fix(api): a non-finite number in a JSON body is refused with a 422, not a 500

The JSON parser under FastAPI accepts `Infinity`/`NaN`, and both progress checkpoints truncate the
position with `int()` — a hand-written checkpoint came back as a 500 with a traceback. Their bodies
now refuse non-finite floats, and the app's validation handler — FastAPI's own shape — echoes a
non-finite input by name: the default one failed to serialise its own error (JSON has no NaN), which
was a 500 by another route. Sibling of the subtitle routes' non-finite query parameters.

backend/tests/test_progress_bounds.py

@@ -0,0 +1,34 @@
+"""A progress checkpoint takes a FINITE position.
+
+The JSON parser under FastAPI accepts `Infinity` and `NaN`, and both progress handlers truncate the
+position with `int()` — so one hand-written checkpoint came back as a 500 with a traceback. Found in
+the 2026-09 bug hunt's second review round, as the sibling of the subtitle routes' non-finite
+`at`/`offset`. Pure: the body is refused before the handler touches the database.
+"""
+
+from types import SimpleNamespace
+
+import pytest
+from fastapi.testclient import TestClient
+
+import app.state as app_state
+from app.auth import current_user
+from app.main import app
+
+
+@pytest.fixture
+def client(monkeypatch, override_dep) -> TestClient:
+    override_dep(current_user, lambda: SimpleNamespace(id=1))
+    monkeypatch.setattr(app_state, "_configured_cache", True)  # pass the instance-setup gate, no DB
+    return TestClient(app)
+
+
+@pytest.mark.parametrize("path", ["/api/videos/v1/progress", "/api/plex/item/98614/progress"])
+@pytest.mark.parametrize("raw", ["Infinity", "-Infinity", "NaN"])
+def test_a_non_finite_position_is_refused(client, path, raw):
+    r = client.post(
+        path,
+        content=f'{{"position_seconds": {raw}, "duration_seconds": 3000}}',
+        headers={"content-type": "application/json"},
+    )
+    assert r.status_code == 422, r.text