{
  "id": "oai:papers.qnfo.org:born-rule-as-haar-measure-on-p-adic-tree-balls-a-structural-derivation-and-its-i",
  "slug": "born-rule-as-haar-measure-on-p-adic-tree-balls-a-structural-derivation-and-its-i",
  "title": "Born Rule as Haar Measure on p‑Adic Tree Balls: A Structural Derivation and Its Implications for Quantum Statistics",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Quantum theory postulates the Born rule without a deeper structural justification, and Bell‑type correlations are usually interpreted as non‑local. This gap hinders deterministic reconstructions of quantum mechanics and inflates the engineering overhead of quantum‑information technologies that must treat randomness as a primitive. We model the static ultrametric Bruhat–Tits tree of p‑adic balls as a branching sample space and derive quantum probabilities from the Haar (topological) measure of these balls. Using the p‑adic norm |x|_p=p^-r, the definition of closed balls B_r(a)=\\xinQ_pmid|x-a|_p≤ p^r\\, and the identification of the ball hierarchy with the tree, we prove that a ball at depth k carries measure p^-k. For p=2, depth 1 yields measure $0.5$, reproducing the Born probability of a balanced qubit; for p=3, depth 1 yields 1/3, reproducing the Born probability of an equal‑amplitude three‑outcome interferometer. A deterministic shared‑ancestor model on the same tree yields CHSH correlations that can be analyzed, but it does not attain the maximal quantum value 2√2; achieving that remains an open question. The result changes practice by providing a geometry‑based foundation for the Born rule, enabling deterministic design of quantum‑information protocols and offering a concrete superdeterministic alternative to non‑local explanations.",
  "abstract_markdown": "Quantum theory postulates the Born rule without a deeper structural justification, and Bell‑type correlations are usually interpreted as non‑local. This gap hinders deterministic reconstructions of quantum mechanics and inflates the engineering overhead of quantum‑information technologies that must treat randomness as a primitive. We model the static ultrametric Bruhat–Tits tree of $p$‑adic balls as a branching sample space and derive quantum probabilities from the Haar (topological) measure of these balls. Using the $p$‑adic norm $|x|_{p}=p^{-r}$, the definition of closed balls $B_{r}(a)=\\{x\\in\\mathbb{Q}_{p}\\mid|x-a|_{p}\\le p^{r}\\}$, and the identification of the ball hierarchy with the tree, we prove that a ball at depth $k$ carries measure $p^{-k}$. For $p=2$, depth 1 yields measure $0.5$, reproducing the Born probability of a balanced qubit; for $p=3$, depth 1 yields $1/3$, reproducing the Born probability of an equal‑amplitude three‑outcome interferometer. A deterministic shared‑ancestor model on the same tree yields CHSH correlations that can be analyzed, but it does not attain the maximal quantum value $2\\sqrt{2}$; achieving that remains an open question. The result changes practice by providing a geometry‑based foundation for the Born rule, enabling deterministic design of quantum‑information protocols and offering a concrete superdeterministic alternative to non‑local explanations.",
  "date": "2026-10-19",
  "first_published": "2026-10-19",
  "last_revised": "2026-10-19T00:00:00Z",
  "modified": "2026-10-19T00:00:00Z",
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  "peer_review_status": "Not independently peer-reviewed",
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  "doi": null,
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  "url": "https://papers.qnfo.org/papers/born-rule-as-haar-measure-on-p-adic-tree-balls-a-structural-derivation-and-its-i",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Born Rule as Haar Measure on p‑Adic Tree Balls: A Structural Derivation and Its Implications for Quantum Statistics"
      }
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    "publicationYear": "2026",
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        "description": "Quantum theory postulates the Born rule without a deeper structural justification, and Bell‑type correlations are usually interpreted as non‑local. This gap hinders deterministic reconstructions of quantum mechanics and inflates the engineering overhead of quantum‑information technologies that must treat randomness as a primitive. We model the static ultrametric Bruhat–Tits tree of p‑adic balls as a branching sample space and derive quantum probabilities from the Haar (topological) measure of these balls. Using the p‑adic norm |x|_p=p^-r, the definition of closed balls B_r(a)=\\xinQ_pmid|x-a|_p≤ p^r\\, and the identification of the ball hierarchy with the tree, we prove that a ball at depth k carries measure p^-k. For p=2, depth 1 yields measure $0.5$, reproducing the Born probability of a balanced qubit; for p=3, depth 1 yields 1/3, reproducing the Born probability of an equal‑amplitude three‑outcome interferometer. A deterministic shared‑ancestor model on the same tree yields CHSH correlations that can be analyzed, but it does not attain the maximal quantum value 2√2; achieving that remains an open question. The result changes practice by providing a geometry‑based foundation for the Born rule, enabling deterministic design of quantum‑information protocols and offering a concrete superdeterministic alternative to non‑local explanations.",
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