{
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  "ref": "R1336",
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  "slug": "line-arrangement-triangular-bounded-completion-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 155196; all comments were checked, including the quadrilateral completion and vertical-line partial results. Cuntz, arXiv:1108.3000, classifies projective simplicial arrangements through 27 lines. That finite classification concerns arrangements already simplicial and does not decide completion of an arbitrary affine subarrangement. Geis, arXiv:1809.09362, gives combinatorial restrictions for simplicial arrangements but no arbitrary completion theorem was found in the dated search. Cuntz, arXiv:2607.17785, studies simplicial arrangements with special vertices. The July 2026 paper does not supply an arbitrary affine completion theorem. A TheoremDB search for triangulating line arrangements, simplicial completion, and triangular bounded cells found no duplicate.\n\nA complete resolution must satisfy: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
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    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/155196/extending-a-line-arrangement-so-that-the-bounded-components-of-its-complement-ar",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
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  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/155196/extending-a-line-arrangement-so-that-the-bounded-components-of-its-complement-ar",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
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      "title": "Complete the stated acceptance conditions",
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      "slug": "R1630",
      "title": "Dated status and exact unresolved remainder",
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    {
      "slug": "line-arrangement-triangular-bounded-completion",
      "title": "line arrangement triangular bounded completion",
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}