{
  "schema_version": 1,
  "problem_number": "OPG-60024",
  "title": "Finite Adjoint-Transform Orbits for Funcoids and Reloids",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a monotone map between complete lattices, let its two adjoint transforms be the supremum of sublevel sets and the infimum of superlevel sets. Porton's diagram problem asks for the missing transform of the outward reloid map and whether its iterated transforms form a finite set. Working in ZFC, we identify the missing map as the join of the induced funcoid with the product of the two Frechet filters. We give a closed transition table containing at most twenty typed maps, thereby proving finiteness for arbitrary source and destination sets. If either set is finite, at most two maps occur. The proof uses Todd Trimble's representation of funcoids by closed subsets of a product of Stone spaces, together with two density statements about nonsingleton fibers of the canonical Stone map. We also give a counterexample to an extra arrow in the published diagram. Twenty is an upper bound, not a claim that all its labels are distinct or that the bound is optimal. This is an unrefereed, AI-assisted preprint.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GN",
    "math.CT"
  ],
  "keywords": [
    "OPG-60024",
    "funcoid",
    "reloid",
    "Stone space",
    "ultrafilter",
    "adjoint transform"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-60024/",
  "pdf_url": "https://eulersolve.org/papers/opg-60024/paper.pdf?v=75a6862ffd38",
  "doi": "10.5281/zenodo.23632224",
  "zenodo_record_url": "https://zenodo.org/records/23632224",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "In ZFC, for arbitrary fixed sets A and B, the missing right adjoint-transform of the outward reloid map is the join of the induced funcoid with the funcoidal product of the two Frechet filters. Its orbit under the two adjoint transforms has at most twenty typed maps when both sets are infinite and at most two when one is finite. The formula and finite transition system resolve the full missing-node and finite-or-infinite orbit question in OPG-60024. An extra arrow in the earlier diagram is disproved. Twenty is an upper bound, not a claim of pairwise distinctness or optimality. The missing-node and finite-or-infinite orbit questions are resolved in ZFC for arbitrary fixed sets. Twenty is an upper bound, not an exact or optimal orbit cardinality. Todd Trimble's closed-subset representation is credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "75a6862ffd381830eec64299a297e6c62f9b76f4b1fe5ca5afc87d99c724f278"
    },
    "source.zip": {
      "sha256": "0741609747ff1c843ac4548d2d37e7109e63e38953be6191e4cb75416fd8fa37"
    },
    "verification_report.md": {
      "sha256": "8657fe4fab6e8fe51813297ff2f155f37ecef5a094ac99640a5b041420dea00c"
    }
  },
  "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."
}