An infinite hierarchy can have no movable parts. OpenAI’s September 24, 2026 paper, Rigidity of the Turing degrees, proves that every symmetry preserving relative computability fixes every degree. The ordering itself leaves no room to exchange computational powers while keeping every relationship intact. That is the answer to the rigidity of the Turing degrees problem. … Read more
An inverse is supposed to undo an operation. OpenAI’s September 23, 2026 preprint constructs two finite algebraic expressions that undo each other in one order and fail in the other. The Kaplansky Direct Finiteness Conjecture predicted that this could never happen in a group algebra. The paper presents a characteristic-two counterexample, with a finitely presented … Read more
A factoring algorithm that almost always works is not the same as one that cannot fail. That distinction drives exact quantum factoring, the subject of OpenAI’s September 25, 2026 preprint. Its claimed advance is a polynomial-time-uniform family of quantum circuits that returns the complete prime factorization of every integer at least two with probability exactly … Read more
A proof can identify an elliptic curve’s rank and still miss the exact arithmetic hiding in its first nonzero L-value. That missing precision is the subject of OpenAI’s October 7 manuscript on the Birch and Swinnerton Dyer Conjecture. It presents the full leading-term formula for elliptic curves over the rationals whose full Selmer group has … Read more
A quantum circuit can spend much of its budget remembering how to undo its own work. The new Claude quantum circuits research attacks that overhead. Announced by Anthropic on October 16, 2026, the project used Claude Mythos 5 to redesign eight computational primitives. Seven designs reduced both Toffoli gates and qubits. The eighth introduced reversible … Read more
An injection can compare two infinities without choosing a single correct representative. That gap is the heart of the Partition Principle vs Axiom of Choice problem. OpenAI’ss September 24, 2026 paper presents a negative answer: assuming Zermelo–Fraenkel set theory is consistent, the Partition Principle can hold while the Axiom of Choice fails. The construction also … Read more
Remove the alternating signs from a determinant, and a familiar matrix formula becomes a central challenge in algebraic complexity. OpenAI’s September 24, 2026 paper advances the permanent vs determinant problem by proving a cubic lower bound on the size of determinant representations, even when coefficientwise limits are allowed. The previous unrestricted complex border bound was … Read more