The increment protocol

The failure mode of naive prompting is not that the model produces wrong mathematics. It is that it produces unfalsifiable mathematics — fluent, confident prose that gestures at an argument without ever committing to a checkable claim. Output of that kind cannot be refuted, which means it also cannot be evaluated.

The protocol removes the option. The model is told it has not proved the problem, that claiming otherwise is an error, and that if its own work appears to resolve the problem the correct response is to hunt for the mistake. What it is asked for instead is an increment: a lemma with a proof, a sharpened bound with the improvement stated, a reformulation with the equivalence justified, or — most often, and most usefully — a precise account of where an approach breaks.

Every step must carry one of three labels. [ESTABLISHED] marks a standard result being cited, and must be accompanied by the result's name; a model that cannot name the theorem is instructed to say so rather than gesture at it. [DERIVED] marks a step justified within the increment itself. [CONJECTURAL] marks anything believed but unjustified.

An unlabelled step is treated as an error. The labels are not verification — the model can mislabel, and does — but they convert vague confidence into a specific claim that a reader can check line by line.

Every increment closes with an explicit statement of what remains unjustified. In practice this line is the most informative part of the output, because it is where the model is forced to be specific about its own gap rather than trailing off into optimism.