{
  "id": "oai:papers.qnfo.org:qec-darwinism-ultrametric",
  "slug": "qec-darwinism-ultrametric",
  "title": "Archimedean Shadows: The QEC-Darwinism Tradeoff in Ultrametric Spaces",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Maity et al. (arXiv:2608.03944, 2026) proved that quantum error correction and Quantum Darwinism cannot coexist above a critical logical fidelity F_L > 0.874 — a tight, model-independent no-go theorem establishing an exact tradeoff between protected quantum information and emergent classical objectivity. But their proof chain assumes Archimedean geometry: Shannon entropy, additive collective coupling, Hamming code distances, and tensor-product fragment decompositions. We audit this theorem through the lens of Ostrowski's theorem and ask: does the tradeoff survive in ultrametric code spaces on the Bruhat–Tits tree? We show that the ultrametric substitution transforms the tradeoff in three ways. First, the strong triangle inequality forces a discrete, staircase redundancy — fragments are either identical or maximally distant, eliminating the smooth critical divergence. Second, the equal-weight coupling ∑_k Z_k becomes a hierarchical weight p^-d(block,k), reducing the effective environment size. Third, the Shannon entropy H₂ is replaced by a valuation-weighted entropy H_v, discretizing the no-go threshold. The Archimedean bound is recovered as the p → ∞ limit, but at small primes the tradeoff admits regimes forbidden by the original theorem. We frame three falsifiable predictions for quantum processors with 1/f^α noise and identify eight open mathematical questions (Q1–Q8) whose resolution would make the ultrametric bound quantitative. A first numerical implementation of the BT-tree code (Q6) confirms the quantized-redundancy prediction and the effective- environment reduction, while revealing an honest caveat: the deep-ultrametric regime collapses logical fidelity before the staircase can resolve in F_L, shifting the experimentally relevant regime to shallow trees or weak hierarchy. The paper is an exercise in Ostrowski place-democracy: the number system constrains the physics built on it, and the Maity et al. theor",
  "abstract_markdown": "Maity et al. (arXiv:2608.03944, 2026) proved that quantum error correction and\nQuantum Darwinism cannot coexist above a critical logical fidelity $F_L > 0.874$\n— a tight, model-independent no-go theorem establishing an exact tradeoff between\nprotected quantum information and emergent classical objectivity. But their\nproof chain assumes Archimedean geometry: Shannon entropy, additive collective\ncoupling, Hamming code distances, and tensor-product fragment decompositions.\nWe audit this theorem through the lens of Ostrowski's theorem and ask: does the\ntradeoff survive in ultrametric code spaces on the Bruhat–Tits tree? We show\nthat the ultrametric substitution transforms the tradeoff in three ways.\nFirst, the strong triangle inequality forces a **discrete, staircase redundancy**\n— fragments are either identical or maximally distant, eliminating the smooth\ncritical divergence. Second, the equal-weight coupling $\\sum_k Z_k$ becomes a\n**hierarchical weight** $p^{-d(\\text{block},k)}$, reducing the effective\nenvironment size. Third, the Shannon entropy $H_2$ is replaced by a\n**valuation-weighted entropy** $H_v$, discretizing the no-go threshold. The\nArchimedean bound is recovered as the $p \\to \\infty$ limit, but at small primes\nthe tradeoff admits regimes forbidden by the original theorem. We frame three\nfalsifiable predictions for quantum processors with $1/f^\\alpha$ noise and\nidentify eight open mathematical questions (Q1–Q8) whose resolution would make the\nultrametric bound quantitative. A first numerical implementation of the BT-tree\ncode (Q6) confirms the quantized-redundancy prediction and the effective-\nenvironment reduction, while revealing an honest caveat: the deep-ultrametric\nregime collapses logical fidelity before the staircase can resolve in $F_L$,\nshifting the experimentally relevant regime to shallow trees or weak\nhierarchy. The paper is an exercise in Ostrowski\nplace-democracy: the number system constrains the physics built on it, and\nthe Maity et al. theor",
  "date": "2026-08-05",
  "first_published": "2026-08-05",
  "last_revised": "2026-08-05T00:00:00Z",
  "modified": "2026-08-05T00:00:00Z",
  "version": "1.11.0",
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  "language": "en",
  "keywords": [
    "quantum error correction",
    "quantum darwinism",
    "ultrametric",
    "bruhat-tits tree",
    "p-adic",
    "ostrowski",
    "no-go theorem",
    "measurement stratigraphy",
    "consilience"
  ],
  "quality_tags": [
    "High-quality scholarly research",
    "open-access scholarly work",
    "versioned research publication"
  ],
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  "doi": null,
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  "url": "https://papers.qnfo.org/papers/qec-darwinism-ultrametric",
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  "license_spdx": null,
  "sha256": "fae1f86f02036c2ac3318d85717392e4f4df987c7c93c209b1da973fa8ee9f27",
  "version_url": "https://papers.qnfo.org/papers/qec-darwinism-ultrametric/v/1.11.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
        "nameType": "Personal",
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    "titles": [
      {
        "title": "Archimedean Shadows: The QEC-Darwinism Tradeoff in Ultrametric Spaces"
      }
    ],
    "publisher": "QNFO",
    "publicationYear": "2026",
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    "subjects": [
      {
        "subject": "quantum error correction"
      },
      {
        "subject": "quantum darwinism"
      },
      {
        "subject": "ultrametric"
      },
      {
        "subject": "bruhat-tits tree"
      },
      {
        "subject": "p-adic"
      },
      {
        "subject": "ostrowski"
      },
      {
        "subject": "no-go theorem"
      },
      {
        "subject": "measurement stratigraphy"
      },
      {
        "subject": "consilience"
      },
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        "date": "2026-08-05",
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        "description": "Maity et al. (arXiv:2608.03944, 2026) proved that quantum error correction and Quantum Darwinism cannot coexist above a critical logical fidelity F_L > 0.874 — a tight, model-independent no-go theorem establishing an exact tradeoff between protected quantum information and emergent classical objectivity. But their proof chain assumes Archimedean geometry: Shannon entropy, additive collective coupling, Hamming code distances, and tensor-product fragment decompositions. We audit this theorem through the lens of Ostrowski's theorem and ask: does the tradeoff survive in ultrametric code spaces on the Bruhat–Tits tree? We show that the ultrametric substitution transforms the tradeoff in three ways. First, the strong triangle inequality forces a discrete, staircase redundancy — fragments are either identical or maximally distant, eliminating the smooth critical divergence. Second, the equal-weight coupling ∑_k Z_k becomes a hierarchical weight p^-d(block,k), reducing the effective environment size. Third, the Shannon entropy H₂ is replaced by a valuation-weighted entropy H_v, discretizing the no-go threshold. The Archimedean bound is recovered as the p → ∞ limit, but at small primes the tradeoff admits regimes forbidden by the original theorem. We frame three falsifiable predictions for quantum processors with 1/f^α noise and identify eight open mathematical questions (Q1–Q8) whose resolution would make the ultrametric bound quantitative. A first numerical implementation of the BT-tree code (Q6) confirms the quantized-redundancy prediction and the effective- environment reduction, while revealing an honest caveat: the deep-ultrametric regime collapses logical fidelity before the staircase can resolve in F_L, shifting the experimentally relevant regime to shallow trees or weak hierarchy. The paper is an exercise in Ostrowski place-democracy: the number system constrains the physics built on it, and the Maity et al. theor",
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