{
  "abstract": "We develop a unified geometric representation of spin-½ Stern–Gerlach measurement and linear photon polarization using nonnegative distributions of individual directional amplitudes with deterministic binary responses. For spin-½, hemispherical classification of individual orientations reproduces the complete rotational transition law, P₊(α) = cos²(α/2). For linear photon polarization, an unoriented transverse axis and deterministic ±45° acceptance sectors reproduce Malus' law, P(β) = cos²β, through an exact angular overlap integral.\n\nThe principal mathematical results are two support-constrained uniqueness theorems. In both geometries, requiring a normalized nonnegative distribution confined to the preparation acceptance region, together with agreement with the complete family of rotational transition probabilities, uniquely determines the corresponding cosine-weighted prepared-state density. The spherical and planar proofs establish how geometric support conditions eliminate ambiguities otherwise left unresolved by binary filtering.\n\nA systematic consistency analysis examines pristine populations, complete transition matrices, analyzer reversal, rotational covariance, immediate repeatability, operational isotropy of equal mixtures, and sequential re-preparation. The constructions reproduce the standard ideal measurement statistics while distinguishing definite individual directional amplitudes from their statistical distributions across prepared ensembles.\n\nTogether, these results establish a common mathematical framework for two physically different binary measurement geometries, identifying the positive directional distributions compatible with their complete quantum transition statistics and providing a basis for investigating higher-spin measurement structures.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Jorgen Karlsen",
      "orcid": "0009-0006-3318-7037"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkwq4sc8xfsrd.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23603945",
  "fulltext_url": null,
  "id": "bkwq4sc8xfsrd",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23603945",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23603945/files/261023-1%20Directional%20Amplitude%20Geometry.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkwq4sc8xfsrd.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:20:52Z",
      "native_id": "23603945",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23603945"
    }
  ],
  "subjects": [],
  "title": "Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency",
  "title_text": "Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency",
  "type": "preprint",
  "updated_at": "2026-11-06T12:20:52Z",
  "url": "https://research.draconitic.com/paper/bkwq4sc8xfsrd",
  "venue": "Zenodo",
  "versions": []
}