{
  "id": "oai:papers.qnfo.org:adelic-shannon-theory",
  "slug": "adelic-shannon-theory",
  "title": "Adelic Shannon Theory: From Problem Statement to Constructive Foundations",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over R. The Adelic Physics Programme's central claim — Q, not R, is the physically accessible base field — suggests that information theory itself may have a p-adic component invisible to the archimedean formulation. We advance beyond the problem statement of the companion research design to construct the foundational elements of an Adelic Shannon Theory: a generalisation of information theory to the adele ring A_Q. Three components are developed. (1) p-Adic Entropy. We formalise the p-adic entropy H_p(X) = -∑_x p(x) v_p(x) as an ultrametric generalisation of Shannon entropy. We prove that H_p(X) satisfies non-negativity, concavity in p, and a generalised data-processing inequality (the ultrametric contraction property). We identify the p-adic maximum-entropy distribution as the geometric distribution on the valuation: P(v_p(X) = k) = (1 - p^-1) p^-k, yielding H_p^max = 1/(p-1). (2) Adelic Channel Capacity. We formalise the Additive Ultrametric (AUM) channel — the p-adic analogue of the AWGN channel — and derive its capacity from first principles: C_p = (1)/(2)log_p(1 + SNR_p), where SNR_p is the ratio of signal valuation scale to noise valuation scale. We prove the product-formula coding theorem for the factorisable A",
  "abstract_markdown": "Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over $\\mathbb{R}$. The Adelic Physics Programme's central claim — $\\mathbb{Q}$, not $\\mathbb{R}$, is the physically accessible base field — suggests that information theory itself may have a $p$-adic component invisible to the archimedean formulation. We advance beyond the problem statement of the companion research design to construct the foundational elements of an Adelic Shannon Theory: a generalisation of information theory to the adele ring $\\mathbb{A}_{\\mathbb{Q}}$.\n\nThree components are developed. **(1) p-Adic Entropy.** We formalise the $p$-adic entropy $H_p(X) = -\\sum_x p(x) v_p(x)$ as an ultrametric generalisation of Shannon entropy. We prove that $H_p(X)$ satisfies non-negativity, concavity in $p$, and a generalised data-processing inequality (the ultrametric contraction property). We identify the $p$-adic maximum-entropy distribution as the geometric distribution on the valuation: $P(v_p(X) = k) = (1 - p^{-1}) p^{-k}$, yielding $H_p^{\\max} = 1/(p-1)$. **(2) Adelic Channel Capacity.** We formalise the Additive Ultrametric (AUM) channel — the $p$-adic analogue of the AWGN channel — and derive its capacity from first principles: $C_p = \\frac{1}{2}\\log_p(1 + \\text{SNR}_p)$, where $\\text{SNR}_p$ is the ratio of signal valuation scale to noise valuation scale. We prove the product-formula coding theorem for the factorisable A",
  "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-shannon-theory",
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  "version_url": "https://papers.qnfo.org/papers/adelic-shannon-theory/v/1.2.0",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Adelic Shannon Theory: From Problem Statement to Constructive Foundations"
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        "description": "Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over R. The Adelic Physics Programme's central claim — Q, not R, is the physically accessible base field — suggests that information theory itself may have a p-adic component invisible to the archimedean formulation. We advance beyond the problem statement of the companion research design to construct the foundational elements of an Adelic Shannon Theory: a generalisation of information theory to the adele ring A_Q. Three components are developed. (1) p-Adic Entropy. We formalise the p-adic entropy H_p(X) = -∑_x p(x) v_p(x) as an ultrametric generalisation of Shannon entropy. We prove that H_p(X) satisfies non-negativity, concavity in p, and a generalised data-processing inequality (the ultrametric contraction property). We identify the p-adic maximum-entropy distribution as the geometric distribution on the valuation: P(v_p(X) = k) = (1 - p^-1) p^-k, yielding H_p^max = 1/(p-1). (2) Adelic Channel Capacity. We formalise the Additive Ultrametric (AUM) channel — the p-adic analogue of the AWGN channel — and derive its capacity from first principles: C_p = (1)/(2)log_p(1 + SNR_p), where SNR_p is the ratio of signal valuation scale to noise valuation scale. We prove the product-formula coding theorem for the factorisable A",
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