fix(deepseek): run the web UI server in the background, not in a shell tab

Clicking "DeepSeek web UI..." opened two tabs: the web tab asked for, and a
shell tab running the server next to it. The shell was deliberate - the server
lived in an ordinary session so it was visible, scrollable, killable and died
with its tab, and nothing new had to supervise a long-lived HTTP server. That
reasoning was sound and the result was still wrong in use: opening a dashboard
should open one tab, and after the first launch the terminal is pure noise.

The server moves to a background child process owned by a new
`src/deepseek-web-server.ts`, behind `POST /api/deepseek/web`. What the session
gave away for free is now explicit, which is most of the module:

- Exactly one server. A second click reuses the running one instead of racing
  it for a port; the session flow could not do this at all, because two clicks
  were simply two sessions.
- Restarted when the requested authority changes. `--trusted-host` fences dsh's
  own /api against the browser authority, and a Codeman reachable at both
  loopback and a tailnet name has two. Reusing a server fenced for the other
  origin renders a page whose every call 403s, which reads as a broken
  dashboard rather than a misconfigured one, so a mismatch restarts instead.
- Killed on shutdown. The child is detached so its whole plugin tree can be
  signalled at once, which also means it would outlive Codeman and hold its
  port against the next start - the exact EADDRINUSE this feature already got
  wrong once.
- Boot output captured and returned. With no shell tab there is nowhere else
  for a stack trace to land, so a failed spawn reports its own tail.

The endpoint is fenced at the same bar as the profile installer and for the
same reason: booting a dsh profile executes the plugin code in it, so this is a
privileged action even though it reads as "open a page". `authority` comes from
the client (`location.host`) because only the browser knows which origin is in
play, and it is regex-confined at the schema boundary - defence in depth behind
the argv-array spawn, admitting host:port in the shapes a browser authority can
take and nothing readable as a second argument.

`GET /api/deepseek/web-port` is gone; port selection moved into the supervisor,
which is the thing that knows whether a server is already running. The two
client-side probe helpers went with it, since the server now owns the wait.

Verified over the tailnet authority end to end: no session is created (session
count unchanged, one tab), the server runs on 3081 beside the user's own dsh
web on 3080, status reports the tailnet authority, and the proxied dashboard
renders with zero 4xx. Full gate green (6148 passed, +6).

docs/deepseek-integration-plan.md

@@ -49,14 +49,30 @@ in the six external CLIs before it.
 The browser UI is the only interactive surface DeepSeek ships itself, so it gets
 a **shortcut, not a run mode**: `Run ▸ DeepSeek web UI…` starts
 `dsh web --no-open --host 127.0.0.1 --port <free> --trusted-host <codeman-authority>`
-in an ordinary shell session and opens the URL as an ordinary web tab.
+as a background process and opens the URL as an ordinary web tab.
+
+The server was a **shell session** first, on the reasoning that Codeman already
+supervises those (visible, scrollable, killable, dies with its tab) so nothing
+new had to own a long-lived HTTP server. That version worked and was still
+wrong in use: clicking "open the DeepSeek web UI" put a terminal tab on screen
+next to the web tab actually asked for, every single time, and after the first
+launch the terminal was pure noise. Opening a dashboard should open one tab.
+
+So `POST /api/deepseek/web` owns it instead (`src/deepseek-web-server.ts`), and
+what the session gave away for free is now explicit: exactly one server, reused
+rather than raced on a second click; restarted when the requested authority
+changes; killed on Codeman shutdown (a detached child would otherwise hold its
+port against the next start — the very EADDRINUSE this feature already got
+wrong once); and boot output captured, since with no shell tab there is nowhere
+else for a stack trace to land. It is fenced at the same bar as the profile
+installer: booting a dsh profile executes the plugin code in it, so it requires
+the privileged grant in multi-user mode.
 
-Built entirely from parts that already exist: the server is a shell session
-(visible, scrollable, killable, dies with its tab) and the UI is a web tab.
-Nothing new supervises a long-lived HTTP server, because Codeman already does.
 `--trusted-host` is load-bearing — dsh fences its `/api` behind a browser-trust
 check on the request authority, and a Codeman web tab reaches it through
-Codeman's own origin via the webview proxy, not directly.
+Codeman's own origin via the webview proxy, not directly. The authority comes
+from the CLIENT (`location.host`) because only the browser knows which of a
+multi-homed Codeman's origins is actually in play.
 
 Three things about this shortcut are load-bearing and each came from it failing
 in exactly that way against a real install:
@@ -88,10 +104,11 @@ listing both showed the same dashboard twice. Being managed is also what lets a
 relaunch repoint the existing row instead of stacking one dead dashboard per
 restart, since the port is now chosen per launch.
 
-⚠️ The authority baked into `--trusted-host` is the one the launch was clicked
-from. Codeman reachable at several authorities (loopback *and* a tailnet name)
-therefore needs the server restarted from whichever one is in use; the reuse
-path checks that the server is reachable, not that it trusts the current origin.
+The authority baked into `--trusted-host` is the one the launch was clicked
+from, and reuse is conditional on it: a running server fenced for a *different*
+origin is stopped and restarted rather than reused, because reusing it renders a
+page whose every API call 403s — which reads as a broken dashboard rather than a
+misconfigured one.
 
 ## 4. Touch points (the checklist)
 
@@ -144,9 +161,9 @@ behaviour covered by 31 new unit tests; and an isolated instance used to exercis
 - A Docker case with `mode: 'deepseek'` (needs a `--no-cache` agent-image
   rebuild — see the `--no-cache` rule in CLAUDE.md).
 - A remote-SSH deepseek case.
-- The web-UI shortcut against a tunnel authority. Loopback is verified end to end
-  through the webview proxy (dashboard renders, its `/api` calls succeed); the
-  cross-authority case above is a known limitation rather than an open question.
+- The web-UI shortcut against a tunnel authority. Loopback and a tailnet name are
+  both verified end to end through the webview proxy (dashboard renders, its
+  `/api` calls succeed, no shell session created).
 
 ## 6. Follow-ups