The Geometry of <i>d</i><sup>2</sup><i>y</i><sup>1</sup>/<i>dt</i><sup>2</sup> = <i>f</i> (<i>y</i>, <i>ẏ</i>, <i>t</i>) and <i>d</i><sup>2</sup><i>y</i><sup>2</sup>/<i>dt</i><sup>2</sup> = <i>g</i>(<i>y</i>, <i>ẏ</i>, <i>t</i>), and Euclidean Spaces
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
Abstract This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d 2 y 1 / dt 2 = f ( y , ẏ , t ) and d 2 y 2 / dt 2 = g ( y , ẏ , t ) be reparameterized by t → T ( y , t ) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.
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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 | 0.009 | 0.003 |
| Meta-epidemiology (narrow) | 0.009 | 0.009 |
| Meta-epidemiology (broad) | 0.010 | 0.005 |
| Bibliometrics | 0.004 | 0.009 |
| Science and technology studies | 0.007 | 0.009 |
| Scholarly communication | 0.005 | 0.003 |
| Open science | 0.010 | 0.005 |
| Research integrity | 0.004 | 0.010 |
| Insufficient payload (model declined to judge) | 0.010 | 0.007 |
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