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/host/apiproxy/src/api/events.ts

@@ -14,6 +14,7 @@ import type { CallId } from '@deepseek-ai/dsh-llm/brand'
 import type { SessionEvent, SessionId } from '@deepseek-ai/dsh-session/types'
 import type { ToolCallView, ToolResultView } from '@deepseek-ai/dsh-tools/presentation'
 import type { RpcError, RpcId, RpcRequest } from './rpc.ts'
+import type { TaskView } from './tasks.ts'
 import type { WorkspaceView } from './workspace.ts'
 
 // Client-side consumers take the render-intent vocabulary from the contract;
@@ -81,6 +82,20 @@ export type MuxFrame =
    * in QueueDock, while pending steering renders at the conversation tail.
    */
   | { type: 'session/queue'; sessionId: SessionId; items: QueuedInboxItem[] }
+  /**
+   * Complete set of background tasks this session can see, after every registry
+   * commit that changes it: registration, the stopping transition, settlement,
+   * and owner-disposal removal. The registry is process-local and holds no
+   * durable event, so — exactly like `session/queue` — the whole snapshot is
+   * what makes a start, a kill, a reconnect, and a second tab converge on one
+   * authoritative value.
+   *
+   * Sent as a subscription baseline only for a session that currently has
+   * tasks; an absent key means an empty set. A change that empties the set
+   * still sends `[]`, since that transition is the only one absence cannot
+   * express.
+   */
+  | { type: 'session/tasks'; sessionId: SessionId; tasks: TaskView[] }
   /**
    * One projection unit's finished value changed (session-projection RFC).
    * Live push state, never logged — replay recomputes on the host (the