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  "id": "oai:papers.qnfo.org:p-adic-ultrametric-embeddings-for-explainable-hierarchical-ai-representations",
  "slug": "p-adic-ultrametric-embeddings-for-explainable-hierarchical-ai-representations",
  "title": "p‑adic Ultrametric Embeddings for Explainable Hierarchical AI Representations",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "High‑stakes AI systems—such as medical diagnosis, autonomous driving, and financial risk assessment—are impeded by opaque internal representations that prevent regulators and users from trusting model decisions. Existing post‑hoc explainability methods rely on Euclidean embeddings that obscure the hierarchical structure inherent in many decision problems. This work addresses the gap by proposing a learnable mapping φ : R^n → Q_p that embeds neural‑network activations into a p‑adic ultrametric space. The mapping is constructed from quantized activations and base‑p digit expansion, guaranteeing that the induced p‑adic distance d_p=p^-k where k is the index of the first differing digit, i.e. the depth of the lowest common ancestor in the induced decision tree. We derive an explicit hierarchical loss, prove analytically that the ultrametric distance corresponds to tree depth, and compute a concrete example with p=3 that yields d_p=1/729≈1.37×10^-3 and a tree depth of 3. Additional derivations include the depth–capacity law p^D leaves at depth D, a prime‑selection overhead rule for a 1000‑class taxonomy, and a storage analysis for a 128‑component, 8‑digit representation showing that the naïve digit-per-byte encoding requires $1024$ bytes versus $512$ bytes for 32‑bit floats, i.e., a 2× increase that motivates a compression scheme. The result implies that explanations reduce to tracing a taxonomy branch rather than interpreting high‑dimensional vectors. If validated, model developers can replace Euclidean latent spaces with p‑adic ones, achieving interpretable hierarchical representations without sacrificing predictive performance.",
  "abstract_markdown": "High‑stakes AI systems—such as medical diagnosis, autonomous driving, and financial risk assessment—are impeded by opaque internal representations that prevent regulators and users from trusting model decisions. Existing post‑hoc explainability methods rely on Euclidean embeddings that obscure the hierarchical structure inherent in many decision problems. This work addresses the gap by proposing a learnable mapping $\\phi\\!:\\!\\mathbb{R}^{n}\\!\\rightarrow\\!\\mathbb{Q}_{p}$ that embeds neural‑network activations into a $p$‑adic ultrametric space. The mapping is constructed from quantized activations and base‑$p$ digit expansion, guaranteeing that the induced $p$‑adic distance $d_{p}=p^{-k}$ where $k$ is the index of the first differing digit, i.e. the depth of the lowest common ancestor in the induced decision tree. We derive an explicit hierarchical loss, prove analytically that the ultrametric distance corresponds to tree depth, and compute a concrete example with $p=3$ that yields $d_{p}=1/729\\approx1.37\\times10^{-3}$ and a tree depth of 3. Additional derivations include the depth–capacity law $p^{D}$ leaves at depth $D$, a prime‑selection overhead rule for a 1000‑class taxonomy, and a storage analysis for a 128‑component, 8‑digit representation showing that the naïve digit-per-byte encoding requires $1024$ bytes versus $512$ bytes for 32‑bit floats, i.e., a $2\\times$ increase that motivates a compression scheme. The result implies that explanations reduce to tracing a taxonomy branch rather than interpreting high‑dimensional vectors. If validated, model developers can replace Euclidean latent spaces with $p$‑adic ones, achieving interpretable hierarchical representations without sacrificing predictive performance.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/p-adic-ultrametric-embeddings-for-explainable-hierarchical-ai-representations",
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        "name": "Rowan Brad Quni-Gudzinas",
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        "title": "p‑adic Ultrametric Embeddings for Explainable Hierarchical AI Representations"
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