Global positioning: The uniqueness question and a new solution method
Bibliographic record
Abstract
We provide a new algebraic solution procedure for the global positioning problem in n dimensions using m satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when m ≥ n + 2 , the solution is unique for almost all user positions. Even better, when m ≥ 2 n + 2 , almost all satellite configurations will guarantee a unique solution for all user positions. Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness. Some of our results are obtained using tools from algebraic geometry.
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How this classification was reachedexpand
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 | 0.001 |
| 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) | 0.000 | 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 teacher head, 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".