{
  "schema_version": 1,
  "problem_number": "AMR-068-0001",
  "title": "The Stress-Sign Stratification of Four Labeled Points in Three Dimensions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We classify the stress-sign stratification of all labeled four-point configurations in R^3 for the complete graph K_4, including coincident points and every affine-rank degeneration. There are exactly 65 connected strata: two of affine rank three, 25 of affine rank two, 37 of affine rank one, and the diagonal. We give their dimensions and the entire closure order, with 800 strict incidences and 207 Hasse edges. Rank-two strata are indexed by mixed affine-dependence sign vectors modulo reversal, and collinear strata by ordered coincidence partitions modulo reversal. Their boundary relation is a closed-interval intersection condition; an explicit matrix perturbation realizes every asserted rank-two degeneration. In each ambient dimension d>=4 the two top strata merge, giving 64 strata. This answers only the B_3(K_4) component of Karpenkov's Problem 1. The B_2(K_6) and B_3(K_5) components are not resolved here. This is a self-audited, unrefereed preprint, with exact-arithmetic supplementary checks; no independent review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "tensegrity",
    "stress-sign stratification",
    "configuration space",
    "four labeled points",
    "affine dependence",
    "closure poset",
    "AMR-068-0001"
  ],
  "manuscript_version_date": "2026-10-06",
  "publication_date": "2026-10-06",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-06",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-068-0001/",
  "pdf_url": "https://eulersolve.org/papers/amr-068-0001/paper.pdf?v=cc6ee44f1ab9",
  "doi": "10.5281/zenodo.23175231",
  "zenodo_record_url": "https://zenodo.org/records/23175231",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete classification of the B_3(K_4) component of Karpenkov's Problem 1: all 65 connected strata, dimensions and closure incidences, including coincidences; also B_d(K_4) for d>=4. The B_2(K_6) and B_3(K_5) components remain unresolved here, so the whole AMR-068-0001 source record is not claimed solved. The stress-sign framework and prior planar classifications are credited. Exact-arithmetic finite checks supplement, not replace, the written all-configuration proof. AI-assisted, self-audited and unrefereed; independent review, formal verification and absolute priority are not claimed. Novelty remains undetermined after a bounded primary-source search.",
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    "source.zip": {
      "sha256": "08bb7087d26a2d90d52ffa49c675e1b18238930f66678772f6aa7c19a09a7ca2"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}