{
  "abstract": "We study two distinguished families of Fibonacci configurations and connect their median geometry with the asymptotic growth of random second-order recurrences. For the degenerate configuration $\\mathcal{C}_d(n)=(F_{n-1},F_{n-2},F_{n-3})$, the largest side is the sum of the other two, and Apollonius' theorem yields the exact rational median $m_a^d=F_{n-4}/2$ for $n\\ge5$. For the isosceles chain $\\mathcal{C}_v(n)=(F_{n-1},F_n,F_n)$, the median to the base is $m_a^v=\\frac12\\sqrt{4F_n^2-F_{n-1}^2}$, and its normalized limit is $\\lim_{n\\to\\infty}m_a^v/F_n=\\cos(\\pi/10)$. Using congruence arguments and Cohn's classical result on Fibonacci numbers that are twice a square, we prove that $4F_n^2-F_{n-1}^2$ is never a perfect square for $n\\ge4$; hence the isosceles median is irrational for every $n\\ge4$. We then examine three classical sign placements in random second-order Fibonacci recurrences, starting from two mutually independent iid Rademacher sequences. We construct exact, time-dependent diagonal orthogonal gauge transformations relating the two-sided-sign matrix cocycle to the two one-sign cocycles. Each transformed coefficient sequence separately retains the iid Rademacher law. Moreover, under the resulting coupling, the Euclidean operator norms of the three finite matrix products coincide exactly at every time. Consequently, the corresponding top logarithmic Furstenberg--Kesten Lyapunov exponents are equal: $\\gamma_A=\\gamma_B=\\gamma_C$. This equality is established under the stated symmetric-sign assumptions and does not assert invariance for arbitrary biased sign laws. In particular, the common exponent is $\\gamma=\\log(1.13198824\\ldots)=0.12397559\\ldots$, where $1.13198824\\ldots$ is Viswanath's classical almost-sure exponential growth factor for the random Fibonacci sequence. Thus, the paper brings together two complementary aspects: the arithmetic and geometric distinction between degenerate and non-degenerate isosceles Fibonacci configurations, and the exact gauge invariance of the asymptotic growth of classical random Fibonacci matrix cocycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Moïse Basse"
    },
    {
      "name": "El Hadji Dème"
    },
    {
      "name": "Mohamed Ben Fraj Ben Maaouia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4md954777p2r6.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "4md954777p2r6",
  "issns": [],
  "landing_url": "https://hal.science/hal-05763646v2",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-nt"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05763646/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4md954777p2r6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "hal-05763646",
      "source": "hal",
      "url": "https://hal.science/hal-05763646v4"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-nt",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Median Duality in Fibonacci Triangles: A Link to Lyapunov Exponents of Random Recurrences",
  "title_text": "Median Duality in Fibonacci Triangles: A Link to Lyapunov Exponents of Random Recurrences",
  "type": "preprint",
  "updated_at": "2026-11-06T12:20:44Z",
  "url": "https://research.draconitic.com/paper/4md954777p2r6",
  "venue": "HAL",
  "versions": []
}