{
  "schema_version": 1,
  "problem_number": "AMR-098-0008",
  "title": "An Allocation-Invariant Diffuse Measure That Is Not Mass-Stationary",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Invariance under every measurable equivariant measure-preserving allocation does not characterize mass-stationarity of a diffuse random measure without an independent stationary background. For every dimension d >= 2, a periodic measure supported on parallel hyperplanes with distinct positive weights gives an explicit obstruction. Horizontal translation symmetry prevents deterministic equivariant allocations from splitting layers, and a finite weighted quotient forces the rooted configurations to be fixed. Every probability law on the finitely many root types is allocation-invariant, whereas exactly the weight-proportional law is mass-stationary. The latter is the normalized Palm law of a stationary ergodic uniform-phase probability with finite intensity. An explicit preserving Markov kernel detects all other root laws, so the separate Markovian-kernel question is not claimed solved. Related atomic, product and invariant-direction methods are credited. This is a self-audited, unrefereed note without a priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.PR"
  ],
  "keywords": [
    "AMR-098-0008",
    "math.PR"
  ],
  "manuscript_version_date": "2026-10-21",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-098-0008/",
  "pdf_url": "https://eulersolve.org/papers/amr-098-0008/paper.pdf?v=1d62df767583",
  "doi": "10.5281/zenodo.23586149",
  "zenodo_record_url": "https://zenodo.org/records/23586149",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The background-free allocation equivalence in Thorisson's Problem 4.1 and Last-Thorisson (2009), Problem 7.5, is false already on R^d for every d>=2. Periodic parallel hyperplanes carrying distinct positive weights give diffuse Radon measures for which all laws on the finitely many rooted types are invariant under every measurable equivariant preserving allocation; only the weight-proportional law is mass-stationary. A stationary ergodic probability parent has finite intensity. The example is singular and has invariant directions. It neither contradicts positive-density theorems nor resolves the separate Markovian-kernel question. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "1d62df76758391f739694ca51bcce4ac159c20c244491801dd54c9f743f3d798"
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    "source.zip": {
      "sha256": "6e8793204f064f406d1daf1459a4b2c1a57944009b3646bc201a581e355a1755"
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    "verification_report.md": {
      "sha256": "5689ff206fc88fdff549631a97ec0b0ae4af223dfe021232761bcf74369ea325"
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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."
}