{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
  "title": "Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8",
  "canonical_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",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:event",
      "problem_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-06-01T00:00:00.000Z",
      "title": "Darboux injections from closed manifolds (Zenodo, June 2026)",
      "summary": "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.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:attempt"
      ],
      "method_family_ids": [
        "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:method"
      ],
      "source_assertion_ids": [
        "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:attempt",
      "problem_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-06-01T00:00:00.000Z",
      "human_collaborators": [
        "Peter L."
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Banakh and Banakh, The continuity of Darboux injections between manifolds",
          "url": "https://doi.org/10.48550/arXiv.1809.00401",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Companion preprint",
          "url": "https://doi.org/10.5281/zenodo.22346412",
          "kind": "paper",
          "license": null
        },
        {
          "label": "MathOverflow 235893",
          "url": "https://mathoverflow.net/questions/235893",
          "kind": "discussion",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Darboux injections from closed manifolds (Zenodo, June 2026)",
          "url": "https://doi.org/10.5281/zenodo.22347647",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8",
          "url": "https://vibemathed.com/problem/is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:verification",
      "problem_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
      "solution_event_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "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.",
      "sources": [
        {
          "label": "VibeMathed: Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8",
          "url": "https://vibemathed.com/problem/is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Darboux injections from closed manifolds (Zenodo, June 2026)",
          "url": "https://doi.org/10.5281/zenodo.22347647",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Darboux injections from closed manifolds (Zenodo, June 2026)",
      "url": "https://doi.org/10.5281/zenodo.22347647",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8",
      "url": "https://vibemathed.com/problem/is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble:signal:verify_now",
      "problem_id": "vibemathed:is-every-darboux-bijection-of-the-4-sphere-problem-1-7-and-of-the-3-torus-proble",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "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."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}