Discontinuities of Two-Sided Specifications

Definitions and mathematical statement

The supplied background identifies the first question as concerning a two-sided specification for the Schonmann projection. The record does not define that projection, its parameters, or the intended specification. Thus the following is a conditional formulation in a standard Ising configuration space; the missing model data must be supplied before it becomes a fully specified problem.

Let $I$ be the intended countable set of sites, let $S=\{-1,+1\}$, and equip $\Omega=S^I$ with the product topology and its Borel sigma-algebra. Let $\mu$ be the intended spin distribution. A two-sided specification $\gamma=(\gamma_\Lambda)_{\Lambda\subset I,\,0<|\Lambda|<\infty}$ is a consistent family of probability kernels giving conditional distributions in finite sets $\Lambda$ given the entire exterior configuration on $I\setminus\Lambda$. Each kernel leaves the exterior configuration fixed. Compatibility with $\mu$ means that, for every finite nonempty $\Lambda$ and measurable event $A$, \[ \gamma_\Lambda(A\mid\omega) =\mu\left(A\mid\sigma(\omega_i:i\in I\setminus\Lambda)\right)(\omega) \quad\text{for $\mu$-almost every $\omega$.} \] A bounded local function is a function $f:\Omega\to{\mathbb{R}}$ depending on only finitely many coordinates. Write \[ (\gamma_\Lambda f)(\omega) =\int_\Omega f(\eta)\,\gamma_\Lambda({\,\mathrm{d}}\eta\mid\omega). \] Under the usual product-topology interpretation of continuity of a specification, define \[ D_\gamma= \left\{\omega\in\Omega: \begin{array}{l} \text{there exist a finite nonempty $\Lambda\subset I$ and a bounded local $f$}\\ \text{such that $\gamma_\Lambda f$ is discontinuous at $\omega$} \end{array} \right\}. \] Here discontinuity means that some sequence $\omega^{(n)}\to\omega$, converging coordinatewise, satisfies $ (\gamma_\Lambda f)(\omega^{(n)})\not\to(\gamma_\Lambda f)(\omega) $.

Question, subject to identifying the intended objects. For the intended Schonmann-projection measure $\mu$ and its intended two-sided specification $\gamma$, does \[ \mu(D_\gamma)=0 \] hold? Does the corresponding assertion hold for the long-range Ising models intended by the source?

The source does not specify the measure meant by "measure zero," the topology, the version of the conditional kernels, or whether discontinuities are ordinary discontinuities of a chosen version or essential discontinuities. The displayed formulation uses $\mu$, the product topology, and ordinary discontinuities only as explicitly stated conventions for a conditional template. It does not resolve whether the source asks about a designated specification or the existence of a compatible specification with a null discontinuity set. The lattice, interaction, temperature, external field, boundary conditions, and quantification over long-range models are also unspecified. Consequently no universal assertion over model parameters or specification versions is justified by this record.

Short English statement

The question asks whether discontinuities of the intended two-sided conditional description occur only on a set of probability zero for the Schonmann projection, and whether an analogous conclusion holds for the intended long-range Ising models. The record leaves the precise measures, specifications, parameters, and continuity convention unresolved. Its supplied dated assessment labels the problem open and describes the one-sided analogue as known; these claims require independent review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004429$, title "Discontinuities of Two-Sided Specifications," exact source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied. The proposed year is recorded as 2020.
• Supplied report citation: Mini-Workshop: One-sided and Two-sided Stochastic Descriptions (2021), Oberwolfach Reports, pp. 601--637, DOI 10.4171/owr/2020/11. Supplied source link: https://doi.org/10.4171/owr/2020/11.
• Supplied literature source, labeled OWR-2020-11: https://ems.press/content/serial-article-files/46847.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It states that the two-sided Schonmann-projection discontinuity-set measure-zero question is explicitly unanswered, that the one-sided analogue is known, and that the two-sided claim remains unresolved. The background describes this as literature triage derived from the supplied Oberwolfach Reports curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies the status "open" and the statement-status label corrected_verified. Its verification note reports manual semantic, typographic, and LaTeX review and claims that the clean statement is self-contained. These are supplied registry assertions; they do not supply the missing mathematical definitions or establish an independent mathematical review.