Project the manager's list snapshot into the store (title derivation is display-only). */
   private projectList(): void {
     const {
-      items, current, phase, subagentsByParent, currentAddress,
+      items, current, phase, subagentsByParent, tasksBySession, currentAddress,
     } = this.manager.getListSnapshot()
     const ids: SessionId[] = []
     const byId: Record<SessionId, SessionSummary> = {}
@@ -675,7 +681,7 @@ export class SessionsService implements ISessions {
         ...(currentAddress === undefined ? {} : { subagentAddress: currentAddress }),
       })
     }
-    this.list.set({ ids, byId, current, phase, subagentsByParent, currentAddress })
+    this.list.set({ ids, byId, current, phase, subagentsByParent, tasksBySession, currentAddress })
     this.pruneScopes()
   }