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  "id": "oai:papers.qnfo.org:emergent-spacetime-and-time-from-a-static-p-adic-relational-graph",
  "slug": "emergent-spacetime-and-time-from-a-static-p-adic-relational-graph",
  "title": "Emergent Spacetime and Time from a Static p‑Adic Relational Graph",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The prevailing view that spacetime and quantum randomness are fundamental continuous entities conflicts with persistent conceptual and technical problems in quantum gravity and cosmology. A static, discrete, non‑Archimedean relational network offers a unified ontological alternative, but quantitative bridges to observable physics are missing. We define a p‑adic relational graph, derive an emergent ultrametric distance by coarse‑graining its adjacency structure, Using the p‑adic Potts model on a Cayley tree we prove that a phase transition occurs if and only if the prime p=3, providing a concrete signature of the underlying p‑adic substrate. From the p‑adic light‑cone construction we compute the length of a cylinder at depth k=2 for p=5 as $20$ (in p‑adic units). An inequality for p‑adic equiangular lines yields a bound on the number of admissible lines in a d‑dimensional p‑adic Hilbert space. Vertex‑counting on the (p+1)‑regular tree gives the exact growth law N_n=(p+1)p^n-1 and the cumulative count V(n)=1+(p+1)(p^n-1)/(p-1). These results furnish testable ultrametric imprints in entanglement clustering and cosmic microwave background fluctuations, thereby converting the static relational hypothesis into a falsifiable scientific program.",
  "abstract_markdown": "The prevailing view that spacetime and quantum randomness are fundamental continuous entities conflicts with persistent conceptual and technical problems in quantum gravity and cosmology. A static, discrete, non‑Archimedean relational network offers a unified ontological alternative, but quantitative bridges to observable physics are missing. We define a $p$‑adic relational graph, derive an emergent ultrametric distance by coarse‑graining its adjacency structure,  Using the $p$‑adic Potts model on a Cayley tree we prove that a phase transition occurs if and only if the prime $p=3$, providing a concrete signature of the underlying $p$‑adic substrate. From the $p$‑adic light‑cone construction we compute the length of a cylinder at depth $k=2$ for $p=5$ as $20$ (in $p$‑adic units). An inequality for $p$‑adic equiangular lines yields a bound on the number of admissible lines in a $d$‑dimensional $p$‑adic Hilbert space. Vertex‑counting on the $(p+1)$‑regular tree gives the exact growth law $N_n=(p+1)p^{n-1}$ and the cumulative count $V(n)=1+(p+1)(p^{n}-1)/(p-1)$. These results furnish testable ultrametric imprints in entanglement clustering and cosmic microwave background fluctuations, thereby converting the static relational hypothesis into a falsifiable scientific program.",
  "date": "2026-10-19",
  "first_published": "2026-10-19",
  "last_revised": "2026-10-19T00:00:00Z",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/emergent-spacetime-and-time-from-a-static-p-adic-relational-graph",
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  "version_url": "https://papers.qnfo.org/papers/emergent-spacetime-and-time-from-a-static-p-adic-relational-graph/v/1.0.0",
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    "titles": [
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        "title": "Emergent Spacetime and Time from a Static p‑Adic Relational Graph"
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    "publicationYear": "2026",
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        "description": "The prevailing view that spacetime and quantum randomness are fundamental continuous entities conflicts with persistent conceptual and technical problems in quantum gravity and cosmology. A static, discrete, non‑Archimedean relational network offers a unified ontological alternative, but quantitative bridges to observable physics are missing. We define a p‑adic relational graph, derive an emergent ultrametric distance by coarse‑graining its adjacency structure, Using the p‑adic Potts model on a Cayley tree we prove that a phase transition occurs if and only if the prime p=3, providing a concrete signature of the underlying p‑adic substrate. From the p‑adic light‑cone construction we compute the length of a cylinder at depth k=2 for p=5 as $20$ (in p‑adic units). An inequality for p‑adic equiangular lines yields a bound on the number of admissible lines in a d‑dimensional p‑adic Hilbert space. Vertex‑counting on the (p+1)‑regular tree gives the exact growth law N_n=(p+1)p^n-1 and the cumulative count V(n)=1+(p+1)(p^n-1)/(p-1). These results furnish testable ultrametric imprints in entanglement clustering and cosmic microwave background fluctuations, thereby converting the static relational hypothesis into a falsifiable scientific program.",
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