Abstract closure scheme for ascent operators

Construct an abstract class of admissible ascent operators, including temporal semantic elevation and hierarchical supervisory ascent, under which the class of non-verifiable properties \(\mathcal{U}\) is closed, thereby establishing a single closure principle encompassing the article’s one-step, iterated, and supervisory results.

Background

The paper proves that the class U\mathcal{U} of non-verifiable properties is stable under a specified temporal elevation operator and that finite supervisory towers inherit undecidability. It asks whether these apparently different forms of ascent can be treated uniformly.

The unresolved task is to define “admissible ascent operators” broadly enough to include both temporal elevation and hierarchical supervision, while keeping the definition sufficiently restrictive that closure of U\mathcal{U} is not trivial. The paper explicitly identifies the abstract scheme subsuming these cases as open, while noting that the individual intensional supervisory-tower result has already been established.

References

There is an abstract class of ``admissible ascent operators''---including temporal elevation and hierarchical supervision---under which $\mathcal{U}$ is closed. The closure theorem of this article and the non-termination of the supervisory tower would be two instances of one principle: every admissible ascent preserves non-verifiability.

— The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation  (2609.11326 - Mezquita, 10 Sep 2026) in Conjecture (closure scheme), Section 7, subsection “A closure scheme for ascent operators”

The conjecture is that $$ forces the non-factorisation through $2$ of its values in $\mathcal{U}$, by an application of Lawvere's theorem to a fixed-point-free endomorphism of $\Omega$, with the point-surjection built from the universal enumeration of PC.

— The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation  (2609.11326 - Mezquita, 10 Sep 2026) in Section 7, subsection “The categorical reading in the effective topos”