Count the quiet after a restart from when the controller first saw the search pending

The node-engine's own since starts again at every restart, so an agent that
restarted the engine in a loop kept agent-can-become-root quiet for ever (the
review of 2026-10-09). The controller now keeps when it first saw the verdict
waiting for the setuid search (migration 0085), forgets it at the next
complete verdict, and raises once the engine's own bound has passed since;
the bound and the pending reason are read from mesh-host's rootsearch.

internal/inventory/nodes.go

@@ -1292,3 +1292,24 @@ func (i *Inventory) Sequence(ctx context.Context, id string) (int64, error) {
 	}
 	return *n, nil
 }
+
+// RootSearchPending keeps when the controller first saw this node's agent account waiting for the node-engine's
+// setuid search (novox/hq ADR 0266) and answers it: the first time it is seen since the last complete verdict,
+// at, kept; seen again, what was kept. Kept across the engine's restarts and the controller's own.
+func (i *Inventory) RootSearchPending(ctx context.Context, node string, at time.Time) (time.Time, error) {
+	var since time.Time
+	err := i.store.Pool().QueryRow(ctx,
+		`update node set agent_root_pending_since = coalesce(agent_root_pending_since, $2)
+		  where name = $1 returning agent_root_pending_since`, node, at).Scan(&since)
+	if errors.Is(err, pgx.ErrNoRows) {
+		return time.Time{}, fmt.Errorf("%w: %s", ErrNoSuchNode, node)
+	}
+	return since, err
+}
+
+// RootSearchJudged forgets when a search began pending: a complete verdict came.
+func (i *Inventory) RootSearchJudged(ctx context.Context, node string) error {
+	_, err := i.store.Pool().Exec(ctx,
+		`update node set agent_root_pending_since = null where name = $1 and agent_root_pending_since is not null`, node)
+	return err
+}