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/ui-task/README.md

@@ -0,0 +1,24 @@
+# @deepseek-ai/dsh-client-ui-task
+
+English | [中文](README.zh.md)
+
+Web background-task feature owner: contributes one entry to `conversation.session.header.actions` listing the `ctx.tasks` records this session can see. The data arrives entirely through the `tasksBySession` list mirror that [`dsh-client-runtime`](../runtime/README.md) folds from `session/tasks` frames, so this package issues no RPC and holds no state beyond popover visibility.
+
+The trigger renders only when the session has at least one task, so an ordinary conversation never grows a control for a capability it is not using. Its badge counts `running` plus `stopping` and is omitted at zero, leaving a session that holds only finished tasks a quiet entry point into its history rather than one advertising a count of nothing. The popover is a flat list: live rows first by `startedAt` ascending, then settled rows by `finishedAt` descending, with a same-millisecond tie broken on start order so the host's map iteration never decides it. A row shows the producer kind, the label, a status marker, the producer's `detail` in place of the generic status word once it has one, and an elapsed duration. That duration advances once per second while the row is live and freezes at `finishedAt`; the clock runs only while an open list holds something that moves. A settled row missing `finishedAt` reads as zero rather than as a negative figure, and a duration past an hour stays in hours rather than growing a day vocabulary no producer currently reaches.
+
+Settled rows stay visible and de-emphasized until the registry drops them at owner disposal. They are in the snapshot, a failed task's `detail` is the only place its failure is legible, and filtering them out here is work the output and cancellation phases would undo. A running one-shot background subagent therefore appears both here and in the [subagent catalog](../ui-subagent/README.md): the catalog navigates into the child's transcript, while this list is the only handle a future cancellation can attach to.
+
+Escape closes the list and returns focus to the trigger, as does a pointer press outside it. The last task disappearing closes the list before the control unmounts, so focus never vanishes from a removed node. Styling uses tokens only; copy goes through the package's own `task` locale namespace. The behavior is specified by the [Web background-task display Agent Note](../../../.agents/notes/implemented/feature/2026-08-08-web-background-task-display.md).
+
+## Model Experience
+
+None, as this package renders host-computed registry state for a human and touches no prompt, message, schema, stream, or tool result. The model's own view of the same tasks stays with [`dsh-tool-tasks`](../../tasks/tool-tasks/README.md).
+
+#### KV Cache effect
+
+None; the package never assembles or sends provider requests.
+
+## Known Limitations and Deferred Work
+
+- **Rows are read-only** — a task's streamed output and a human-initiated cancellation are separate phases. Cancellation additionally owes a model-facing decision the seam does not answer today: `kill()` marks terminal delivery reported, so an interrupt written against the current contract would leave the model believing its task is still running.
+- **The list is not the registry's own set** — it shows what the owning session can see through the wire view, so an unowned task (one started without a live `Agent`) is invisible here while `task_list` still reports it to the model.