counterexample

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The Maxwell Conjecture Is False (GPT 5.6 Sol)
2 months ago
- The Maxwell conjecture claims that the electrostatic potential of n point charges has at most (n-1)^2 non-degenerate critical points.
- The authors exhibit a configuration of five point charges whose potential has at least 24 critical points, all non-degenerate.
- Since 24 > 16 = (5-1)^2, this counterexample proves the Maxwell conjecture is false.
Locally everywhere does not imply everywhere
2 months ago
- Levent Alpöge discovered a counterexample to the Jacobian conjecture using Claude Fable 5.
- The Jacobian conjecture claims that a polynomial function with constant nonzero Jacobian determinant has a polynomial inverse.
- Alpöge's counterexample is a polynomial from ℝ³ to ℝ³ with constant Jacobian determinant -2 but is not globally invertible (multiple inputs map to same output).
- This disproves the conjecture for n ≥ 3, but the n = 2 case remains open.
- Claude (the AI) was unaware of its role in the discovery, as it was just a tool used by the mathematician.
LLMs break down in funny ways when told the Jacobian Conjecture counterargument
2 months ago
- Anthropic researcher Levent Alpöge tweeted a counterargument to the Jacobian Conjecture, verified via Claude Fable 5 and empirically confirmed.
- The counterargument creates a paradox for LLMs: they can verify it mathematically but their knowledge cutoff says the conjecture is unsolved, leading to confusion.
- The author tested 14 LLMs with the same query about the counterargument, observing varied responses: some verified, some rejected, some got confused.
- A variant prompt attributing the discovery to a cat elicited humorous and snarky responses from the same LLMs.
- LLMs handled the cognitive dissonance relatively gracefully, contrary to the 'DOES NOT COMPUTE' trope from science fiction.
Jacobian Conjecture for Baby
2 months ago
- A function is a machine that maps each input to exactly one output, and can be visualized as moving every point on a number line.
- A function is undoable if there are no collisions (two inputs to the same output) and no gaps (outputs without an input), making it invertible.
- Polynomials are machines built only from addition and multiplication, the simplest concretely checkable functions.
- A plane map moves every point of the plane using two polynomial rules; it can be undone if its Jacobian determinant (local area factor) is nonzero everywhere.
- Local undoability (nonzero Jacobian everywhere) does not guarantee global undoability; collisions can occur between distant points unseen by a local microscope.
- The Jacobian Conjecture asks: if a polynomial map's local area factor is a constant nonzero everywhere, must it be globally undoable with a polynomial inverse?
- The conjecture was unsolved for 87 years, with all tests confirming it, partial proofs for low degrees, and obstacles including real-number counterexamples and clock arithmetic failures.
- In July 2026, an explicit degree-7 counterexample in three dimensions was discovered: a polynomial map with constant Jacobian determinant -2 but three different points mapping to the same output, disproving the conjecture for dimensions 3 and higher, while the original two-dimensional case remains open.
A digestion of the Jacobian conjecture counterexample
2 months ago
- The Jacobian conjecture asserts that a polynomial map with non-zero constant Jacobian is globally invertible.
- A counterexample in three dimensions (and higher) has been found, disproving the conjecture for dimensions ≥3.
- The counterexample is constructed using a multiplication map of homogeneous polynomials in two variables.
- Local injectivity is achieved by normalizing via resultants and using the PSL(2,C) action.
- Global non-injectivity arises from the fact that a cubic polynomial can factor into linear and quadratic parts in three ways.
- A three-dimensional slice of a four-dimensional variety is shown to be isomorphic to affine space, yielding the desired polynomial map.
With the Help of Codex: A Reproducible Proof Against the Jacobian Conjecture
2 months ago
- The post presents a polynomial map from C³ to C³ with constant Jacobian determinant -2 that maps three distinct points to the same point, contradicting the Jacobian Conjecture.
- The map is credited to Levent Alpoge, and the post provides an independent symbolic verification using SymPy, with exact rational arithmetic and no floating-point numbers.
- Attached files (PDF and Python script) allow readers to reproduce the verification, which includes computing the Jacobian determinant, evaluating the map at three points, and solving for the fiber.
- The verification confirms F(0,0,-1/4) = F(1,-3/2,13/2) = F(-1,3/2,13/2) = (-1/4,0,0), with all three inputs distinct and no other solutions.
- The post emphasizes that this extraordinary claim requires independent reproduction and expert scrutiny, and provides clear instructions to run the script and verify the checksum.

counterexample

The Maxwell Conjecture Is False (GPT 5.6 Sol)

Locally everywhere does not imply everywhere

LLMs break down in funny ways when told the Jacobian Conjecture counterargument

Jacobian Conjecture for Baby

A digestion of the Jacobian conjecture counterexample

With the Help of Codex: A Reproducible Proof Against the Jacobian Conjecture