Mass Growth Versus Categorical Entropy

Definitions and mathematical statement

Let $\mathcal D$ be a triangulated category and let $F\colon\mathcal D\to\mathcal D$ be an exact endofunctor. A Bridgeland stability condition $\sigma=(Z,\mathcal P)$ consists of a central charge $Z\colon K_0(\mathcal D)\to{\mathbb{C}}$ and a slicing satisfying the stability-condition axioms; in particular, every nonzero object has a finite Harder--Narasimhan filtration with semistable factors of strictly decreasing phases. If these factors for $E\ne0$ are $A_1,\ldots,A_r$, define its mass by \[ m_\sigma(E)=\sum_{j=1}^r |Z(A_j)|, \qquad m_\sigma(0)=0. \] A standard mass-growth convention at parameter zero is \[ h_\sigma(F)=\sup_{E\in\mathcal D,\ E\ne0} \limsup_{n\to\infty}\frac{1}{n}\log m_\sigma(F^nE), \] with $\log 0=-\infty$.

To give a concrete standard meaning to categorical entropy, suppose additionally that $\mathcal D$ is nonzero and essentially small and has a split generator $G$, meaning that the smallest full triangulated subcategory containing $G$ and closed under direct summands is $\mathcal D$. These are explicit working hypotheses for the formulation below; they are not specified in the supplied statement. Define $\delta_0(G,E)$ to be the infimum of all integers $r\geq0$ for which there are an object $E'$, integers $k_1,\ldots,k_r$, and a tower \[ 0=E_0,\ E_1,\ldots,E_r\cong E\oplus E' \] with distinguished triangles \[ E_{j-1}\longrightarrow E_j\longrightarrow G[k_j] \longrightarrow E_{j-1}[1] \qquad(1\leq j\leq r). \] Use $\inf\varnothing=+\infty$. The corresponding zero-parameter categorical growth rate is \[ h_0(F)=\limsup_{n\to\infty} \frac{1}{n}\log\delta_0(G,F^nG). \] This displays the growth quantities explicitly using a limsup convention. The supplied record does not specify its definitions, generator hypotheses, or limit conventions; their agreement with the intended cited formulation requires review.

Question. For a triangulated category $\mathcal D$, an exact endofunctor $F$, and a Bridgeland stability condition $\sigma$ for which the intended entropy quantities are defined, determine natural hypotheses on $(\mathcal D,F,\sigma)$ guaranteeing \[ h_\sigma(F)=h_0(F). \] This is a request for conditions ensuring equality, rather than an assertion of equality for every such triple. The supplied verification note says that the report establishes an inequality between mass growth and categorical entropy before posing the equality question; that description has not been independently checked here.

Unresolved source qualification. The supplied original statement instead asks for a geometric interpretation of $h_t(S)$, where $S$ is a Serre functor on a category of graded matrix factorizations, and for conditions on an unspecified associated entropy behavior. It does not specify the matrix-factorization data, the parameter range, or that behavior. The registry's correction note identifies this as a preceding, distinct problem and selects the equality question above. The two formulations are therefore not asserted to be equivalent.

Short English statement

Determine natural conditions under which the exponential growth of Harder--Narasimhan mass under an exact endofunctor equals its categorical entropy at parameter zero. The supplied corrected formulation selects this equality question; the original Serre-functor question is distinct. Precise categorical hypotheses and entropy conventions still require source review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002401$, source problem code catalogue entry, title Mass Growth Versus Categorical Entropy; supplied complete source record. The project also identifies catalog/problems.json as a supplementary registry. Neither registry file was independently inspected for this draft.
• Supplied report citation: Matrix Factorizations in Algebra, Geometry, and Physics (2014), Oberwolfach Reports, pp. 2501--2552, DOI 10.4171/owr/2013/44. https://doi.org/10.4171/owr/2013/44. The supplied record gives proposed year 2013 and names no individual proposer.
• Supplied literature source, "Oberwolfach Report": https://ems.press/content/serial-article-files/46474.
• Supplied literature source, Mass growth of triangulated autoequivalences: https://link.springer.com/article/10.1007/s00229-023-01533-z.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not an independently verified literature assessment. The supplied status is "partially solved": the record reports equality in several important settings while describing natural general hypotheses guaranteeing equality as open. No particular settings or theorem statements are supplied.
• The supplied statement status is corrected_verified. Its verification note attributes the equality question to the official report and distinguishes it from a preceding Serre-functor problem. This is a supplied source-verification claim, not independent verification performed for this draft.