[antonrh]

Four bugs the extension review found, each reproduced first.

A mounted sub-application was invisible

_iter_routes walked included routers but not Mount, so a sub-app's routes
were never validated, and get_container read connection.app.state, which
for a mounted app is the sub-app's own state that nobody filled:

/sub/greet raised AttributeError: 'State' object has no attribute 'container'

install() now walks mounted applications and gives each one the container,
and the container is looked up in the ASGI scope first, since a mounted app
replaces scope["app"] with itself. When there is no container anywhere, the
failure says so instead of surfacing a Starlette attribute error.

A route added after install() failed at request time

install() walked app.routes once, so anything registered later reached the
handler unwired and died with TypeError: Dependency type is not set., naming
neither the route nor the parameter. install() now watches
add_api_route, add_api_websocket_route, include_router and mount, so a
late route is validated as it is added, with the same message a route present
at install time gets.

Only the first handler of a subscriber was injected

_get_broker_handlers read subscriber.calls[0].handler. A subscriber
decorated more than once, the normal way to attach filtered handlers to one
topic, left every handler after the first without an owner, and the failure
appeared only when a message routed to it.

pydantic_settings.install() could not run twice

A second call, which an app factory called per test does, died with
LookupError: The provider Annotated[str, 'settings.name'] is already registered — naming neither the settings class nor the field. It now names
both, and takes override=True for the case where installing again is the
point.

Both landing changes are documented: "Mounted applications" on the FastAPI
page and "Installing twice" on the pydantic-settings page.

[codspeed]

Merging this PR will not alter performance

✅ 17 untouched benchmarks

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_{Comparing fix-extensions (7381caa) with main (f481f80)}