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/tasks.schema.ts

@@ -0,0 +1,33 @@
+/**
+ * tasks domain zod schemas: the branded task id and the wire view carried by
+ * `session/tasks` frames.
+ */
+
+import { z } from 'zod'
+import type { TaskId } from '@deepseek-ai/dsh-tasks/brand'
+import type { TaskView } from './tasks.ts'
+import type { Wire } from './rpc.schema.ts'
+
+/** TaskId: one brand cast after non-empty string validation. */
+export const taskIdSchema = z.string().min(1) as unknown as z.ZodType<TaskId>
+
+/**
+ * One wire task view. `kind` stays an open string because producer plugins
+ * extend the registry's kind map by declaration merging, so the closed set is
+ * not knowable at this boundary.
+ */
+export const taskViewSchema = z.object({
+  id: taskIdSchema,
+  kind: z.string().min(1),
+  label: z.string().min(1),
+  status: z.union([
+    z.literal('running'),
+    z.literal('stopping'),
+    z.literal('completed'),
+    z.literal('killed'),
+    z.literal('failed'),
+  ]),
+  detail: z.string().optional(),
+  startedAt: z.number().int().nonnegative(),
+  finishedAt: z.number().int().nonnegative().optional(),
+}) satisfies z.ZodType<Wire<TaskView>>