Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

For every integer $n\ge 2$ we construct a bijection of $\mathbb{R}^n$ that maps every connected set to a connected set but whose inverse does not. Its discontinuity set is exactly the closed positive coordinate ray, and the inverse image of a straight segment is the middle-thirds Cantor set on that ray. This answers in the negative the question posed by Willie Wong on MathOverflow (question 235893, 2016): must the inverse of a connectedness-preserving bijection of $\mathbb{R}^n$ be connectedness-preserving? By Tanaka's theorem and invariance of domain, this is equivalent to the existence of a connectedness-preserving bijection of $\mathbb{R}^n$ that is discontinuous. The construction is uniform in $n$. A thin solid tube is extended indefinitely by finger moves so that its cross-sections return near every point of its complementary compactum; a nonproper collapse identifies the ray with the ideal end of the tube while spreading the point at infinity over the rest of the ideal boundary; a single boundary line of the tube, cut into countably many disjoint compact arcs each nearly dense in the complement, supplies countably many disjoint dense connected subsets, in every dimension including two; and a separation argument (the gate lemma) certifies the connectedness-preserving property on arbitrary connected sets, without compactness or path-connectedness assumptions. The map can be taken Borel measurable. The compact situation is rigid by contrast: every connectedness-preserving injection from a connected closed $n$-manifold into an $n$-manifold is a homeomorphism onto a component (companion note, doi:10.5281/zenodo.22347647). The paper explains the relation with an earlier planar construction based on prime ends, and includes a provenance appendix tracing the origin of each idea. Disclosure of AI assistance. The mathematics was developed autonomously by AI systems — OpenAI's Codex (GPT-6, Ultra effort) for the structural theory, constructions, adversarial audits, verification and drafting, and Anthropic's Claude Fable 5.1 (Extra effort) for planning, review of run reports, and the single-line coloring that makes the construction uniform in the dimension — in a research programme designed and directed by the author, who chose the problems and their order, wrote the briefs, ran the audits between systems, and takes responsibility for the result. The AI systems are tools, not authors. Complete records of all runs are retained by the author. Files: main.pdf (the paper), main.tex and references.bib (LaTeX source).