{
  "id": "oai:papers.qnfo.org:factorization-barriers-as-trapping-of-random-walks-on-bruhat-tits-trees",
  "slug": "factorization-barriers-as-trapping-of-random-walks-on-bruhat-tits-trees",
  "title": "Factorization Barriers as Trapping of Random Walks on Bruhat–Tits Trees",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Hard integer factorization underlies RSA security, yet current primality testing relies on probabilistic algorithms that retain a non‑zero error probability. A concrete gap exists: no deterministic, efficiently computable test guarantees primality without residual risk. We address this gap by modeling the p‑adic boundary as the infinite (p + 1)‑regular Bruhat–Tits tree t_p and interpreting factorization difficulty as geometric trapping of a simple random walk in the tree bulk. Our analytical approach (i) derives the exact vertex count V_D of a depth‑D truncation of t_p, (ii) applies the mixing‑time lower bound for d‑regular graphs to obtain explicit step bounds, and (iii) incorporates the deterministic spectral‑gap dichotomy observed for prime‑induced versus composite‑induced subtrees. For p=2 and D=7 we compute V₇=382, obtain a mixing‑time lower bound T_mix≥4.33ln n (≈ 25.7 steps for n=382 and ≈ 59.8 steps for n=10⁶), and show that a prime yields λ₁≥0.1716 while a composite satisfies λ₁≤ C/D (projection with C=1 gives λ₁≤0.1429). If the spectral‑gap dichotomy holds, RSA key generation could replace probabilistic tests with a deterministic, tree‑based protocol, eliminating certification risk and reducing computational overhead.",
  "abstract_markdown": "Hard integer factorization underlies RSA security, yet current primality testing relies on probabilistic algorithms that retain a non‑zero error probability. A concrete gap exists: no deterministic, efficiently computable test guarantees primality without residual risk. We address this gap by modeling the $p$‑adic boundary as the infinite $(p\\!+\\!1)$‑regular Bruhat–Tits tree $\\mathfrak{t}_p$ and interpreting factorization difficulty as geometric trapping of a simple random walk in the tree bulk. Our analytical approach (i) derives the exact vertex count $V_D$ of a depth‑$D$ truncation of $\\mathfrak{t}_p$, (ii) applies the mixing‑time lower bound for $d$‑regular graphs to obtain explicit step bounds, and (iii) incorporates the deterministic spectral‑gap dichotomy observed for prime‑induced versus composite‑induced subtrees. For $p=2$ and $D=7$ we compute $V_7=382$, obtain a mixing‑time lower bound $T_{\\mathrm{mix}}\\ge4.33\\ln n$ ($\\approx$ 25.7 steps for $n=382$ and $\\approx$ 59.8 steps for $n=10^{6}$), and show that a prime yields $\\lambda_1\\ge0.1716$ while a composite satisfies $\\lambda_1\\le C/D$ (projection with $C=1$ gives $\\lambda_1\\le0.1429$). If the spectral‑gap dichotomy holds, RSA key generation could replace probabilistic tests with a deterministic, tree‑based protocol, eliminating certification risk and reducing computational overhead.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
  "modified": "2026-10-20T00:00:00Z",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/factorization-barriers-as-trapping-of-random-walks-on-bruhat-tits-trees",
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  "version_url": "https://papers.qnfo.org/papers/factorization-barriers-as-trapping-of-random-walks-on-bruhat-tits-trees/v/1.0.0",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Factorization Barriers as Trapping of Random Walks on Bruhat–Tits Trees"
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    "publicationYear": "2026",
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        "description": "Hard integer factorization underlies RSA security, yet current primality testing relies on probabilistic algorithms that retain a non‑zero error probability. A concrete gap exists: no deterministic, efficiently computable test guarantees primality without residual risk. We address this gap by modeling the p‑adic boundary as the infinite (p + 1)‑regular Bruhat–Tits tree t_p and interpreting factorization difficulty as geometric trapping of a simple random walk in the tree bulk. Our analytical approach (i) derives the exact vertex count V_D of a depth‑D truncation of t_p, (ii) applies the mixing‑time lower bound for d‑regular graphs to obtain explicit step bounds, and (iii) incorporates the deterministic spectral‑gap dichotomy observed for prime‑induced versus composite‑induced subtrees. For p=2 and D=7 we compute V₇=382, obtain a mixing‑time lower bound T_mix≥4.33ln n (≈ 25.7 steps for n=382 and ≈ 59.8 steps for n=10⁶), and show that a prime yields λ₁≥0.1716 while a composite satisfies λ₁≤ C/D (projection with C=1 gives λ₁≤0.1429). If the spectral‑gap dichotomy holds, RSA key generation could replace probabilistic tests with a deterministic, tree‑based protocol, eliminating certification risk and reducing computational overhead.",
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