Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property

Definitions and mathematical statement

Let $L$ be an abstract model-theoretic logic with sentences over vocabularies and an isomorphism-invariant satisfaction relation. The source does not fix its closure properties, whether it extends first-order logic, or the allowed vocabulary sizes.

Under full compactness, any set $T$ of $L$-sentences has a model whenever every finite subset has a model. This is distinct from a restriction to countable $T$ or another cardinal-specific compactness property.

A standard theory-level Beth property says: whenever an $L$-theory in vocabulary $\tau\cup\{R\}$ implicitly defines a relation $R$ over $\tau$ (two models with the same $\tau$-reduct must agree on $R$), there is an $L$-formula over $\tau$ explicitly defining $R$ in all models of the theory. The source does not specify theory-versus-sentence or relation-versus-function variants.

Under the Craig-interpolation interpretation identified by the research summary, interpolation requires that whenever $\alpha\in L(\tau_1)$ and $\beta\in L(\tau_2)$ satisfy $\alpha\models\beta$, there be $\gamma\in L(\tau_1\cap\tau_2)$ with \[ \alpha\models\gamma\models\beta. \] Failure means there are such $\alpha,\beta$ with no interpolant in the common vocabulary.

The existence question is whether some $L$ simultaneously has compactness, the Beth property, and the source's $\Delta$-interpolation property, while failing this interpolation property. The specialized $\Delta$-interpolation condition is not defined in the record. It cannot be silently replaced by ordinary Craig interpolation or a guessed definability closure. Its exact definition, the logic framework, and the preceding property variants must be fixed to obtain a determinate existence problem.

Short English statement

Can a compact abstract logic have both Beth definability and $\Delta$-interpolation while failing Craig interpolation? The requested combination is preserved, but the specialized $\Delta$ property and abstract-logic conventions are missing.

The supplied summary places the question in the Makowsky--Shelah abstract-model-theory program, mentions fixed-point-logical results satisfying subsets of the properties, and reports no verified construction or impossibility theorem for this exact combination. This is an OPEN-TRIAGE assessment. The pinned Wikipedia primary-source audit remains required; unpublished background conflicts with published registry metadata.

Sources

• Local UnsolvedMath registry, permanent ID 7600022, source code catalogue entry, Beth/$\Delta$-interpolation/compactness question. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Wikipedia contributors, List of unsolved problems in mathematics, Model theory and formal languages, bullet 22, pinned revision 1366636767, https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages; MediaWiki-source extraction, accessed 2026-07-29.
• Makowsky, "Compactness, embeddings and definability," in Model-Theoretic Logics, Springer (1985), as cited in the supplied summary; no pages or exact problem reference are supplied.
• The summary mentions Shelah's "Beth property in inflationary fixed point logics" and monotone fixed-point logics without complete bibliographic details or exact results matching this question.
• Source-specific provenance: primary-source status audit NEEDS_REVIEW; background says unpublished pending audit although published metadata is true. Wikipedia text is CC BY-SA 4.0, with attribution and ShareAlike handling required for release, separately from dataset attribution.