Sizes of spaces of triangulations of 4-manifold and balanced presentations of the trivial group
Bibliographic record
Abstract
Let [Formula: see text] be any compact four-dimensional PL-manifold with or without boundary (e.g. the four-dimensional sphere or ball). Consider the space [Formula: see text] of all simplicial isomorphism classes of triangulations of [Formula: see text] endowed with the metric defined as follows: the distance between a pair of triangulations is the minimal number of bistellar transformations required to transform one of the triangulations into the other. Our main result is the existence of an absolute constant [Formula: see text] such that for every [Formula: see text] and all sufficiently large [Formula: see text] there exist more than [Formula: see text] triangulations of [Formula: see text] with at most [Formula: see text] simplices such that pairwise distances between them are greater than [Formula: see text] ([Formula: see text] times). This result follows from a similar result for the space of all balanced presentations of the trivial group. (“Balanced” means that the number of generators equals to the number of relations). This space is endowed with the metric defined as the minimal number of Tietze transformations between finite presentations. We prove a similar exponential lower bound for the number of balanced presentations of length [Formula: see text] with four generators that are pairwise [Formula: see text]-far from each other. If one does not fix the number of generators, then we establish a super-exponential lower bound [Formula: see text] for the number of balanced presentations of length [Formula: see text] that are [Formula: see text]-far from each other.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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".