You say that this is its intent. But perhaps the intent is (that it shall not be found) in your houses for seven days, and in your boundaries for all time? It is, therefore, written (Ibid. 13:7) "and se'or shall not be seen in all of your boundaries for seven days." Just as in your houses, for seven days, so, in your boundaries.
The Chametz Ban in Your Boundaries Lasts Seven Days Not Forever
Mekhilta Tractate Pischa 17:18