Equality of Canonical Models from Universal Projective Sets

Definitions and mathematical statement

Let $P$ denote the good universal $\Pi^1_2$ set referred to in the supplied record, and let $\widetilde\varphi=(\varphi_n)_{n<\omega}$ be a $\Delta^1_3$-scale on $P$. A scale is a sequence of ordinal-valued norms satisfying the usual scale convergence conditions; the precise projective complexity convention and the meaning of "good universal" must be taken from the source's intended framework. Write $O_{\Sigma^1_3,\widetilde\varphi}$ for the corresponding universal $\Sigma^1_3$ object constructed relative to this scale, and $O_{\Sigma^1_3}$ for the designated object appearing without a scale parameter in the source. For an ordinal $\alpha$ and a predicate $A$, $L_\alpha[A]$ denotes the height-$\alpha$ relativized constructible hierarchy, subject to the source's intended coding of $A$.

The supplied question, with $\widetilde\varphi$ arbitrary among the scales allowed by that framework, is whether \[ L_{u_\omega}[O_{\Sigma^1_3}] = L_{u_\omega}[O_{\Sigma^1_3,\widetilde\varphi}]. \] Here $u_\omega$ retains its source notation. The record does not define this ordinal, the two $O$-constructions, their underlying domains or codings, or the background axioms under which the scale and constructions exist. Consequently, this is a conditional transcription of the intended equality question, not a self-contained formulation with those choices resolved. In particular, arbitrary universal projective sets must not be substituted for the designated $O$-objects without their missing construction rules.

Short English statement

For an arbitrary admissible $\Delta^1_3$-scale on the designated good universal $\Pi^1_2$ set, do the designated universal $\Sigma^1_3$ object and its scale-dependent counterpart generate the same constructible hierarchy up to $u_\omega$? The construction conventions and background assumptions require clarification from the source.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003406$, source problem code catalogue entry, title "Equality of Canonical Models from Universal Projective Sets." The supplied collection is "Oberwolfach Reports Open Problems." No individual proposer is supplied.
• Supplied source citation: Set Theory (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/11, pp. 527--589. https://doi.org/10.4171/owr/2017/11. The record separately gives proposed year 2017; the year discrepancy is unresolved.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46674.
• Supplied literature source, "Zhu, higher sharp IV": https://arxiv.org/abs/1706.00660.
• Supplied literature source, "Zhu, higher sharp I": https://arxiv.org/abs/1604.00481.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes the equality as a uniqueness question. It reports that the checked higher-sharp literature establishes related representation theorems but does not establish this equality. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.