On an invariance property of the space of smooth vectors
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Bibliographic record
Abstract
Let (π,H) be a continuous unitary representation of the (infinite-dimen- sional) Lie group G, and let γ:R→Aut(G) be a group homomorphism which defines a continuous action of R on G by Lie group automorphisms. Let π#(g,t)=π(g)Ut be a continuous unitary representation of the semidirect product group G⋊γR on H. The first main theorem of the present note provides criteria for the invariance of the space H∞ of smooth vectors of π under the operators Uf=∫Rf(t)Utdt for f∈L1(R) and f∈S(R), respectively. When g is complete and the actions of R on G and g are continuous, we use the above theorem to show that, for suitably defined spectral subspaces gC(E), E⊆R, in the complexified Lie algebra gC and H∞(F), F⊆R, for Ut in H∞, we have dπ(gC(E))H∞(F)⊆H∞(E+F).
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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