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.schema.ts

@@ -13,6 +13,7 @@ import { approvalRequestIdSchema } from './approvals.schema.ts'
 import {
   contentBlockSchema, messageIdSchema, sessionEventSchema, sessionIdSchema, toolEventViewSchema,
 } from './sessions.schema.ts'
+import { taskViewSchema } from './tasks.schema.ts'
 import { workspaceIdSchema, workspaceViewSchema } from './workspace.schema.ts'
 
 /** Question shape validated strictly against core dsh-user-interaction. */
@@ -58,6 +59,7 @@ export const muxFrameSchema = z.discriminatedUnion('type', [
       message: messageSchema,
     })),
   }),
+  z.object({ type: z.literal('session/tasks'), sessionId: sessionIdSchema, tasks: z.array(taskViewSchema) }),
   // value stays wide: it already passed its unit's own schema on the host,
   // and deep-validating here would import every domain's schema into the carrier.
   z.object({ type: z.literal('session/projection'), sessionId: sessionIdSchema, key: z.string().min(1), value: z.unknown(), seq: z.number().int().nonnegative() }),

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

packages/host/apiproxy/src/api/index.ts

@@ -44,6 +44,7 @@ export type { DirectoryEntry, DirectoryListing, HostApi } from './host.ts'
 export type {
   SubagentAddress, SubagentCatalog, SubagentListEntry, SubagentPromptReceipt, SubagentsApi,
 } from './subagents.ts'
+export type { TaskView } from './tasks.ts'
 export type { WorkspaceApi, WorkspaceId, WorkspaceView } from './workspace.ts'
 export type { CommandsApi, CommandDescriptor } from './commands.ts'
 export type { SkillsApi, SkillEntry } from './skills.ts'

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>>

packages/host/apiproxy/src/api/tasks.ts

@@ -0,0 +1,36 @@
+/**
+ * Browser-safe background-task domain contract. The registry's live records
+ * never cross the wire; a view is the subset a human list needs, minted fresh
+ * per push.
+ */
+
+import type { TaskId } from '@deepseek-ai/dsh-tasks/brand'
+
+/**
+ * One background task as the client sees it.
+ *
+ * Three registry fields are deliberately absent. `ownerSession` is redundant
+ * beside the frame's own `sessionId`; `reported` is an internal notice-delivery
+ * bit with no user meaning; `outputLimitBytes` is producer-owned model
+ * presentation policy that never reaches a human surface.
+ */
+export interface TaskView {
+  /** Registry-issued `<kind>-N` identity, stable for the task's whole life. */
+  id: TaskId
+  /**
+   * Producer kind (`bash`, `pwsh`, `pty-send`, `subagent`, …). Kept as a bare
+   * string because producer plugins extend the kind map by declaration merging,
+   * so no client build can enumerate the closed set.
+   */
+  kind: string
+  /** Producer-supplied one-line label: the command, or the delegation description. */
+  label: string
+  /** Current lifecycle state. */
+  status: 'running' | 'stopping' | 'completed' | 'killed' | 'failed'
+  /** Kind-specific status detail ('exit code: 3'), present once the producer supplied one. */
+  detail?: string
+  /** Epoch ms when the task was registered. */
+  startedAt: number
+  /** Epoch ms when the task settled; absent while live. */
+  finishedAt?: number
+}