Resolved Directed-Landscape Converse Lead-In

Definitions and mathematical statement

The supplied record concerns the KPZ fixed point, denoted by $H(x,t;f)$, and the directed landscape, denoted by $L$. It uses a class of initial data denoted by $\mathrm{UC}$, but does not define that class, its allowed function values, or its growth conditions. Neither random object is defined sufficiently in the supplied excerpt to give a self-contained formulation.

The accompanying background asserts that, for every fixed $f\in\mathrm{UC}$ and every fixed $s\in{\mathbb{R}}$, \[ H(x,t;f)\overset{\mathrm d}{=} \sup_{y\in{\mathbb{R}}}\left\{f(y)+L(y,s;x,s+t)\right\}, \qquad (x,t)\in{\mathbb{R}}\times(0,\infty). \tag{3} \] The excerpt explicitly describes this as equality in law as functions of $(x,t)$, rather than merely equality of individual marginal distributions. It does not specify the function space or measurable structure for this assertion, or the hypotheses ensuring that the supremum is well defined and has the intended range. This is a transcription of a supplied background assertion, not a proposed new theorem or a verified result.

The extracted question is "Is the opposite true?" Its antecedent is absent, so neither the proposition to be reversed nor the meaning of "opposite" can be recovered. In particular, the excerpt does not identify a converse to (3) as the intended question.

The background also states that, unlike the KPZ fixed point, the directed landscape does not have exact formulas for all statistics, and describes its construction through a scaling limit of an unspecified aspect of the classical Robinson--Schensted--Knuth (RSK) correspondence. These are supplied contextual claims; their scope and hypotheses are not provided.

Unresolved formulation and classification. The registry labels the item open, and its dated literature assessment treats the isolated question as unclear and normalizes it to open. However, its separate statement-verification field says that the official report immediately answers this lead-in question with a theorem and that the extraction is not an open problem. The statement status is recorded as unrecoverable. These supplied qualifications prevent a precise standalone open-problem statement; no missing antecedent, converse, or theorem is reconstructed here.

Short English statement

The excerpt supplies a variational identity connecting the KPZ fixed point and the directed landscape, but does not define all of its hypotheses. The question "Is the opposite true?" lacks its antecedent. The supplied verification says it is an answered lead-in, while the registry retains an open label, so no precise standalone open problem can be stated.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006284$, exact source problem code catalogue entry; supplied title: "Resolved Directed-Landscape Converse Lead-In." The record belongs to the Oberwolfach Reports Open Problems collection. No proposer is supplied; the proposed year is 2025.
• Supplied report citation: Statistical Physics Out of Equilibrium: Quantitative Results and Universality (2025), Oberwolfach Reports, pp. 1333--1388, DOI 10.4171/owr/2025/26. Source URL and supplied literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2025/26. This report is supplied as the authoritative provenance; it has not been independently consulted for this draft.
• The excerpt attributes the connection between the directed landscape and the KPZ fixed point to references [3, 6]. Their bibliographic identities and authors are not supplied, so those attributions cannot be resolved here.
• Supplied literature assessment, checked exactly 2026-08-22: the question lacks a specified proposition or negation and was therefore normalized to open. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• Separate supplied statement-verification note: the official report uses the converse question as a lead-in and immediately answers it with a theorem; consequently, the extraction is not an open problem. The record retains the original text publicly for provenance and promotes no separate clean statement. This verification claim is supplied metadata, not an independent finding of this draft.