{
  "id": "oai:papers.qnfo.org:valuation-independent-foundations",
  "slug": "valuation-independent-foundations",
  "title": "Valuation Without R: A Category-Theoretic Foundation for Finite Measurement",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Measurement in physics is always finite and approximate — yet its mathematical foundations are embedded in the Archimedean continuum of real numbers and the cumulative hierarchy of set theory. We invert this dependency: the act of measurement is axiomatized as a graded distinguishability map v: S² → N ∪ \\∞\\ over a poset of states satisfying the ultrametric inequality, with no dependence on R or set-theoretic foundations. The category Val of valuation spaces with non-expansive maps is self-contained — R appears only as a limit idealization. The effective spatial dimension d emerges as the exponent in the asymptotic growth N(r) ∼ q^d · r of the distinguishability graph at resolution r, constrained by the cohomological consistency of the refinement sheaf across resolutions. Falsifiable predictions are pre-registered at Planck-scale resolution: ultrametric clustering in the distinguishability graph G_r supersedes Euclidean nearest-neighbor structure at r > r_c. WBS: QNFO.UMP.004 | Status: published",
  "abstract_markdown": "Measurement in physics is always finite and approximate — yet its mathematical foundations are embedded in the Archimedean continuum of real numbers and the cumulative hierarchy of set theory. We invert this dependency: the act of measurement is axiomatized as a graded distinguishability map $v: \\mathcal{S}^2 \\to \\mathbb{N} \\cup \\{\\infty\\}$ over a poset of states satisfying the ultrametric inequality, with no dependence on $\\mathbb{R}$ or set-theoretic foundations. The category Val of valuation spaces with non-expansive maps is self-contained — $\\mathbb{R}$ appears only as a limit idealization. The effective spatial dimension $d$ emerges as the exponent in the asymptotic growth $N(r) \\sim q^{d \\cdot r}$ of the distinguishability graph at resolution $r$, constrained by the cohomological consistency of the refinement sheaf across resolutions. Falsifiable predictions are pre-registered at Planck-scale resolution: ultrametric clustering in the distinguishability graph $G_r$ supersedes Euclidean nearest-neighbor structure at $r > r_c$. **WBS: QNFO.UMP.004** | **Status: published**",
  "date": "2026-08-04",
  "first_published": "2026-08-04",
  "last_revised": "2026-08-04T00:00:00Z",
  "modified": "2026-08-04T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/valuation-independent-foundations",
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  "version_url": "https://papers.qnfo.org/papers/valuation-independent-foundations/v/0.8.1",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Valuation Without R: A Category-Theoretic Foundation for Finite Measurement"
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