I had to look up the name for this and it's called the correlative comparative or the conditional comparative. The whole clause you need to consider is "the better that experiment goes tomorrow, the better for all of us", not just the first part.
It's that "the… the…" form that makes it what it is. Here are some other examples of it:
- The sooner, the better.
- The more, the merrier.
- The more you learn, the better your grades will be.
- The greater the risk, the greater the return.