How Space and Time Might Come from a Static Network

In the paper's words

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.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Application of the $p$‑adic equiangular‑line bound to a concrete $d=2$, $\gamma=0.5$ example; the bound constrains only the $p$‑adic norm $|n|_3\le1$ and yields no finite bound on the ordinary count of lines (Section 4).

   from 1. Introduction

2. In a $2$‑dimensional $p$‑adic Hilbert space with $\gamma=0.5$, the bound constrains only the $p$‑adic norm $|n|_3\le1$ and does not impose a finite upper limit on the ordinary number of lines.

   from 5. Results

3. Solving $\varepsilon(d,7)\le\eta$ for $d$ yields $$ \frac{14}{d-14}\le5\times10^{-3} \;\Longrightarrow\; d-14\ge\frac{14}{5\times10^{-3}}=2800 \;\Longrightarrow\; d\ge2814 . $$ Thus any pair of effective points separated by fewer than $2814$ graph edges would violate the $0.5\%$ error budget.

   from 5. Coarse‑graining error bound for a target tolerance

4. Substituting $|d|_{p}=1$ (any $d$ not divisible by $p$) and $\gamma^{2}$ with $p$‑adic norm $1$ (e.g. $\gamma=0.5$ for $p=3$) reduces the bound to $|n|^{2}\le\max\{|n|,1\}$.

   from Equiangular‑line bound under dimension and angle variations

5. A prototype ion‑trap system with $10$ qubits and full control electronics would require a cost estimate that depends on specific hardware choices; the performance metric will be determined by experimental measurement of ultrametric violation probability.

   from (e) Projected cost and performance

6. The concrete numerical results (phase transition at $p=3$, cylinder length $20$ for $p=5$, $k=2$, the bound constrains only the $p$‑adic norm $|n|_3\le1$ and does not impose a finite upper limit on the ordinary number of lines in $\mathbb{Q}_3^2$, and explicit vertex‑counting formulas) provide falsifiable predictions.

   from 10. Conclusion

How it is done

1. 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 Abstract

2. Proof that a phase transition in the three‑state $p$‑adic Potts model on a Cayley tree occurs iff $p=3$ (Section 4).

   from 1. Introduction

3. This inequality links the number of admissible directions to the dimension $d$ and the common angle $\gamma$, mirroring the real‑space van Lint–Seidel bound. $p$‑adic quantum mechanics. $p$‑adic and adelic quantum mechanics extend ordinary quantum theory by allowing complex‑valued wave functions on $p$‑adic arguments $$3$$ .

   from 2. Background and Related Work

4. Given a partition $\mathcal{P}=\{B_\alpha\}$ of $V$ into disjoint blocks of size $m$, we define the effective distance between blocks $B_\alpha$ and $B_\beta$ as $$ d_{\alpha\beta}= \Bigl|\,\sum_{i\in B_\alpha}\sum_{j\in B_\beta} A_{ij}\,\Bigr|_p . $$ Using the $p$‑adic norm $|\cdot|_p$ from $$5$$ , $d_{\alpha\beta}$ satisfies the ultrametric inequality $$ d_{\alpha\gamma}\le\max\{d_{\alpha\beta},d_{\beta\gamma}\}, $$ ensuring that the coarse‑grained space is an ultrametric space.

   from 3.2 Coarse‑graining and emergent distance

What would falsify it

1. A single observation that would falsify the main claim is the absence of ultrametric clustering in high‑resolution entanglement data from a quantum simulator.

   from 8.2 Falsifiability

2. Specifically, if the pairwise $p$‑adic distances inferred from measured correlations fail to satisfy the strong triangle inequality beyond statistical noise, the static $p$‑adic relational hypothesis would be refuted.

   from 8.2 Falsifiability

3. Can the $p$‑adic light‑cone structure be linked to observable causal horizons in cosmology?

   from 8.3 Open Questions

Where the claim stops

1. The concrete numerical results (phase transition at $p=3$, cylinder length $20$ for $p=5$, $k=2$, the bound constrains only the $p$‑adic norm $|n|_3\le1$ and does not impose a finite upper limit on the ordinary number of lines in $\mathbb{Q}_3^2$, and explicit vertex‑counting formulas) provide falsifiable predictions.

   from 10. Conclusion

2. 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 Abstract

3. This reliance incurs concrete costs: (i) the cosmological constant problem remains unsolved because vacuum energy is computed in a continuum field theory; (ii) the measurement problem persists because the Born rule is postulated rather than derived; (iii) numerical simulations of quantum many‑body systems scale exponentially with the assumed continuum degrees of freedom, limiting predictive power for large‑scale structure formation.

   from 1. Introduction

Where to read

1. Abstract175 words · 1 min
2. 1. Introduction266 words · 1 min
3. 2. Background and Related Work359 words · 2 min
4. 3. Methods349 words · 2 min
5. 4. Analysis583 words · 3 min
6. 5. Results169 words · 1 min
7. Appendix C. Extended Derivations and Worked Examples834 words · 4 min
8. Appendix D. Sensitivity, Limiting Cases and Robustness Checks668 words · 3 min
9. 8. Discussion221 words · 1 min
10. 7. Practical Implications and Engineering Pathways225 words · 1 min
11. 10. Conclusion107 words · 1 min
12. Provenance and AI Involvement225 words · 1 min