{
  "abstract": "The quantum harmonic oscillator (QHO) is normally built on a structure that contains a Hilbert space, ladder operators, and a Born rule. Here we derive it from counting. We adopt three information-theoretic postulates: information is carried by binary sequences of length $n$; only symbol counts, not the sequences themselves, are observable; and the transition measure must account for exactly the $2^n$-configuration capacity of the register. The first two postulates force the counting problem into the Hamming/Delsarte association scheme and endow it with an exact symmetry under relabeling of the two symbols. Given an exchange-symmetric bilinear transition measure we prove a rigidity theorem: relabeling symmetry and normalization together force the alternating sign $(-1)^t$ which drives interference. The variable that carries the sign, i.e. the overlap between input and output sequences, or equivalently their Hamming distance $d_{ab}$, is a hidden quantum number of the transition as a whole, defined only by the two endpoints jointly and unknowable from either alone. Interference thus enters as alternating-sign bookkeeping over a nonlocal, relational variable $d_{ab}$, rather than as a separate dynamical ingredient. This forced weighting makes the transition probability proportional to the square of a Krawtchouk polynomial, whose difference equation has an exactly equally spaced spectrum and which converges to the QHO as $n\\to\\infty$. At fixed excitation, deviations from the QHO are $\\mathcal{O}(1/n)$, and are testable in laboratory realizations of the oscillator, since any real system has finite information capacity. In this framework, interference, quadratic (Born-rule-like) probabilities, and the particle-hole symmetric oscillator spectrum are consequences of counting; only the bilinear form of the measure is assumed (though well motivated) rather than derived.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10568",
  "author_total": 3,
  "authors": [
    {
      "name": "Samuel Powers"
    },
    {
      "name": "Dino Skrgic"
    },
    {
      "name": "Dejan Stojkovic"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/09bzqw9n6ymjw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10568",
  "fulltext_url": null,
  "id": "09bzqw9n6ymjw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10568",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.gen-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7212269586",
  "pdf_url": "https://arxiv.org/pdf/2609.10568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09bzqw9n6ymjw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.10568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10568"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3107|Quantum Mechanics and Applications",
      "subfield": "amo"
    }
  ],
  "title": "The quantum harmonic oscillator from binary sequences",
  "title_text": "The quantum harmonic oscillator from binary sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:26Z",
  "url": "https://research.draconitic.com/paper/09bzqw9n6ymjw",
  "venue": "arXiv",
  "versions": []
}