Heyde theorem for locally compact Abelian groups containing no subgroups topologically isomorphic to the 2-dimensional torus
Bibliographic record
Abstract
We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let $X$ be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$ and let $α$ be a topological automorphism of the group $X$ such that ${\rm Ker}(I+α)=\{0\}$. Let $ξ_1$ and $ξ_2$ be independent random variables with values in $X$ and distributions $μ_1$ and $μ_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form $L_2 = ξ_1 + αξ_2$ given $L_1 = ξ_1 +ξ_2$ is symmetric, then $μ_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in $G$. We also prove that this theorem is false if $X$ is the 2-dimensional torus.
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How this classification was reachedexpand
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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".