feat: show a conversation its own background tasks, and give it back every outcome

An `execute_task mode="async"` was invisible from the chat that started it.
The only trace was the receipt in the transcript and a row on the Tasks page
— which does not say *which* of those rows the assistant just spawned — so
"is it still going?" had no answer where the question is asked.

Worse, a task that did not simply succeed never came back at all. `run_job`
branched on `Ok`/`Err` first and routed by `job.kind` only inside the `Ok`
arm, so a failure or a kill left through the `Err` arm's unconditional
`hub.notify` — the home source (`/sethome`), worded "Cron job … failed" —
while the parent conversation sat waiting for a `task_completed` that would
never arrive. The wrong chat, and a wedged one.

The fix is a shape, not a branch: one `JobOutcome` classification, then one
`match job.kind` delivery site for every ending. An async task now ends in
its parent conversation whatever happened to it. The sink has a single
channel deliberately — to the model reading it, "it broke" is a result like
any other and must not be overlookable — so a failure is delivered as prose,
carrying whatever partial output the run produced, which is usually the only
clue about why. A cron job keeps the home notification: it belongs to nobody's
conversation. Cancellation becomes a third outcome rather than a flavour of
failure (`job_runs.status` has always had `'cancelled'` in its CHECK and
nothing ever wrote it), classified off the new typed `TurnCancelled` error so
nothing keys on a message string.

The strip above the composer is the visible half. `ServerEvent::TaskUpdate`
announces state to the source of the parent conversation only; the list is
`renderTaskStrip` (shared by the desktop copilot and the mobile chat), fed by
state on `ChatSession`. Each row links to `#session/{id}` — the page that
already shows, live, what a background agent is doing, and without which
"a task is running" is a fact you can do nothing with. Stopping is the
existing kill endpoint. A finished row clears itself after 20 s (its result
is in the conversation by then); a failed one stays until dismissed, and the
dismissal is remembered across reloads.

`GET /api/{source}/tasks` is what makes the strip survive a browser refresh:
the event is a broadcast with no replay, so without a load-time read a reload
would empty a chat that still has work running under it. It answers with the
running tasks plus failures from the last 30 minutes — the two states a person
can still act on. Successes are absent on purpose. Its window compares through
`datetime()` on both sides: `completed_at` is RFC 3339 and the cutoff is
SQLite-shaped, and `'T' > ' '` would let every same-day row through a window
meant to exclude it.

Not addressed, and worth doing next: a cron job's result should go where its
creator says, not always to the home chat.

docs/index.md

@@ -4,7 +4,7 @@ This folder is written for **you, the assistant**, not for the human directly. I
 
 Keep answers grounded in what's actually enabled and configured for this instance — check with the relevant tool (e.g. list installed/enabled plugins) rather than assuming everything described here is turned on. A feature documented here may not be enabled on this particular instance.
 
-This index will grow over time. Right now it covers memory, projects, system agents, access grants, voice input and plugins; more sections (agents, connectors, security groups, shared folders…) will be added later.
+This index will grow over time. Right now it covers memory, projects, background tasks, system agents, access grants, voice input and plugins; more sections (agents, connectors, security groups, shared folders…) will be added later.
 
 ## Features
 
@@ -13,6 +13,7 @@ This index will grow over time. Right now it covers memory, projects, system age
 | [memory.md](memory.md) | Private and shared memory: what goes where, the indexes and history log, why some shared facts can't be changed on request |
 | [projects.md](projects.md) | Projects: shared folders with their own assistant chat, a live file explorer, and member sharing |
 | [system-agents.md](system-agents.md) | Background agents that run on a schedule (event triage, the two memory lints, the nightly conversation review of a supervised account): what they watch, why they only ever report, why a run can be skipped, and their settings |
+| [tasks.md](tasks.md) | Background tasks: the strip above the message box, following one live, stopping one, and how every outcome comes back to the conversation |
 | [settings.md](settings.md) | The admin's Config page: interface language, the compaction model picker, debug mode |
 | [access.md](access.md) | Who can use which plugin or connector: the open default, removing access per person, and the role switch that keeps children out of it |
 | [voice.md](voice.md) | Voice input: configuring a transcription model, and why the microphone button does nothing unless the page is served over HTTPS or localhost |