Universal Integral Relations for Closed Polygons

Definitions and mathematical statement

Fix a polygon size $n$ in the range intended by the source; that range is not specified in the supplied record. Let $\mathcal C_n$ denote the source's space of closed $n$-gons, and write $I_j(P),J_j(P)$ for its monodromy integrals at $P\in\mathcal C_n$. Denote their unspecified index set by $\mathcal J_n$. The ambient geometry, equivalence relation defining this space, admissibility conditions, and definitions and normalizations of these integrals are not supplied. Thus these symbols retain their source-dependent meanings.

For $j\in\mathcal J_n$, define the supplied weight by \[ w_n(j)= \begin{cases} 3j-k,&n=2k,\\ 3j-k+1,&n=2k+1. \end{cases} \] The record presents the following as identities holding for every $P\in\mathcal C_n$, with every sum taken over $j\in\mathcal J_n$: \[ \begin{aligned} \sum_j I_j(P)&=3,&\sum_j J_j(P)&=3,\\ \sum_j w_n(j)I_j(P)&=0,&\sum_j w_n(j)J_j(P)&=0,\\ \sum_j w_n(j)^2\left(I_j(P)-J_j(P)\right)&=0. \end{aligned} \] These identities are reproduced as supplied premises, not independently verified assertions.

The source poses three questions:

1. Are the displayed identities the only universal relations among the $I_j,J_j$ on $\mathcal C_n$? Here a universal relation is an equality holding for every polygon in the intended space. The allowed class of relations and the meaning of being a consequence of the displayed identities are not specified; consequently, no precise algebraic or functional completeness assertion can be extracted without additional definitions.
2. Does every inscribed polygon in the intended domain satisfy \[ I_j(P)=J_j(P)\qquad\text{for every }j\in\mathcal J_n? \] The record does not define the inscribing curve, the ambient geometry, or any genericity or nondegeneracy requirements. Its wording does not resolve whether this question is restricted to $\mathcal C_n$ or concerns a larger polygon space.
3. Is the induced Poisson map on twisted $n$-gons in $\mathbb{RP}^{1}$ completely integrable? The map, its inducing construction, the definition of a twisted polygon, the relevant configuration space or quotient, and the Poisson structure are not supplied. Neither its rank nor the intended criterion of complete integrability is specified, so this question remains a source-dependent formulation.

A fully specified mathematical statement requires these missing definitions and hypotheses; they are not inferred here.

Short English statement

The source asks whether five displayed identities exhaust the universal relations among polygon monodromy integrals, whether inscribed polygons have matching integrals $I_j=J_j$, and whether an associated Poisson map on twisted polygons in $\mathbb{RP}^{1}$ is completely integrable. Essential definitions and domains are missing. These are supplied questions labeled open by the record, and this statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001292$, title Universal Integral Relations for Closed Polygons, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; the supplied evidence does not identify which contains this record. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied report citation: Dynamische Systeme (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/32, pp. 1787--1842. Report and literature-source link: https://doi.org/10.4171/owr/2009/32. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature-source label: Fuchs--Tabachnikov on self-dual polygons, https://arxiv.org/abs/0707.1048. The record does not provide a particular result or passage supporting any of the three questions.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes explicit open conjectures about universal relations, inscribed polygons, and integrability, and reports that no general resolution was found. It lists the preceding report and Fuchs--Tabachnikov link as checked sources. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The record labels its statement corrected_verified and attributes the restored identities, parity-dependent weight, and two associated conjectures to the official OWR report. That verification claim is supplied metadata, not verification performed for this draft. The report title's year 2010, the DOI containing 2009, and the proposed year 2009 are retained without reconciliation.