In branches of insights – brains, each case is a unit that exposes an answer, a reason, and a check. The aim is to write compact, trustworthy programs that read like a short abstract. The program computes a clear answer to a clear question; it emits a formally constrained account of why the answer holds and it runs checks that can fail loudly if an assumption is wrong or an edge case matters. The point is not only to compute a result but to carry an auditable trail that shows what was done, why it was warranted under the declared rules, and how the artifact verifies itself.
The discipline behind these cases is a pragmatic prompt-program-proof at design time followed by a mechanical runtime. At design time, a large language model helps synthesize the program from a prompt that bundles the question, the data, and the admissible rules. At runtime, only the program executes. It produces the answer, emits the reason, and evaluates the checks. Proof accountability is realized as reasons plus checks, not as external proof artifacts.
Each branch is self-contained, readable, and repeatable. A sensible way to read one is to locate the question and the final output, examine the reason to see how the premises support the conclusion, and then look at what the check actually enforces, whether that is a known identity, a conservation law, or a bound on error. If a check fails, the branch reports a precise violation and that failure becomes part of the learning rather than something to hide.
As these cases accumulate, their answers can feed subsequent programs while their reasons and checks preserve the discipline that keeps the whole tree of insights coherent.