{
  "id": "oai:papers.qnfo.org:p-adic-lift-and-shift-transplanting-elementary-calculus-to-the-2-adic-tree-and-a",
  "slug": "p-adic-lift-and-shift-transplanting-elementary-calculus-to-the-2-adic-tree-and-a",
  "title": "p‑Adic Lift‑and‑Shift: Transplanting Elementary Calculus to the 2‑adic Tree and a Primary‑Education Blueprint",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Elementary analysis is taught exclusively on the Archimedean real line, yet many scientific domains—hierarchical data mining, ultrametric phylogenetics, and quantum error correction—operate naturally on non‑Archimedean structures. This paper closes the gap by (i) formally identifying the points at which real‑calculus proofs fail in the field Q_p, (ii) proving a Fundamental Theorem of Calculus for the Volkenborn integral on Mahler bases and verifying it on the basis function binomx1=x and the Bernoulli antiderivative relation Δ B₁(x)=x, (iii) deriving concrete numerical bounds from the p‑adic van Lint–Seidel inequality, and (iv) proposing a six‑year primary‑curriculum built on the binary (p=2) tree metaphor, complete with a manipulatives set of $63$ tiles, a $480$‑hour instructional budget, and a falsifiable pilot protocol. Using the inequality from [8] we compute a bound n_max=10 for a $10$‑dimensional $2$‑adic space with common angle γ=0.5. We also compute the Volkenborn integrals ∫_Z₂x dx=-1/2 and ∫_Z₂x² dx=1/6, and verify Δ B₁(x)=x for the Bernoulli antiderivative. The result changes practice by providing a mathematically rigorous pathway to teach ultrametric intuition as a native number sense, thereby expanding the pedagogical toolkit beyond the Archimedean monopoly.",
  "abstract_markdown": "Elementary analysis is taught exclusively on the Archimedean real line, yet many scientific domains—hierarchical data mining, ultrametric phylogenetics, and quantum error correction—operate naturally on non‑Archimedean structures. This paper closes the gap by (i) formally identifying the points at which real‑calculus proofs fail in the field $\\mathbb{Q}_{p}$, (ii) proving a Fundamental Theorem of Calculus for the Volkenborn integral on Mahler bases and verifying it on the basis function $\\binom{x}{1}=x$ and the Bernoulli antiderivative relation $\\Delta B_{1}(x)=x$, (iii) deriving concrete numerical bounds from the $p$‑adic van Lint–Seidel inequality, and (iv) proposing a six‑year primary‑curriculum built on the binary ($p=2$) tree metaphor, complete with a manipulatives set of $63$ tiles, a $480$‑hour instructional budget, and a falsifiable pilot protocol. Using the inequality from [8] we compute a bound $n_{\\max}=10$ for a $10$‑dimensional $2$‑adic space with common angle $\\gamma=0.5$. We also compute the Volkenborn integrals $\\int_{\\mathbb{Z}_{2}}x\\,dx=-\\tfrac12$ and $\\int_{\\mathbb{Z}_{2}}x^{2}\\,dx=\\tfrac16$, and verify $\\Delta B_{1}(x)=x$ for the Bernoulli antiderivative. The result changes practice by providing a mathematically rigorous pathway to teach ultrametric intuition as a native number sense, thereby expanding the pedagogical toolkit beyond the Archimedean monopoly.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
  "modified": "2026-10-20T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/p-adic-lift-and-shift-transplanting-elementary-calculus-to-the-2-adic-tree-and-a",
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        "name": "Rowan Brad Quni-Gudzinas",
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        "title": "p‑Adic Lift‑and‑Shift: Transplanting Elementary Calculus to the 2‑adic Tree and a Primary‑Education Blueprint"
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        "description": "Elementary analysis is taught exclusively on the Archimedean real line, yet many scientific domains—hierarchical data mining, ultrametric phylogenetics, and quantum error correction—operate naturally on non‑Archimedean structures. This paper closes the gap by (i) formally identifying the points at which real‑calculus proofs fail in the field Q_p, (ii) proving a Fundamental Theorem of Calculus for the Volkenborn integral on Mahler bases and verifying it on the basis function binomx1=x and the Bernoulli antiderivative relation Δ B₁(x)=x, (iii) deriving concrete numerical bounds from the p‑adic van Lint–Seidel inequality, and (iv) proposing a six‑year primary‑curriculum built on the binary (p=2) tree metaphor, complete with a manipulatives set of $63$ tiles, a $480$‑hour instructional budget, and a falsifiable pilot protocol. Using the inequality from [8] we compute a bound n_max=10 for a $10$‑dimensional $2$‑adic space with common angle γ=0.5. We also compute the Volkenborn integrals ∫_Z₂x dx=-1/2 and ∫_Z₂x² dx=1/6, and verify Δ B₁(x)=x for the Bernoulli antiderivative. The result changes practice by providing a mathematically rigorous pathway to teach ultrametric intuition as a native number sense, thereby expanding the pedagogical toolkit beyond the Archimedean monopoly.",
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