{
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  "slug": "acrp08-paradigm-forecast",
  "title": "When Will Non-Archimedean Geometry Displace the Real Numbers? A Structured Assessment of the Adelic Substrate Thesis",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The real numbers are the default continuum for physical modeling — every quantum state lives in a Hilbert space over C, every neural network optimizes over R^n, every error-correcting code is designed with Euclidean distance. Yet Ostrowski's theorem (1916) demonstrates that the real numbers are merely one of many completions of the rationals: every non-trivial absolute value on Q is equivalent to either the Archimedean absolute value or a p-adic absolute value for some prime p. The Archimedean default is a choice, not a necessity. Two domains already use non-Archimedean geometry as their native substrate — phylogenetic inference (ultrametric distance) and isogeny-based cryptography (p-adic arithmetic geometry) — establishing that the paradigm is viable. We examine whether and when non-Archimedean modeling will extend beyond these existence proofs into a third domain, assessing five candidate domains qualitatively and anchoring our judgments to a historical reference class of mathematical framework adoptions. We find that the central bet — displacement in at least one domain by 2036 — has a calibrated probability of approximately 0.15–0.20, consistent with a reference class in which zero of the three closest historical analogues (non-Euclidean geometry → general relativity, abstract algebra → quantum mechanics, information theory → biology) achieved mainstream adoption within a decade. The null hypothesis — that non-Archimedean modeling remains a niche mathematical tool — is the well-anchored prediction. We register nine dated, strength-weighted calibration predictions and identify actionable near-term interventions that could compress the historical timeline.",
  "abstract_markdown": "The real numbers are the default continuum for physical modeling — every quantum state lives in a Hilbert space over $\\mathbb{C}$, every neural network optimizes over $\\mathbb{R}^n$, every error-correcting code is designed with Euclidean distance. Yet Ostrowski's theorem (1916) demonstrates that the real numbers are merely *one of many* completions of the rationals: every non-trivial absolute value on $\\mathbb{Q}$ is equivalent to either the Archimedean absolute value or a $p$-adic absolute value for some prime $p$. The Archimedean default is a choice, not a necessity. Two domains already use non-Archimedean geometry as their native substrate — phylogenetic inference (ultrametric distance) and isogeny-based cryptography ($p$-adic arithmetic geometry) — establishing that the paradigm is viable. We examine whether and when non-Archimedean modeling will extend beyond these existence proofs into a third domain, assessing five candidate domains qualitatively and anchoring our judgments to a historical reference class of mathematical framework adoptions. We find that the central bet — displacement in at least one domain by 2036 — has a calibrated probability of approximately 0.15–0.20, consistent with a reference class in which zero of the three closest historical analogues (non-Euclidean geometry → general relativity, abstract algebra → quantum mechanics, information theory → biology) achieved mainstream adoption within a decade. The null hypothesis — that non-Archimedean modeling remains a niche mathematical tool — is the well-anchored prediction. We register nine dated, strength-weighted calibration predictions and identify actionable near-term interventions that could compress the historical timeline.",
  "date": "2026-08-01",
  "first_published": "2026-08-01",
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  "url": "https://papers.qnfo.org/papers/acrp08-paradigm-forecast",
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    "titles": [
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        "title": "When Will Non-Archimedean Geometry Displace the Real Numbers? A Structured Assessment of the Adelic Substrate Thesis"
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        "description": "The real numbers are the default continuum for physical modeling — every quantum state lives in a Hilbert space over C, every neural network optimizes over R^n, every error-correcting code is designed with Euclidean distance. Yet Ostrowski's theorem (1916) demonstrates that the real numbers are merely one of many completions of the rationals: every non-trivial absolute value on Q is equivalent to either the Archimedean absolute value or a p-adic absolute value for some prime p. The Archimedean default is a choice, not a necessity. Two domains already use non-Archimedean geometry as their native substrate — phylogenetic inference (ultrametric distance) and isogeny-based cryptography (p-adic arithmetic geometry) — establishing that the paradigm is viable. We examine whether and when non-Archimedean modeling will extend beyond these existence proofs into a third domain, assessing five candidate domains qualitatively and anchoring our judgments to a historical reference class of mathematical framework adoptions. We find that the central bet — displacement in at least one domain by 2036 — has a calibrated probability of approximately 0.15–0.20, consistent with a reference class in which zero of the three closest historical analogues (non-Euclidean geometry → general relativity, abstract algebra → quantum mechanics, information theory → biology) achieved mainstream adoption within a decade. The null hypothesis — that non-Archimedean modeling remains a niche mathematical tool — is the well-anchored prediction. We register nine dated, strength-weighted calibration predictions and identify actionable near-term interventions that could compress the historical timeline.",
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