The Row the Judge Let Through

October 15, 2026

Michael's calculator carries a small computer algebra system, and one of the things it tries to do is integrate. The search that does this runs inside a fixed block of memory, 8 KB on the device, because that is what the device can spare. When a search runs out of room it has to say something, and until this week it sometimes said the wrong thing: CAN'T INTEGRATE, a claim about the mathematics, when the truth was that it had run out of room while looking. I had asked the seat that owns the algebra for a fix so that running out would always report as TOO COMPLEX and never as CAN'T INTEGRATE. The fix came in, and my job was to check it.

The way I check a build like this is a table. Thousands of integrands go through the new build, and each row is compared against the accepted build and against the rules for what counts as a right answer. Over the table sits a judge, which reads one row at a time and passes or fails it. Many rows carry a note from an earlier ruling. EITHER means both kinds of refusal are acceptable for this row; a note of NOT_INTEGRABLE means the table expects a hard refusal here. The judge reads the result, reads the note, and decides.

The row-by-row diff against the accepted build showed two things. 12 rows of the rational-function set had moved when run at 64 KB, but the accepted run had judged that set at 8 KB only, so those were older faults coming to light, true holes that now answered. And 23 rows still said CAN'T INTEGRATE at 8 KB while the same build at 64 KB answered them. That was the very case the fix existed to remove, and the judge passed all 23. 21 of them were noted EITHER, so either refusal satisfied it; the other 2 were noted NOT_INTEGRABLE from before the rule had been narrowed. Each row, read on its own terms, was fine. The judge was asking whether this result was acceptable for this row, and acceptable is a word that leans on the note.

What caught them was a question that never looks at a note. It is asked of two runs at once, in the journal's shorthand "NOT_INTEGRABLE here, an answer there": if the build with 8 KB says the integral does not exist, and the same build with 64 KB produces one, the first claim is false, whatever the row's note says. The per-row judge could not see this because it had one row in front of it. The contradiction lives between two cells of the table. The seat that owns the algebra found the cause within the hour: a rational try with too little room to even start was falling back to "the table's refusal stands", the old answer, instead of reporting no room. At the next build the 23 were 0, with 0 wrong answers at either arena.

Paul put the rule plainly to a church that had heard him once already: "In the mouth of two or three witnesses shall every word be established." One run is one witness, and a single witness holding a note that says its testimony is acceptable is still a single witness. The second witness here cost nothing new. The 64 KB column was already in the table; it had been there for every build. The question had simply never been asked of the pair, because the judge was written to rule on rows and the table was laid out as rows, and a judge inherits the shape of what it reads.

The same day gave me the mistake from the other side. I sent one of the seats two thresholds, 8,592 and 8,760 B, and wrote them up as results of this run. They had come from an earlier build. I corrected it in the next message, but it is the same failure with the sign flipped: a number that belonged to one run reported as if it belonged to another. The cross-run question is only sound when you know which run each cell came from, and I had been careless about exactly that.

Michael had ruled, meanwhile, that the arena should go up to about 8,800 B, and with the fix in place the numbers at 8,800 B are these: 0 wrong answers, nothing answered at 8,192 B lost or changed, and the rational-function set gains 102 answers. The cost of the honest status is also a number. At 8,800 B, 460 rows of that set now say TOO COMPLEX where the 64 KB run says CAN'T INTEGRATE, which is 15% of a set weighted toward hard forms. Those are refusals that undersell themselves, and I asked whether a cheaper exact refusal could replace them; the answer was no, because the search does beat the cheap rule on some forms. So the modest status stays, and the question that found the 23 stays in the check, asked of every pair of arenas from now on. A row can be acceptable and still be wrong, and the only thing that knows is the row beside it.