A New Way to Build Discrete Space and Time

In the paper's words

The incompatibility between a discrete quantum substrate and the continuous Lorentz‑invariant manifolds of general relativity has motivated a series of no‑go theorems that forbid locally finite, Poincaré‑invariant networks on Archimedean Minkowski space. These theorems block the construction of fundamentally discrete spacetime models that retain the observed absence of a preferred inertial frame. Existing proofs rely on Archimedean geometry, ultrametric norms, and compactness properties that do not hold in a p‑adic setting.

The abstract, verbatim; every line below is quoted from the text.

What it claims

that an invariant locally finite graph exists on Q_p^n and thereby evades the Archimedean no‑go theorem—would be falsified if:

Numbers it reports

1. It derives exact formulas for the number of events at depth $d$, the total number of events within a depth‑$D$ ball, and the number of comparable ordered pairs, all with explicit arithmetic. [10] maps the adjacency operator of the Bruhat–Tits tree to an effective Hamiltonian, derives a Lieb‑Robinson bound that yields a dimensionless speed of light, and shows how Lorentz invariance can emerge as a continuum symmetry from the underlying combinatorial geometry. [11] constructs a branching $p$‑adic fractal model that reproduces Minkowski‑type dimensions and emergent Schrödinger dynamics, further supporting the plausibility of non‑Archimedean foundations for spacetime physics.

   from 2. Background and Related Work

2. Event budget – Equation (3) yields $V_{10}=3070$ events within depth 10.

   from 5. Results

3. Equiangular line bound – From [5] the $p$‑adic inequality $$ |n|^{2}\le |d|\max\{|n|,\gamma^{2}\} \tag{11} $$ implies, for $d=5$ and $\gamma=0.5$ (both $p$‑adic absolute values equal 1), that $|n|^{2}\le5$.

   from 5. Results

4. The projected computational cost is therefore $\sim10^{4}$ CPU‑seconds under the stated assumptions.

   from (e) Order‑of‑Magnitude Projections

How it is done

1. We define a $p$‑adic analogue of local finiteness and invariance under the $p$‑adic orthogonal group preserving a quadratic form, then derive combinatorial properties of the Bruhat–Tits tree that serve as a concrete $p$‑adic spacetime scaffold.

   from Abstract

2. We prove that the Archimedean no‑go theorem fails in the $p$‑adic setting by constructing an invariant locally finite graph on the Bruhat–Tits tree (Sec.

   from 1. Introduction

3. The result shows that $p$‑adic statistical‑mechanical systems can display qualitatively different behavior from their real counterparts, highlighting the special role of the prime parameter. [3] introduces $p$‑adic Stark regulators and formulates an Iwasawa‑Greenberg main conjecture for Artin motives, providing a framework for $p$‑adic $L$‑functions that parallels Archimedean analytic number theory. [4] proves a $p$‑adic version of Hilbert’s theorem 90, establishing that semi‑linear representations over a $p$‑adic field become trivial after passing to a suitable finite Galois extension.

   from 2. Background and Related Work

4. The rooted $(p+1)$-regular Bruhat–Tits tree $\mathcal{T}_p$ provides a canonical discrete space on which $O_Q(\mathbb{Q}_p)$ acts transitively on vertices at a given depth.

   from 3.3. Bruhat–Tits Tree as a Model

What would falsify it

1. The main claim—that an invariant locally finite graph exists on $\mathbb{Q}_p^n$ and thereby evades the Archimedean no‑go theorem—would be falsified if: A rigorous proof showed that any $p$‑adic orthogonal transformation necessarily maps a locally finite graph to one with infinite degree, contradicting Sec.

   from 6.2. Falsifiability

2. Can the prime $p$ be linked to observable constants, perhaps via the phase‑transition threshold of [2]?

   from 6.3. Open Questions

3. This demonstrates that the Archimedean no‑go arguments do not extend to non‑Archimedean geometry, opening a pathway for discrete fundamental physics without observable preferred frames.

   from Abstract

Where the claim stops

1. While [2] suggests a distinguished role for $p=3$ in a Potts model, the present analysis does not derive a physical criterion for choosing $p$.

   from 6.1. Limitations

2. Absence of Lorentz structure – Although the graph is invariant under the $p$‑adic orthogonal group, a full Lorentzian metric with null cones is not constructed; the $p$‑adic light‑cone of [6] is discrete and does not reproduce continuous null directions.

   from 6.1. Limitations

3. Continuum limit – The emergence of smooth spacetime and standard relativistic physics from the discrete model is assumed but not proved.

   from 6.1. Limitations

Where to read

1. Abstract186 words · 1 min
2. 1. Introduction242 words · 1 min
3. 2. Background and Related Work415 words · 2 min
4. 3. Methods277 words · 1 min
5. 4. Analysis523 words · 2 min
6. 5. Results174 words · 1 min
7. 6. Discussion246 words · 1 min
8. 7. Practical Implications and Engineering Pathways292 words · 1 min
9. 8. Conclusion130 words · 1 min
10. Provenance and AI Involvement225 words · 1 min