Semiclassical quantum gravity: obtaining manifolds from graphs
Bibliographic record
Abstract
We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a unique manifold in an appropriate sense, and constructing the manifold when it exists. This problem arises in a variety of approaches to quantum gravity that assume a discrete structure at the fundamental level; the present work is motivated by the semiclassical sector of loop quantum gravity, so we will take the discrete structure to be a graph and the manifold to be a spatial slice in spacetime. We identify a class of graphs, those whose vertices have a fixed valence, for which such a construction can be specified. We define a procedure designed to produce a cell complex from a graph and show that, for graphs with which it can be carried out to completion, the resulting cell complex is in fact a PL-manifold. Graphs of our class for which the procedure cannot be completed either do not arise as edge graphs of manifold cell decompositions, or can be seen as cell decompositions of manifolds with structure at small scales (in terms of the cell spacing). We also comment briefly on how one can extend our procedure to more general graphs.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".