Bibliographic record
Abstract
We define compactness of a gravitational lens as the scaled closest distance of approach (i.e., r0/ M) of the null geodesic giving rise to an image. We model 40 supermassive dark objects as Schwarzschild lenses and compute compactness of lenses (determined by the formation of the first-order relativistic image). We then obtain a novel formula for the compactness of a lens as a function of mass to the distance ratio ( M/ Dd) and the ratio of lens–source to the observer–source distances ( Dds/ Ds). This formula yields a very important result: Just an observation of a relativistic image would give an incredibly accurate upper bound to the physical compactness (the ratio of the radius to mass) of the lens without having any knowledge of mass of the lens, angular source position, and observer–source and lens–source distances. Similarly, we show that the observation of the second-order relativistic image would give a lower value of upper bound to the physical compactness. These results, though obtained for supermassive dark objects at galactic centers, are valid for any object compact enough to give rise to relativistic images.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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 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".