Integrity is often used to mean keeping one's promises to others and/or with oneself. This is an important aspect of integrity, but it is not fundamental. If fact, in some circumstances integrity will include breaking one's promises to others or with oneself. This "integrity" (the integrity of keeping promises) is just one of the branches of the "tree of integrity" that is mistaken for the roots of integrity.
Full integrity consists of two interrelated types of integrity: Now-Next integrity (NNI), and Oneself-Others integrity (OOI).
Now-Next integrity means that your Now and your Next are working together and on the same page. Now-Next integrity is more fundamental than Oneself-Others integrity. Without having a measure of Now-Next integrity first, creating and maintaining Oneself-Others integrity is more problematic.
Oneself-Others integrity means that your Oneself and your Others are working together and on the same page, with little or no conflict.
If you focus on creating and maintaining Now-Next integrity as well as Oneself-Others integrity, then the "integrity" of keeping your promises to others and with yourself is, generally, a slam dunk.