Geometric completion of differential systems using numeric-symbolic continuation
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Bibliographic record
Abstract
Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and under-determined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases the application of exact or numerical integration methods.Motivated to avoid expression swell of pure symbolic approaches and with the desire to handle systems with approximate coefficients, we propose the use of homotopy continuation methods to perform the differential-elimination process on such non-square systems. Examples such as the classic index 3 Pendulum illustrate the new procedure. Our approach uses slicing by random linear subspaces to intersect its jet components in finitely many points. Generation of enough generic points enables irreducible jet components of the differential system to be interpolated.
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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 | 0.000 | 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 | 0.001 | 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 it