Proved, Certified, Swept, Sampled
A claim in a paper can be backed by very different things. It can have a written proof. It can have a theorem the Lean kernel has checked. It can have a certificate that ran exact rational arithmetic over an entire region and never once used a floating-point number. Or it can have "I sampled ten million configurations and none of them broke it". All four print the same way: a sentence in a serif…
The author of the paper discusses the different ways a claim can be backed up, such as having a written proof, a theorem checked by the Lean kernel, a certificate that runs exact rational arithmetic over an entire region, or sampling ten million configurations and finding none that break it. However, the latter is not as reliable as the former, as evidenced by a certificate that lied to the author three times.
One such instance involved a claim about a quintet of rings in a four-dimensional domain of parameters. Initially, the claim was certified computationally by subdividing the domain into boxes and evaluating at each box, but this was refuted by an adversarial reviewer. The second attempt used rational-directed arithmetic and was also refuted, highlighting the issue with tolerances.
A tolerance of 1e-12 was used as slack, which, at the golden point where the parameters equal the golden ratio, caused the tangency to be replaced by a tolerance, rendering the certificate not evidence of the geometry. The author emphasizes the importance of relying on written proofs and exact symbolic computations rather than verification workflows and numerical sweeps, which are quality control and evidence, not substitutes for proof.
Written by urgent.news from Dev.to's reporting — not their text. Machine-written — may contain errors; check the original before relying on it.
