fix(dbmanager): keep the pool alive across errors and restarts

Implements the fixes from audit/pkg/dbmanager.audit.md.

- Stop closing the shared *sql.DB to recover from errors. Adapter
  factories and the health checker no longer call Reconnect; Reconnect is
  atomic and operator-only.
- Postgres uses a custom driver.Connector: Reconnect retires pooled
  connections by generation without closing the pool, so held Bun/GORM
  handles keep working. Verified against a live server restart.
- Add TCP keepalive, TCP_USER_TIMEOUT, a bounded reuse ping and
  statement_timeout as a runtime parameter; drop the 2 min timeout floor.
- Health check pings without holding the connection lock.
- Listener: single goroutine pair, bounded Close without UNLISTEN, and
  serialised use of the pgx connection (fixes conn busy and a close race).
- Fix Connect/Close/Connect/Close panic, idempotent Connect, dial outside
  the manager lock, clean up on partial failure.
- SQLite: pin :memory: to one connection, pragmas via DSN.
- Escape credentials in Postgres/MSSQL/Mongo DSNs; sslmode defaults to
  prefer. Wire retry settings, publish metrics, fix logger calls.
- NewConnectionFromDB: Close is a no-op with a warning (caller owns the
  pool); Reconnect only pings.
- Document correct usage in the README; mark the audit with what was done.

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>

pkg/dbmanager/providers/tcp_timeout_linux.go

@@ -0,0 +1,24 @@
+//go:build linux
+
+package providers
+
+import (
+	"syscall"
+	"time"
+
+	"golang.org/x/sys/unix"
+)
+
+// setTCPUserTimeout returns a net.Dialer Control func setting TCP_USER_TIMEOUT.
+func setTCPUserTimeout(d time.Duration) func(network, address string, c syscall.RawConn) error {
+	return func(network, address string, c syscall.RawConn) error {
+		var sockErr error
+		err := c.Control(func(fd uintptr) {
+			sockErr = unix.SetsockoptInt(int(fd), unix.IPPROTO_TCP, unix.TCP_USER_TIMEOUT, int(d.Milliseconds()))
+		})
+		if err != nil {
+			return err
+		}
+		return sockErr
+	}
+}