Induced spatial geometry from causal structure
Bibliographic record
Abstract
Abstract Motivated by the Hawking–King–McCarthy–Malament (HKMM) theorem and the associated reconstruction of spacetime geometry from its causal structure and local volume element , we define a one-parameter family of spatial distance functions on a Cauchy hypersurface using only and . The parameter corresponds to a ‘mesoscale’ cut-off which, when appropriately chosen, provides a distance function which approximates the induced spatial distance function to leading order. This admits a straightforward generalisation to the discrete analogue of a Cauchy hypersurface in a causal set. For causal sets which are approximated by continuum spacetimes, this distance function approaches the continuum induced distance when the mesoscale is much smaller than the scale of the extrinsic curvature of the hypersurface, but much larger than the discreteness scale. We verify these expectations by performing extensive numerical simulations of causal sets which are approximated by simple spacetime regions in 2 and 3 spacetime dimensions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".