Proof of Atiyah's conjecture for two special types of configurations
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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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 it