Internal mode-induced growth in $3$d nonlinear Klein-Gordon equations
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Bibliographic record
Abstract
This note complements the paper \cite{LP} by proving a scattering statement for solutions of nonlinear Klein-Gordon equations with an internal mode in $3$d. We show that small solutions exhibit growth around a one-dimensional set in frequency space and become of order one in $L^{\infty}$ after a short transient time. The dynamics are driven by the feedback of the internal mode into the equation for the field (continuous spectral) component. The main part of the proof consists of showing suitable smallness for a "good" component of the radiation field. This is done in two steps: first, using the machinery developed in \cite{LP}, we reduce the problem to bounding a certain quadratic normal form correction. Then we control this latter by establishing some refined estimates for certain bilinear operators with singular kernels.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| 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