What breakthroughs has Jerry achieved by the story's end?

On a basic level, Jerry has accomplished the physical feat he set out to complete. He has conditioned himself to be able to hold his breath for several minutes and overcome his fears to swim through the tunnel in the rock: an unbelievably dangerous and difficult thing.

Further, Jerry no longer craves the acceptance of the older, local boys with whom he so much wanted to fit when he first went to the wild bay. This is pretty significant. He attempted such a dangerous feat because they were doing it, and he felt childish and rejected when he could not do it before. However, he no longer links this experience to their recognition of him as an equal. This lack of concern for what they think is an achievement, as it shows the maturation that has resulted from Jerry's dangerous experience. He has also developed the ability to delay gratification, something he couldn't do at the beginning (consider his behavior when he wanted goggles). But now, he's developed "A curious, most unchildlike persistence, a controlled patience," and this is definitely the sign of a mature person.