Proof of Atiyah's conjecture for two special types of configurations
Bibliographic record
Abstract
Abstract. To an ordered N-tuple (x1,..., xN) of distinct points in R 3, Atiyah [1, 2] has associated an ordered N-tuple of homogeneous polynomials (p1,..., pN) in C[x,y] of degree N − 1, each pi determined only up to a scalar factor. He has conjectured that these polynomials are linearly independent. We show that his conjecture is true for two special configurations of N points. Moreover we show that, for one of these configurations, the stronger conjecture [3, Conjecture 2] is valid. Key words. Atiyah’s conjecture, the Hopf map, configuration of N points in R 3, projective line PC 1. AMS subject classifications. Primary 51M04, 51M16, Secondary 70G25 1. Two conjectures. Let (x1,..., xN) be an ordered N-tuple of distinct points in R 3. Each ordered pair (xi, xj) with i ̸ = j determines a point xj − xi |xj − xi| on the unit sphere S2 ⊂ R3. Identify S2 with the complex projective line PC 1 by using a stereographic projection. Hence one obtains a point (uij, vij) ∈ PC 1 and a nonzero linear form lij = uijx + vijy ∈ C[x, y]. Define homogeneous polynomials pi ∈ C[x, y] of degree N − 1 by pi = ∏ lij(x, y), i = 1,..., N. (1.1) j̸=i Conjecture 1.1. (Atiyah [2]) The polynomials p1,..., pN are linearly independent. Atiyah [1, 2] has observed that his conjecture is true if the points x1,..., xN are collinear. He has also verified the conjecture for N = 3. The case N = 4 has been verified by Eastwood and Norbury [4]. For additional information on the conjecture (further conjectures, generalizations, and numerical evidence) see [2, 3]. In order to state the second conjecture, one has to be more explicit. Identify R 3 with R × C and denote the origin by O. Following Eastwood and Norbury [4], we make use of the Hopf map h: C 2 \\ {O} → (R × C) \\ {O} defined by: h(z, w) = ((|z | 2 − |w | 2)/2, z ¯w). This map is surjective and its fibers are the circles {(zu, wu) : u ∈ S 1}, where S 1 is the unit circle in C. If h(z, w) = (a, v), we say that (z, w) is a lift of (a, v). For
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.014 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".