Independence of synthetic Curvature Dimension conditions on transport\n distance exponent
Bibliographic record
Abstract
The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes\nsynthetic notions of a Ricci curvature lower bound $K$ joint with an upper\nbound $N$ on the dimension. Their condition, called the Curvature-Dimension\ncondition and denoted by $\\mathrm{CD}(K,N)$, is formulated in terms of a\nmodified displacement convexity of an entropy functional along\n$W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance\nfunction as transport cost does not influence the theory. By denoting with\n$\\mathrm{CD}_{p}(K,N)$ the analogous condition but with the cost as the\n$p^{th}$ power of the distance, we show that $\\mathrm{CD}_{p}(K,N)$ are all\nequivalent conditions for any $p>1$ -- at least in spaces whose geodesics do\nnot branch. We show that the trait d'union between all the seemingly unrelated\n$\\mathrm{CD}_{p}(K,N)$ conditions is the needle decomposition or localization\ntechnique associated to the $L^{1}$-optimal transport problem. We also\nestablish the local-to-global property of $\\mathrm{CD}_{p}(K,N)$ spaces.\n
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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 | 0.000 | 0.000 |
| 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) | 0.000 | 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 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".