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/README.md

@@ -36,6 +36,8 @@ Session model routing is a session-domain contract. `session.models` returns the
 
 Pending queued input is a live control-plane contract, not conversation history. The gateway derives the complete `next-turn` queue from durable `agent/inbox/spliced` mutations and broadcasts authoritative `session/queue` snapshots after each change and on reconnect; pending `next-step` steering stays outside this Web projection. Within `next-step`, user-origin messages carry the `steering` placement while injected context (approval notices, task completion, attached snapshots) carries `context` and is not surfaced until claimed. The message-local `agent/inbox/inserted`, `claimed`, and `discarded` notifications remain available to lifecycle observers but do not build the queue view. `session.updateQueue` addresses one `MessageId`; edit and remove mutate the attached Agent through `Inbox.splice()`. A claim's pure deletion splice wins races before pre-step admission, so a later operation returns `queue-item-not-found`. `session.cancel` aborts only the active turn and preserves pending inbox work; after cancellation reaches quiescence and the closing turn flushes, AgentLoop claims the next waking message in FIFO order, and the browser never resends or promotes it. Queue operations never resume a cold session, and the client never infers retirement from turn or status events.
 
+Background tasks ride the same live-push posture. When `ctx.tasks` is composed, the gateway subscribes to its change feed and broadcasts a whole `session/tasks` snapshot after every registry commit that alters what a session can see — registration, the stopping transition, settlement, and owner-disposal removal — plus a subscription baseline for each session that already has tasks (an absent baseline is the empty set; a change that empties a set still sends `[]`). A change carrying an owner reads through that exact `Agent`, so a push stays correct while its scope tears down; the baseline reads `ctx.agents.get(sessionId)`, which yields only unowned tasks for a session with no live Agent and never resumes a cold one. An unowned change fans out to every subscribed session, because unowned tasks are visible to every caller. The wire `TaskView` drops `ownerSession`, `reported`, and `outputLimitBytes`: the frame's own `sessionId` carries the first, and the other two are internal notice and model-presentation policy. A composition without the registry emits no such frames.
+
 Workspace and Session lists are separate reconnect baselines. `workspace.create({ name })` creates a uniquely titled directory under the configured root, while `workspace.create({ path })` adopts an existing canonical directory and permits basename-derived titles to repeat. `workspace.delete` removes only the Workspace registration, `session.create` accepts an optional preallocated Session id, and `host/workspace-changed`, `host/workspace-removed`, plus `host/session-added` carry committed increments in either arrival order. `workspace.archiveSession` adds one session to the registry-global archive set and answers the full updated set; `workspace.list` carries that set as the reconnect baseline and `host/archived-sessions-changed` pushes the full snapshot after every durable change. Archiving hides the session from grouping surfaces without touching its log or its workspace account; a session neither live nor persisted fails with `session-not-found`. Registration deletion preserves the directory and session logs; its Sessions remain in `session.list` and become Ungrouped. `SessionSummary.blank` and the `host/session-added` frame carry the derived zero-events bit: clients hide blank sessions and reuse them per workspace, flip blank on the first `host/session-status(running:true)`, and treat `session.list` as the reconnect authority; cold summaries are never blank because lazy persistence keeps never-appended sessions out of `list()`.
 
 `session.search` is a bounded content-search projection over the sessions visible through `session.list`. The gateway asks the optional `ctx.sessionQuery` service for globally ranked current-surface user, assistant, and steering matches, consumes that stream until it has at most 20 visible session/snippet pairs plus one lookahead, and revalidates every hit against the list-derived authorization set before returning it. Provider pages start at 20 hits; when a first-page request rejects that limit, the gateway probes 10, 5, 2, then 1 and retains the learned size for continuation and stale-generation restarts. Returned snippets contain at most 240 Unicode code points, and the response schema independently enforces that bound at each client boundary. Keeping the authorization set in Host memory avoids SQLite's variable ceiling for large valid corpora without weakening visibility or ranking.