The saturation conjecture for even spin groups
From OpenAI Aftermath
The saturation conjecture for even spin groups is the type-D case of the conjecture of Kapovich and Millson on tensor products of representations of simply laced groups: if three dominant weights sum to the root lattice and the tensor product of the representations with highest weights stretched by a positive integer N contains an invariant, then the original tensor product already contains one. A manuscript in OpenAI's public mathematics collection, dated 24 September 2026, claims the conjecture for every even spin group Spin(2n), including the half-integral spin weights. The collection lists no Lean formalization of this result.
Background
- As recounted in the manuscript's introduction:
- Eigenvalues and tensor products. Klyachko connected the Hermitian eigenvalue problem with tensor occurrence after dilation, and Berenstein and Sjamaar placed the relation between moment polytopes and occurrence after dilation in a general branching framework. Saturation asks whether real feasibility together with the central congruence already detects an invariant at the original weights.
- Type A and the conjecture. Knutson and Tao proved saturation in type A with honeycombs, Derksen and Weyman gave a quiver proof and Belkale a geometric proof. Kapovich and Millson formulated the root-lattice assertion for simply laced groups.
- Spin groups and saturation factors. Kapovich, Kumar and Millson proved saturation for Spin(8), and Kiers for Spin(10) and Spin(12). Saturation factors were obtained by Kapovich and Millson, a factor of four in type D, by Belkale and Kumar and by Sam, factor two for symplectic and orthogonal groups and four for spin groups, and by Hong and Shen, factor two for odd spin groups under the root-lattice congruence; these do not give the conclusion at scale one. Min proved saturation for Newell–Littlewood numbers of partitions of even total size.
The original result
- Claim, 24 September 2026. The main theorem of Tensor saturation for even spin groups states that for G = Spin(2n), n ≥ 2, and dominant integral weights λ, μ, ν whose sum lies in the root lattice of type Dn, the tensor product of the irreducible representations with these highest weights contains a nonzero invariant if and only if the tensor product with highest weights Nλ, Nμ, Nν does, for every positive integer N. The manuscript describes this as the type-D case of the conjecture of Kapovich and Millson; it includes the half-integral spin weights.
- Scope, as stated: the result concerns three tensor factors and the even spin groups; the root-lattice condition is necessary, and the stretching parameter is a positive integer.
- Method, as described: an invariant at a stretched triple gives a real solution of the corresponding eigenvalue problem. From it the manuscript builds a schedule, a polygonal path that follows Weyl translates of the increments of a path to μ and changes direction by root reflections; an area maximum and a torus-orbit polytope argument give a real schedule whose switch roots form a basis. In type D the only obstruction to integral switch coefficients comes from two unbalanced components of a signed root graph, and motion along half of a primitive relation removes it; rank-two reorderings and a spectral multiplicity count keep the path admissible. Exterior and spin columns then convert integral schedules into tensor intertwiners through a common-wall invariant covector, and a preprojective-module doubling construction removes a common terminal block of zero coordinates.
- Sources. Commit
adc7f12of the openai/math repository: Tensor saturation for even spin groups. - Formalization. The collection lists no Lean scope note or Comparator challenge for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 1 November 2026) |
|---|---|---|---|---|
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_096 in JJardine919/openai-math-audit | Review, AI-assisted audit | Reports no gap and lists no issues. It states that the reviewer read all six sections and rederived the load-bearing steps by hand, among them the index of the signed root lattice, the bound on the half-relation, the common-wall lemma and the preprojective doubling, and that the analytic details of a symmetric-space triangle lemma and a cited compatibility of the semicanonical-basis filtration were not independently re-verified. | Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0. |
| 2026-10-17 | AI Proved What?, episode 204 Only after stretching?, remade version of 24 October 2026 | Reading, exposition | Video explainer of the 46-page preprint. It recalls Knutson and Tao, the conjecture of Kapovich and Millson and the earlier cases Spin(8), Spin(10) and Spin(12), and gives an example in which three copies of the five-dimensional representation of the rotation group in five dimensions pass the root-lattice test without an invariant, while an invariant appears after doubling. It states that the exceptional groups, formulas for the number of invariants and fractional stretches are not claimed. It reports its own scans: among 76,615 small triples for Spin(6) through Spin(16) none fails at the stretches tried, while the odd orthogonal and symplectic groups show 226 failures, all at stretch two; it says that these small ranks were mostly known and that the scans do not test the proof. | Episode status, last checked 30 October 2026: posted as a preprint, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, not formalized; not among the manuscripts that OSVerify reviewed; no expert reaction found. The site states that it is made with Anthropic's Claude. Read on 1 November 2026. |
| 2026-10-18 | AI Proof Watch, record OpenAI reports tensor saturation for even spin groups | Catalogue record, assessment | Rates the importance as notable and the AI involvement as AI-led. It lists as limitations that the root-lattice condition is essential and that the claim is for three tensor factors and type D only, and it rates the technical verification as reported and the human understanding as few, the defaults when no completed check or named explanation is documented. | Curated public tracker; record published 18 October 2026, assessment reviewed 31 October 2026. Read on 1 November 2026. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
