Realization of Persistence Pairings by Filtration Functions

Definitions and mathematical statement

Let $K$ be the filtered simplicial complex in the source, and let $P$ be its persistence pairing. In the standard finite, field-coefficient computation, $P$ records specific birth/death simplex pairs $(\sigma^q,\tau^{q+1})$ obtained from the filtered boundary reduction, with unpaired simplices representing essential classes. Coefficients, treatment of simultaneous filtration values, and the pairing convention are not specified in the record, and finiteness of $K$ is not explicit.

A discrete vector field pairs simplices along codimension-one face relations, with each simplex used in at most one pair. In the usual finite setting, a discrete gradient field is such a matching with no nontrivial closed gradient path. Write $V_P$ for the associated discrete gradient field constructed in the report. This association is not supplied, and it must not be defined simply by declaring the persistence pairs themselves to be gradient pairs: persistence pairs need not be incident simplex/facet pairs.

An integer-valued filtration function ordinarily means a map $f:\operatorname{Simp}(K)\to{\mathbb{Z}}$ with \[ \sigma\subseteq\tau\quad\Longrightarrow\quad f(\sigma)\leq f(\tau), \] so $K_a=\{\sigma:f(\sigma)\leq a\}$ is a subcomplex. Let $\operatorname{Adm}(V_P)$ denote the source's additional, unstated conditions on such a function, and let $P_{V_P,f}$ be its report-specific induced persistence pairing. The conjecture asks \[ \forall(K,P)\text{ in the intended setting}\quad \exists f\in\operatorname{Adm}(V_P):\qquad P_{V_P,f}=P. \] The equality is equality of the specified combinatorial pairings, not merely equality of Betti numbers or isomorphic barcode/persistence-module data. The exact handling of unpaired simplices must agree on both sides.

The constructions $P\mapsto V_P$ and $(V_P,f)\mapsto P_{V_P,f}$, and their admissibility constraints, are essential missing definitions. Simply reusing the original filtration does not settle the question unless those constructions are proved compatible with it. The supplied assessment dated 2026-08-21 identifies this equality as Conjecture 3 and reports no authoritative resolution located; it does not provide the missing algorithm or hypotheses.

Short English statement

Can the gradient-field construction attached to a persistence pairing reproduce that exact pairing after choosing a suitable integer filtration? The issue is compatibility of the source's two constructions, not merely realizing the same barcode. Those constructions and their tie and coefficient conventions are absent.

Sources

• Local UnsolvedMath registry, permanent ID $30000991$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Computational Algebraic Topology (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/29, pp. 1603--1654. https://doi.org/10.4171/owr/2008/29.
• The supplied corrected_verified note reports editorial and LaTeX cleanup without added source-dependent content. The assessment checked 2026-08-21 identifies the assertion as Conjecture 3 and reports no authoritative resolution located. This and the open label are dated supplied triage, not an independent literature search.