feat(server): public GET /health endpoint for uptime monitors

Returns 200 with {status: ok, version, db: ok} when SQLite is reachable,
503 when the DB probe fails. Unauthenticated so external monitors can
poll without credentials.

server/lib/server_web/controllers/health_controller.ex

@@ -0,0 +1,35 @@
+defmodule ServerWeb.HealthController do
+  use ServerWeb, :controller
+
+  @moduledoc """
+  Public health check for uptime monitors.
+  Returns 200 when the process is up AND SQLite is reachable, 503 otherwise.
+  """
+
+  def show(conn, _params) do
+    case probe_db() do
+      :ok ->
+        json(conn, %{
+          status: "ok",
+          version: Application.spec(:server, :vsn) |> to_string(),
+          db: "ok"
+        })
+
+      {:error, reason} ->
+        conn
+        |> put_status(:service_unavailable)
+        |> json(%{status: "degraded", db: "error", reason: inspect(reason)})
+    end
+  end
+
+  defp probe_db do
+    try do
+      case Ecto.Adapters.SQL.query(Server.Repo, "SELECT 1", []) do
+        {:ok, _} -> :ok
+        {:error, e} -> {:error, e}
+      end
+    rescue
+      e -> {:error, e}
+    end
+  end
+end