Completely Positive Separation for the Integer Lattice

Definitions and mathematical statement

Let ${\mathbb{Z}}^2=\{(m,n):m,n\in{\mathbb{Z}}\}$ be the integer lattice with componentwise addition. The supplied record asks the following methodological question:

Can the finite-dimensional completely-positive-functional and separation argument used for free groups be extended to ${\mathbb{Z}}^2$, so as to better understand this group?

This question does not specify a mathematical conclusion to be proved. In particular, the supplied record does not identify the free-group argument, its hypotheses, the algebra and matrix-positive cones on which the completely positive functionals are defined, what is required to be finite-dimensional, or the objects and topology involved in separation. It also does not specify the ranks of the free groups under consideration or the intended conclusion for ${\mathbb{Z}}^2$. Consequently, no fully quantified theorem or equivalence can be extracted without additional source information. The question above retains the methodological target; it does not assert any particular positivity or sums-of-squares characterization.

Short English statement

The question asks whether a method involving finite-dimensional completely positive functionals and separation, used for free groups, can also clarify the case of ${\mathbb{Z}}^2$. The argument and desired conclusion are unspecified, so a precise theorem remains to be identified.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002565$, title "Completely Positive Separation for the Integer Lattice," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2014; no proposer is named.
• Supplied report citation: Real Algebraic Geometry With A View Toward Systems Control and Free Positivity (2015), Oberwolfach Reports, pp. 977--1045, DOI 10.4171/owr/2014/17. Source URL and supplied literature-source link, labelled "Oberwolfach Report 2014/17": https://doi.org/10.4171/owr/2014/17. No passage, contribution, or theorem locating the referenced argument is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that Scheiderer proves the relevant sums-of-squares characterization for ${\mathbb{Z}}^2$, while the extracted methodological request is not a precise theorem and no definitive resolution of that request is identifiable. This is supplied dated curation, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims; it supplies no separate citation for Scheiderer's result.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied editorial assertions and do not resolve the missing mathematical specifications.