…582)

Issue #582 item 4.11 follow-up. PR #602 shipped the client-side
``STATE.indexing`` flag and ``#indexing-indicator`` pill but documented
one remaining gap in its Limitations section: the indicator is session-
bound, so a page reload mid-indexing resets it to ``false`` even though
the run continues server-side, and a second tab opened during the run
shows nothing. Closing that gap is what this change does.

## Server

``IndexEngine`` gets an ``_active_runs: int`` counter and an ``is_active``
read-only property. The three public entry points (``index_path``,
``index_file``, ``index_path_stream``) bracket their work with
``try/finally`` to increment on entry and decrement on exit.

The counter is deliberately separate from ``_index_lock``:

- ``index_path_stream`` runs **outside** the lock by design (so progress
  events can interleave). A lock-only state probe would miss every SSE
  run, which is the dominant indexing surface.
- ``asyncio.Lock.locked()`` is documented as racy.
- A counter (not boolean) lets concurrent stream + locked runs both keep
  the flag on; matches the engine's actual concurrency model.

New route ``GET /api/indexing/active`` returns ``{"active": bool}`` —
intentionally minimal to match the client's single-bool ``STATE.indexing``
model. Adding ``started_at`` / progress fields later is purely additive.
``require_configured`` is wired so the gate matches sibling indexing
routes (``/index``, ``/index/stream``, ``/reindex``).

## Client

``app.js`` gains two helpers:

- ``_indexingHydrateFromServer()`` fires once on ``DOMContentLoaded`` and
  again on ``visibilitychange`` → visible. If the server reports active
  and ``STATE.indexing`` is false, it calls ``_indexingTryStart()`` and
  starts polling.
- ``_indexingPollUntilIdle()`` runs a single-flight 3 s ``setTimeout``
  loop, calling ``_indexingEnd()`` once the server reports inactive.
  ``_indexingEnd()`` also cancels any pending poll handle so the in-tab
  SSE-complete path closes the loop too.

``index.html`` bumps ``app.js?v=78`` → ``?v=79`` per the static-asset
cache-bust convention.

## Tests

Engine — ``TestActiveRunsCounter`` (5 tests):
- idle, during ``index_path`` (gated), during ``index_path_stream``
  (lock-bypass path), two concurrent ``index_path`` runs both bumping
  the counter to 2 outside the lock, and ``finally``-decrement on the
  inner work raising.

Route — ``TestIndexingActive`` (idle / running) plus a
``TestRequireConfigured`` parity test that asserts the same 409 + body
the rest of the indexing surface returns when ``mm init`` hasn't run.

3337 passed, 46 deselected. Ruff clean. mypy advisory clean on changed
files. ``node --check`` clean on ``app.js``.

## References

- Umbrella: #582 (item 4.11)
- Predecessor: #602 (client-side ``STATE.indexing`` + indicator pill)
- Plan: ``~/.claude/plans/4-11-server-bound-api-indexing-active-fo-reflective-swing.md``

Co-Authored-By: Claude <[email protected]>