Linear bounds for constants in Gromov’ssystolic inequality and related results
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Bibliographic record
Abstract
Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6c(n)vol(M^n)^{1\over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible closed curve in $M^n$. We prove that these results hold with $c(n)=({n!\over 2})^{1\over n}\leq {n\over 2}$. We demonstrate that for essential Riemannian manifolds $sys_1(M^n) \leq n\ vol^{1\over n}(M^n)$. All previously known upper bounds for $c(n)$ were exponential in $n$. Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius $r$ is less than $({r\over c(n)})^n$ can be replaced by a weaker assumption that for every point $x\in M^n$ there exists a positive $ρ(x)\leq r$ such that the volume of the metric ball of radius $ρ(x)$ centered at $x$ is less than $({ρ(x)\over c(n)})^n$ (for $c(n)=({n!\over 2})^{1\over n}$). Also, if $X$ is a boundedly compact metric space such that for some $r>0$ and an integer $n\geq 1$ the $n$-dimensional Hausdorff content of each metric ball of radius $r$ in $X$ is less than $({r\over 4n})^n$, then there exists a continuous map from $X$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$.
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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