Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8
candidateconfidence 50%
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Banakh and Banakh (2020) proved that connectedness-preserving (Darboux) injections are continuous in several compact settings — into 1-manifolds from compact sources, from closed surfaces into surfaces, and from closed 3-manifolds with finite $H_1$ into 3-manifolds — and asked whether every Darboux bijection of $\mathbb S^4$ (Problem 1.7) and of $\mathbb T^3$ (Problem 1.8) is a homeomorphism. Answer: yes. For every $n\ge2$, every Darboux injection from a connected closed $n$-manifold into an $n$-manifold is a homeomorphism onto a connected component of the target; no homology hypothesis and no surjectivity are needed
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.5 Pro
The note's acknowledgment: it was produced with substantial assistance from large language models, principally GPT-5.5 Pro, in a research program directed by the author, who selected, checked and assembled the arguments. The proof uses Banakh–Banakh's framework of $n$-varieties and componnectedness, Alexander–Lefschetz duality with $\mathbb F_2$ coefficients, and an induction on minimal carriers of nonzero Čech cohomology classes; a separate proof that metrizable $n$-manifolds are $n$-varieties, and a one-dimensional base case, are supplied. The same theorem was later re-derived by the same method, independently and without access to the note, by GPT-6 (Codex) in a subsequent phase of the program; that re-derivation is in the program's records.
Unreviewed. The 17-page note (Zenodo 10.5281/zenodo.22347647, dated June 2026, posted 5 September) was read here in full; the theorem, the method (Alexander–Lefschetz duality with F2 coefficients, induction on minimal carriers of Čech cohomology classes, in Banakh–Banakh's framework of n-varieties) and the disclosure match the submission, and Problems 1.7 and 1.8 were confirmed verbatim in arXiv 1809.00401. Nobody outside the author's program has read the argument; the re-derivation by a second model inside the same program is internal corroboration. Candidate as submitted.
Darboux injections from closed manifolds (Zenodo, June 2026)
Both problems are answered affirmatively, in every dimension at once: every connectedness-preserving injection from a connected closed $n$-manifold into an $n$-manifold is a homeomorphism onto a component, so in particular every Darboux self-bijection of $\mathbb S^n$ and of every closed manifold is a homeomorphism. This removes the finite-$H_1$ hypothesis of the 2020 theorem for 3-manifolds and extends it above dimension 3. Compactness of the source is essential: the companion preprint (Zenodo 10.5281/zenodo.22346412) shows the corresponding statement fails for $\mathbb R^n$, $n\ge2$. The note does not address noncompact sources, manifolds with boundary, or targets of different dimension.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
The source status is candidate and must not be represented as solved.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.