Classifying Minimum Energy States for Interacting Particles: Spherical Shells
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems and to predict or explain many of the patterns they display. Apart from rare values of the attractive and repulsive exponents $(\alpha,\beta)$, the energy minimizing configurations of particles are not explicitly known, although simulations and local stability considerations have led to conjectures with strong evidence over a much wider region of parameters. For dimension $n\ge 2$ and for a segment $\beta=2<\alpha<4$ on the mildly repulsive frontier we employ strict convexity to conclude that the energy is uniquely minimized ($d_\infty$-locally, up to translation) by a spherical shell. If $n=1$ and $\beta=2<\alpha-1$, we prove that the spherical shell is (i) the unique global energy minimizer and (ii) the unique $d_\infty$-local energy minimizer in the class of even, compactly supported probability measures. In a companion work, we show that, in the mildly repulsive range $\alpha>\beta\ge2$, a unimodal threshold $2<\alpha_{\Delta^n}(\beta) \le \max\{\beta,4\}$ exists such that equidistribution of particles over a unit diameter regular $n$-simplex minimizes the energy if and only if $\alpha \ge \alpha_{\Delta^n}(\beta)$ (and minimizes uniquely up to rigid motions if strict inequality holds). For $n\ge 2,$ the point $(\alpha,\beta)=(2,4)$ separates these regimes. At this point we show the minimizers all lie on a sphere and are precisely characterized by sharing all first and second moments with the spherical shell. Although the minimizers need not be asymptotically stable, our approach establishes $d_\alpha$-Lyapunov nonlinear stability of the associated ($d_2$-gradient) aggregation dynamics near the minimizer in both of these adjacent regimes---without reference to linearization. The $L^\alpha$-Kantorovich--Rubinstein--Wasserstein distance $d_\alpha$ which quantifies stability is chosen to match the attraction exponent.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 it