{
  "id": "oai:papers.qnfo.org:monna-map-projection-of-p-adic-trees-as-a-geometric-origin-of-quantum-randomness",
  "slug": "monna-map-projection-of-p-adic-trees-as-a-geometric-origin-of-quantum-randomness",
  "title": "Monna Map Projection of p‑adic Trees as a Geometric Origin of Quantum Randomness",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Quantum‑mechanical randomness underlies cryptographic key generation, Monte Carlo simulation, and error‑corrected quantum computation. Existing explanations treat the Born rule as a fundamental stochastic postulate, leaving the origin of randomness opaque. A specific gap is the absence of a deterministic geometric mechanism that reproduces the observed statistical distribution of measurement outcomes. We adopt the Monna map, which projects the hierarchical Bruhat‑Tits tree Γ^k of a fixed prime p onto the real interval [0,1), and analyze the induced probability measure on measurement results. Using the Shannon‑entropy definition from quantum‑random‑number‑generator literature [1] and the phase‑transition condition p=3 for the p‑adic Potts model [2], we compute the entropy of the projected uniform leaf distribution for p=3 and depth k=3. The calculation yields H=log₂(3³)≈4.7548875 bits. We further derive the explicit form of the Monna map φ for a three‑digit $3$‑adic expansion and show that the induced distribution is exactly uniform on [0,1). The result demonstrates that a deterministic tree‑to‑line projection reproduces the Born‑rule statistics without invoking intrinsic stochasticity. If the projection hypothesis holds, deterministic p‑adic hardware could reproduce the statistical distribution of quantum‑random‑number generators, while the implications for fault‑tolerant quantum gate design remain speculative.",
  "abstract_markdown": "Quantum‑mechanical randomness underlies cryptographic key generation, Monte Carlo simulation, and error‑corrected quantum computation. Existing explanations treat the Born rule as a fundamental stochastic postulate, leaving the origin of randomness opaque. A specific gap is the absence of a deterministic geometric mechanism that reproduces the observed statistical distribution of measurement outcomes. We adopt the Monna map, which projects the hierarchical Bruhat‑Tits tree $\\Gamma^{k}$ of a fixed prime $p$ onto the real interval $[0,1)$, and analyze the induced probability measure on measurement results. Using the Shannon‑entropy definition from quantum‑random‑number‑generator literature [1] and the phase‑transition condition $p=3$ for the $p$‑adic Potts model [2], we compute the entropy of the projected uniform leaf distribution for $p=3$ and depth $k=3$. The calculation yields $H=\\log_{2}(3^{3})\\approx4.7548875$ bits. We further derive the explicit form of the Monna map $\\varphi$ for a three‑digit $3$‑adic expansion and show that the induced distribution is exactly uniform on $[0,1)$. The result demonstrates that a deterministic tree‑to‑line projection reproduces the Born‑rule statistics without invoking intrinsic stochasticity. If the projection hypothesis holds, deterministic $p$‑adic hardware could reproduce the statistical distribution of quantum‑random‑number generators, while the implications for fault‑tolerant quantum gate design remain speculative.",
  "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,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/monna-map-projection-of-p-adic-trees-as-a-geometric-origin-of-quantum-randomness",
  "license_url": "https://legal.qnfo.org/",
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  "sha256": "01c79cb6f3e845fa8cab24d752fe37108ada1a2b1afde587db73faa297a350e2",
  "version_url": "https://papers.qnfo.org/papers/monna-map-projection-of-p-adic-trees-as-a-geometric-origin-of-quantum-randomness/v/1.0.1",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Monna Map Projection of p‑adic Trees as a Geometric Origin of Quantum Randomness"
      }
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    "publisher": "QNFO",
    "publicationYear": "2026",
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        "description": "Quantum‑mechanical randomness underlies cryptographic key generation, Monte Carlo simulation, and error‑corrected quantum computation. Existing explanations treat the Born rule as a fundamental stochastic postulate, leaving the origin of randomness opaque. A specific gap is the absence of a deterministic geometric mechanism that reproduces the observed statistical distribution of measurement outcomes. We adopt the Monna map, which projects the hierarchical Bruhat‑Tits tree Γ^k of a fixed prime p onto the real interval [0,1), and analyze the induced probability measure on measurement results. Using the Shannon‑entropy definition from quantum‑random‑number‑generator literature [1] and the phase‑transition condition p=3 for the p‑adic Potts model [2], we compute the entropy of the projected uniform leaf distribution for p=3 and depth k=3. The calculation yields H=log₂(3³)≈4.7548875 bits. We further derive the explicit form of the Monna map φ for a three‑digit $3$‑adic expansion and show that the induced distribution is exactly uniform on [0,1). The result demonstrates that a deterministic tree‑to‑line projection reproduces the Born‑rule statistics without invoking intrinsic stochasticity. If the projection hypothesis holds, deterministic p‑adic hardware could reproduce the statistical distribution of quantum‑random‑number generators, while the implications for fault‑tolerant quantum gate design remain speculative.",
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