Wrong oath

Regarding this: “Beyond this, when the intention of the heart does not contradict what has been uttered by the mouth, one does not say that things in the heart are not things.”

After all, there is usually no direct contradiction between the things in the heart and the things in the mouth. The things in the heart usually only qualify the things in the mouth for certain cases. Here is an example of the source in the Gemara:

In the Gemara, in Tractate Kiddushin[1], a case is presented in which a man, living abroad, sold his assets with the intention of immigrating to Eretz Yisrael, but did not mention this at the time of the sale, and later the case occurred and he was prevented from immigrating[2] and now he wants to cancel the sale on the grounds that since he sold his assets while relying on his immigrating, and this immigrating did not occur, it is a mistaken transaction and must be annulled. Regarding this case, Rava said that it is “things in the heart, and things in the heart are not things” – His purpose at the time of the sale has no halakhic significance because “things in the heart are not things” and he cannot demand the cancellation of the sale.

How is this different from the case regarding the contradiction between the heart and the mouth?

This is precisely the issue in which the first and last men establish this rule itself (that when there is no contradiction, things in the heart are things). One must examine the various cases and divisions there.
On the face of it, in the sale of one's assets with the knowledge of the deceased, if one does not specify this, it means that one is selling one's assets as usual (without qualification), and therefore when one says that one wants to cancel because one has not ascended to the deceased, this is a thought that contradicts one's speech. But when a person forbids oneself from saying a word, and in one's mind he only means certain contexts of that word, this is not a frontal contradiction, and one's thought merely interprets the speech.
It can also depend on the fact that the sale is made in front of another person, and therefore there, if you did not specify, it is as if there is no qualification on the sale. An oath is between a person and himself. This consideration was brought up in my words above as a separate section, but I believe it is also related to this distinction.

Thank you very much, Your Honor! May God bless you for all your investment.