A complete classification of homogeneous plane continua
Why this work is in the frame
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Bibliographic record
Abstract
We show that the only compact and connected subsets (i.e. continua) X of the plane R2 which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle S1, the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps f,g:C→X such that f(C)⊂g(C) there exists c0∈C so that f(c0) = g(c0). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 it