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  "id": "oai:papers.qnfo.org:physical-irrelevance-of-uncountability-computable-reals-suffice-for-continuum-ph",
  "slug": "physical-irrelevance-of-uncountability-computable-reals-suffice-for-continuum-ph",
  "title": "Physical Irrelevance of Uncountability: Computable Reals Suffice for Continuum Physics",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Continuum physics is traditionally formulated over the uncountable real line R, yet every laboratory measurement, record, and computation is finite. This paper asks whether the uncountability of R—as distinct from its Archimedean completeness—has any observable consequence. The gap is formal: the physics literature treats R as indispensable, while the computability literature has only recently begun to study the computational content of uncountability, and no self‑contained result connects the two. We (i) prove that the computable reals R_c are countable, with at most P(n)=2^ n+1-1 programs of length n bits (e.g. P(100)=2¹01-1=2 535 301 200 456 458 802 993 406 410 751) and dense in R; (ii) derive an observability bound N=bigllfloor L/δbigrrfloor for an instrument of range L and resolution δ, giving N=10¹8≈2⁵9.79 for a meter‑scale position measurement at 10^-18 m resolution; and (iii) prove an indistinguishability theorem: for every finite experiment the uncountable continuum model and a computable surrogate assign the same distribution over the finite record set, rendering the Continuum Hypothesis empirically vacuous. The result licenses model‑builders to restrict to computable reals without empirical loss. The work is analytical and resides at NASA Technology Readiness Level 1–2; the next step is a proven‑exact benchmark calculation.",
  "abstract_markdown": "Continuum physics is traditionally formulated over the uncountable real line $\\mathbb{R}$, yet every laboratory measurement, record, and computation is finite. This paper asks whether the uncountability of $\\mathbb{R}$—as distinct from its Archimedean completeness—has any observable consequence. The gap is formal: the physics literature treats $\\mathbb{R}$ as indispensable, while the computability literature has only recently begun to study the computational content of uncountability, and no self‑contained result connects the two. We (i) prove that the computable reals $\\mathbb{R}_c$ are countable, with at most $P(n)=2^{\\,n+1}-1$ programs of length $n$ bits (e.g. $P(100)=2^{101}-1=2 535 301 200 456 458 802 993 406 410 751$) and dense in $\\mathbb{R}$; (ii) derive an observability bound $N=\\bigl\\lfloor L/\\delta\\bigr\\rfloor$ for an instrument of range $L$ and resolution $\\delta$, giving $N=10^{18}\\approx2^{59.79}$ for a meter‑scale position measurement at $10^{-18}\\,\\mathrm{m}$ resolution; and (iii) prove an indistinguishability theorem: for every finite experiment the uncountable continuum model and a computable surrogate assign the same distribution over the finite record set, rendering the Continuum Hypothesis empirically vacuous. The result licenses model‑builders to restrict to computable reals without empirical loss. The work is analytical and resides at NASA Technology Readiness Level 1–2; the next step is a proven‑exact benchmark calculation.",
  "date": "2026-10-13",
  "first_published": "2026-10-13",
  "last_revised": "2026-10-20T08:01:15Z",
  "modified": "2026-10-20T08:01:15Z",
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  "url": "https://papers.qnfo.org/papers/physical-irrelevance-of-uncountability-computable-reals-suffice-for-continuum-ph",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Physical Irrelevance of Uncountability: Computable Reals Suffice for Continuum Physics"
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    "publisher": "QNFO",
    "publicationYear": "2026",
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    "dates": [
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        "date": "2026-10-13",
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        "description": "Continuum physics is traditionally formulated over the uncountable real line R, yet every laboratory measurement, record, and computation is finite. This paper asks whether the uncountability of R—as distinct from its Archimedean completeness—has any observable consequence. The gap is formal: the physics literature treats R as indispensable, while the computability literature has only recently begun to study the computational content of uncountability, and no self‑contained result connects the two. We (i) prove that the computable reals R_c are countable, with at most P(n)=2^ n+1-1 programs of length n bits (e.g. P(100)=2¹01-1=2 535 301 200 456 458 802 993 406 410 751) and dense in R; (ii) derive an observability bound N=bigllfloor L/δbigrrfloor for an instrument of range L and resolution δ, giving N=10¹8≈2⁵9.79 for a meter‑scale position measurement at 10^-18 m resolution; and (iii) prove an indistinguishability theorem: for every finite experiment the uncountable continuum model and a computable surrogate assign the same distribution over the finite record set, rendering the Continuum Hypothesis empirically vacuous. The result licenses model‑builders to restrict to computable reals without empirical loss. The work is analytical and resides at NASA Technology Readiness Level 1–2; the next step is a proven‑exact benchmark calculation.",
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