{
  "id": "oai:papers.qnfo.org:adelic-rate-distortion-theory",
  "slug": "adelic-rate-distortion-theory",
  "title": "Adelic Rate-Distortion Theory: The p-Adic Distortion Measure and the Adelic Information-Rate Function",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The Adelic Shannon Theory [1] generalised channel capacity to the adele ring A_Q. The companion bridge paper [2] established that the entropic number $(x, S)$ from Measurement Stratigraphy [3] is precisely the adelic information vector I(X) = (I_∞, I₂, I₃, ldots). This paper completes the trilogy by generalising rate-distortion theory to the adelic setting. We define the p-adic distortion measure d_p(x, x) = p^-v_p(x - x) — the natural ultrametric distortion derived from the p-adic absolute value — and derive the adelic rate-distortion function R(D_∞, D₂, D₃, ldots): the minimum rate (in adelic bits) required to represent a source with distortion at most D_v at each place v. We prove three theorems: (1) the adelic Shannon lower bound — R(D) ≥ I(X) - max_p(x|x): E[d] ≤ D I(X|X), where the inequality is componentwise; (2) the factorisable source theorem — for a source with independent archimedean and p-adic components, the rate-distortion function factorises as R(D) = R_∞(D_∞) × ∏_p R_p(D_p); (3) the Gaussian entropic number is the hardest source — among all sources with the same second-moment constraints at each place, the Gaussian entropic number maximises the rate-distortion function for all distortion levels simultaneously, establishing it as the universal adversarial source for adelic compres",
  "abstract_markdown": "The Adelic Shannon Theory [1] generalised channel capacity to the adele ring $\\mathbb{A}_{\\mathbb{Q}}$. The companion bridge paper [2] established that the entropic number $(x, S)$ from Measurement Stratigraphy [3] is precisely the adelic information vector $\\mathbf{I}(X) = (I_\\infty, I_2, I_3, \\ldots)$. This paper completes the trilogy by generalising rate-distortion theory to the adelic setting. We define the $p$-adic distortion measure $d_p(x, \\hat{x}) = p^{-v_p(x - \\hat{x})}$ — the natural ultrametric distortion derived from the $p$-adic absolute value — and derive the adelic rate-distortion function $R(D_\\infty, D_2, D_3, \\ldots)$: the minimum rate (in adelic bits) required to represent a source with distortion at most $D_v$ at each place $v$. We prove three theorems: (1) the adelic Shannon lower bound — $R(\\mathbf{D}) \\geq \\mathbf{I}(X) - \\max_{p(\\hat{x}|x): \\mathbb{E}[\\mathbf{d}] \\leq \\mathbf{D}} \\mathbf{I}(X|\\hat{X})$, where the inequality is componentwise; (2) the factorisable source theorem — for a source with independent archimedean and $p$-adic components, the rate-distortion function factorises as $R(\\mathbf{D}) = R_\\infty(D_\\infty) \\times \\prod_p R_p(D_p)$; (3) the Gaussian entropic number is the hardest source — among all sources with the same second-moment constraints at each place, the Gaussian entropic number maximises the rate-distortion function for all distortion levels simultaneously, establishing it as the universal adversarial source for adelic compres",
  "date": "2026-07-30",
  "first_published": "2026-07-30",
  "last_revised": "2026-07-30T00:00:00Z",
  "modified": "2026-07-30T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/adelic-rate-distortion-theory",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Adelic Rate-Distortion Theory: The p-Adic Distortion Measure and the Adelic Information-Rate Function"
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    "publicationYear": "2026",
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        "description": "The Adelic Shannon Theory [1] generalised channel capacity to the adele ring A_Q. The companion bridge paper [2] established that the entropic number $(x, S)$ from Measurement Stratigraphy [3] is precisely the adelic information vector I(X) = (I_∞, I₂, I₃, ldots). This paper completes the trilogy by generalising rate-distortion theory to the adelic setting. We define the p-adic distortion measure d_p(x, x) = p^-v_p(x - x) — the natural ultrametric distortion derived from the p-adic absolute value — and derive the adelic rate-distortion function R(D_∞, D₂, D₃, ldots): the minimum rate (in adelic bits) required to represent a source with distortion at most D_v at each place v. We prove three theorems: (1) the adelic Shannon lower bound — R(D) ≥ I(X) - max_p(x|x): E[d] ≤ D I(X|X), where the inequality is componentwise; (2) the factorisable source theorem — for a source with independent archimedean and p-adic components, the rate-distortion function factorises as R(D) = R_∞(D_∞) × ∏_p R_p(D_p); (3) the Gaussian entropic number is the hardest source — among all sources with the same second-moment constraints at each place, the Gaussian entropic number maximises the rate-distortion function for all distortion levels simultaneously, establishing it as the universal adversarial source for adelic compres",
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