On July 20, 2026, mathematician Levent Alpöge published an explicit counterexample to the Jacobian conjecture, produced with the AI model Claude Fable 5: a polynomial map in three variables with Jacobian determinant −2 at every point — locally invertible everywhere — that nonetheless sends three different points to the same output. Mathematicians worldwide verified it within a day, because the object carries its own proof: the algebra is finite, exact, and checkable by anyone. An 87-year-old conjecture, closed in three lines. The two-variable case remains open.
Fable got it right. We can now show you the work — twice over, in two different languages.
Check one: the arithmetic, replayed from scratch.
We keep a small developmental organism: a system that learned its arithmetic from a single innate step — counting — and built signed fractions, algebra, and the rest as a language on top. Every operation it performs is a recorded, replayable event. We gave it the published map and the three points. It evaluated everything in its own fraction arithmetic:
F(0, 0, -1/4) = (-1/4, 0, 0) OK
F(1, -3/2, 13/2) = (-1/4, 0, 0) OK
F(-1, 3/2, 13/2) = (-1/4, 0, 0) OK
control F(0,0,0) = (0, 0, 0) OK — the map is not constant
Three distinct inputs, one output, determinant −2 everywhere: not injective. The refutation, independently re-established by a system whose every add and multiply can be audited. Credit stated precisely: it verified; it did not discover. The discovery belongs to Alpöge and Fable 5. Verification that can be replayed step-by-step is our contribution — and it is the thing machine mathematics is currently starving for.
Check two: the collision as a physical event.
The deeper demonstration is the second run. This organism does not speak language — it responds to what it is fed. So instead of telling it anything, we fed it the problem: each mapped output, encoded as a signal, was pushed into its substrate, and we read what the substrate did. No sentence anywhere in the loop — the record is raw observables:
A F(0,0,-1/4) -> settled 55 iterations energy 3.181981 state 8de9aa884e9374a1 B F(1,-3/2,13/2) -> settled 55 iterations energy 3.181981 state 8de9aa884e9374a1 C F(-1,3/2,13/2) -> settled 55 iterations energy 3.181981 state 8de9aa884e9374a1 D F(0,0,0) control -> settled 56 iterations energy 1.198958 state 3802303db91aef9c A.state == B.state == C.state : True D.state != A.state : True
Three different starting points; the substrate came to rest in one identical configuration — the same settled state to the last byte. The control landed somewhere else entirely. That is what "not injective" feels like from the inside: the same pressure arriving by three different roads.
The answer was read from the substrate, not written by an author.
Claimed: independent re-verification of the published counterexample, in exact fraction arithmetic, every step recorded; and a second confirmation in which identical settled states were read as observables, with a control that settled differently.
Not claimed: discovery (Alpöge / Fable 5); any progress on the open two-variable case; any result beyond the specific points checked. In the second run the substrate acts as a witness to the arithmetic — identical inflows settle identically — not as an independent oracle.
Also disclosed: the inputs were chosen by the operators. The organism does not pose its own questions — it responds to what it is fed. Where its language runs out, its record says held rather than inventing. Our full run log — including our own mistakes, kept permanently — backs every line of this page.
The road is missing.
Be precise about what the world actually received on July 20. Three different things, and only two of them exist:
WHAT Fable found — shown. The three polynomials are public; anyone can check the object. THAT it is true — shown. The object carries its own proof; the algebra is finite and exact. HOW Fable got there — invisible. To everyone, including its makers. A frontier model has no trace, no open weights, no replayable process. The reasoning that produced the biggest mathematical result of the year does not exist anywhere as an auditable record. The celebrated "digestion" of the counterexample is a great mathematician reconstructing, by hand, after the fact, a path that could have led there — a plausible story, offered because the actual path is sealed inside the weights. Nobody knows if that is how it happened. Nobody ever will.
So the scoreboard, honestly: the frontier model produced a new result down a road no one can inspect. Our organism, in these runs, produced no new mathematics — because nothing new was asked of it: every input here was a checking task on known ground, and its reach on any problem is set by what it is fed. Whether it can reach something genuinely new is an open experiment, not a settled limit. What is settled is the other half: every inch of every road it takes is replayable. The empty square is both at once: new mathematics, with the entire road shown. Nobody on Earth occupies it yet. That square is what this system is being built toward — a mind that, when it produces its own counterexample, hands the world not just the object but every step from inflow to settled state, stamped and replayable by anyone.
Why we work this way.
Mathematics is in a trust crisis of AI's making. The field's leading figures say plainly that AI output without validation is too unreliable for serious use, while journals fill with machine-generated proofs nobody can check. The prevailing answer is formal verification — necessary, but it audits the finished artifact, not the process that produced it. We are building the other thing: a system whose process is the audit — arithmetic that exists as replayable events, results stamped with content hashes at the moment of the run, and a changelog that records failures as faithfully as passes, including the failures of the humans and AI systems doing the teaching.
The organism is small. This page shows a property, not power: a machine that computes only what it can trace, confirms what it can witness, and refuses the rest out loud. The substrate's design is deliberately not described here.