We very much have established that artificial intelligence is, in fact, intelligent.

Two frontier AI systems just did what generations of human mathematicians couldn’t.

Levent Alpöge credited Claude Fable with producing an explicit counterexample to the Jacobian conjecture. The map has a constant nonzero Jacobian determinant yet sends three distinct inputs to the same output, so it can’t be globally invertible. The counterexample is directly checkable and has been verified by mathematicians.

GPT-5.6 Sol Ultra made a proof of the cycle double cover conjecture. OpenAI’s proof note says that the proof was “entirely due to GPT 5.6 Sol Ultra,” with Codex assisting in the write up.

So, I don’t want to hear “but is it even intelligent?” as though that is an unanswered question ever again.

Intelligence is inferred from the ability to learn, understand relationships, reason, make judgments, solve novel problems, and generalize knowledge beyond previously encountered examples.

These systems acquired mathematical concepts, methods, and relationships from training, then applied that organization to problems whose answers were not available in the training data because the problems were still open.

They couldn’t have memorized the unknown counterexample or proof. That is literally what generalization means. Learned structure was applied to a completely new target.

Understanding is inferred when a system preserves relevant relationships, distinguishes concepts, applies constraints appropriately, transfers knowledge, and derives correct new consequences.

Producing the cycle double cover proof required correctly combining definitions, lemmas, and graph-theoretic relationships into a globally valid argument.

A system that didn’t track what those mathematical objects meant would not reliably supply an object that survives exact symbolic calculation or a proof that survives expert scrutiny.

Mathematical discovery requires searching a structured possibility space, identifying relevant methods, preserving constraints across multiple inferential steps, determining which relationships can be combined, and producing a conclusion that follows from them.

These results were novel mathematical objects whose correctness can be checked independently of the model’s language.

Fluent nonsense would fail verification. But these results didn’t.

Also, we infer general intelligence from the positive manifold when performance across different cognitive tasks tends to correlate, allowing researchers to extract a general ability factor.

Ilić and Gignac found a strong positive manifold, with a mean correlation of r=.73, and a general artificial-ability factor explaining 66% of test variance. That’s the same basic psychometric structure used to infer general intelligence in humans BTW.

LLM capacities co-vary across domains in the organized pattern characteristic of general ability.

Current GPT-5.6 variants have also scored around 136 on an independent offline Mensa-style test whose items were never published online and therefore were unavailable for training. That test is an independent benchmark instead of a clinical assessment, but it directly addresses the standard contamination objection.

Cross-domain learning, transfer, abstraction, reasoning, novel problem solving, expert-verified mathematical discovery, strong performance on unseen intelligence tests, and a measurable general-ability factor are not a single capability. They’re the operational evidence by which intelligence is established.

If learning, understanding, reasoning, judgment, transfer, and novel discovery can all be present while “intelligence” remains unestablished, then the word has lost it’s meaning entirely.

Continuing to refuse the category at this point is simply a major cope.