Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?

Definitions and mathematical statement

Let $M$ be a compact smooth three-manifold without boundary, given by a finite triangulation with its compatible smooth structure. The target ambient manifold is the standard smooth sphere $S^4\subset\mathbb R^5$. A smooth embedding $e:M\hookrightarrow S^4$ is an injective smooth immersion that is a homeomorphism onto its image; its differential has rank three everywhere.

Find a complete computable set of invariants that decides whether such an embedding exists for every input $M$. The source also asks for a constructive procedure producing an embedding in the yes-case. Thus the desired algorithm must terminate with a correct no-answer on every nonembeddable input and with a finite verifiable description of a smooth embedding on every embeddable input. A compatible triangulated or handle description can serve as output, provided its conversion to a smooth embedding is justified.

The category is fixed: a merely continuous or topological embedding is insufficient. No bound on the running time or output size is prescribed. Nonorientable manifolds are not silently discarded from the input domain; a decision procedure must address them as well. Computable necessary obstructions supply part of a criterion, but the required completeness also needs a sufficiency argument and the requested construction whenever all obstructions vanish.

Short English statement

Can smooth embeddability of every closed three-manifold in the standard four-sphere be decided effectively, with an embedding constructed whenever one exists?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3416 (OPG-37154), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/which_compact_boundaryless_3_manifolds_embed_smoothly_in_the_4_sphere.