Bibliographic record
Abstract
In this paper we study spiderweb central configurations for the N-body problem, i.e configurations given by $$N=n \times \ell +1$$ masses located at the intersection points of $$\ell $$ concurrent equidistributed half-lines with n circles and a central mass $$m_0$$ , under the hypothesis that the $$\ell $$ masses on the i-th circle are equal to a positive constant $$m_i$$ ; we allow the particular case $$m_0=0$$ . We focus on constructive proofs of the existence of spiderweb central configurations, which allow numerical implementations. Additionally, we prove by a rigorous numerical method the uniqueness of such central configurations when $$\ell \in \{2,\ldots ,9\}$$ and arbitrary n and $$m_i$$ ; under the constraint $$m_1\ge m_2\ge \cdots \ge m_n$$ we also prove uniqueness for $$\ell \in \{10,\ldots ,18\}$$ and n not too large. We also give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of n, $$\ell $$ and $$m_0, \ldots ,m_n$$ . Finally, our numerical simulations highlight some interesting properties of the mass distribution.
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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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".