On rank-2 Toda systems with arbitrary singularities: local mass and new estimates
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Bibliographic record
Abstract
For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:(1) The pair of local masses . 1 ; 2 / at each blowup point has the expressionwhere N ij 2 , i D 1; 2, j D 1; 2; 3.(2) At each vortex point p t if .1t ; 2 t / are integers and i 4 , then all the solutions of Toda systems are uniformly bounded.(3) If the blowup point q is a vortex point p t and 1 t ; 2 t and 1 are linearly independent over Q, thenThe Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point. where g is the Laplace-Beltrami operator ( g 0), S is a finite set on M, h 1 ; : : : ; h n are positive and smooth functions on M, i t > 1 is the strength of the Dirac mass p t and D . 1 ; : : : ; n / is a constant vector with nonnegative components.Here for simplicity we just assume that the total area of M is 1.
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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 | 0.000 |
| 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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