Capacity-Achieving Input Distributions of Additive Vector Gaussian Noise Channels: Even-Moment Constraints and Unbounded or Compact Support
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Bibliographic record
Abstract
We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in Rn or is restricted to a given compact subset of Rn. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of n-1 or less. When the input is restricted to a compact subset of Rn, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in Rn.
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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 | 0.001 | 0.002 |
| 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) | 0.001 | 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