Emergent Minkowski Spacetime from Products of Bruhat–Tits Trees: Dimensional Selection, Horizon Geometry, and Scaling‑Limit Analysis

In the paper's words

The dimensionality and Lorentzian signature of spacetime are imposed rather than derived in most quantum‑gravity frameworks, leaving model‑building costly and background‑dependent. We close this gap by constructing a product of D Bruhat–Tits trees, orienting a single tree to supply a directed causal order, and analysing the adelic scaling limit of the resulting forest. Consistency conditions—world‑sheet anomaly cancellation, supersymmetric continuum requirements, and automorphism‑group gauge‑symmetry matching—force D=10 or D=11, the critical dimensions of superstring and M‑theory.

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

Numbers it reports

1. In the limit $n\to\infty$ and with a mixing of primes (e.g., taking a sequence of primes $\{p_{i}\}$ that grows without bound), the product space acquires a topology equivalent to $\mathbb{R}^{D-1,1}$.

   from 3.2 Scaling Limit and Metric Extraction

2. The covering dimension is $$ \dim\partial T_{p} =\lim_{n\to\infty}\frac{\log N_{n}}{\log(1/\delta_{n})} =\lim_{n\to\infty}\frac{\log(p+1)+(n-1)\log p}{n\log p}=1. \tag{12} $$ For the product boundary $\partial\mathcal{F}=\prod_{i=1}^{D}\mathbb{P}^{1}(\mathbb{Q}_{p_{i}})$ with the max‑metric, an $\varepsilon$‑cover requires depth $n_{i}=\lceil\log_{p_{i}}(1/\varepsilon)\rceil$ in each factor.

   from 4.4 Covering Dimension of the Product Boundary

How it is done

1. Correction (version 1.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place.

   from Emergent Minkowski Spacetime from Products of Bruhat–Tits Trees: Dimensional Selection, Horizon Geometry, and Scaling‑Limit Analysis

2. By constructing the product forest $\mathcal{F}=T_{c}\times T_{2}\times\cdots\times T_{D}$ and analysing its boundary $\partial\mathcal{F}=\prod\mathbb{P}^{1}(\mathbb{Q}_{p_{i}})$, we derive a scaling limit that reproduces a Lorentzian metric.

   from 1. Introduction

3. Invariant‑pattern thinking posits that physical laws are statements about quantities unchanged under specified transformations; Ostrowski’s theorem guarantees that the only completions of $\mathbb{Q}$ are the real and the $p$-adic fields [7].

   from 2. Background and Related Work

4. In the limit $n\to\infty$ and with a mixing of primes (e.g., taking a sequence of primes $\{p_{i}\}$ that grows without bound), the product space acquires a topology equivalent to $\mathbb{R}^{D-1,1}$.

   from 3.2 Scaling Limit and Metric Extraction

What would falsify it

1. If a simulation of the product forest using the toolkit of [8] produces a scaling limit whose metric fails to exhibit a Lorentzian signature (e.g., all eigenvalues of the emergent metric are positive) or if the covering dimension of the product boundary deviates from $D$, the main claim would be falsified.

   from 6.2 Falsification Criterion

2. Observation of a spectral dimension approaching $3$ in the scaling limit would support the hypothesis; a persistent deviation would refute it.

   from 7.3 Minimal Next Experiment

3. To mitigate this, any public dissemination must emphasise the provisional, theoretical nature of the construction and the necessity of the falsification tests described in Section 6.

   from 7.6 Risks and Misuse

Where the claim stops

1. The scaling limit assumes an infinite sequence of primes and idealised mixing; any practical implementation must truncate the prime set.

   from 6.1 Limitations

2. The horizon‑radius computation uses illustrative parameters (assumption A1); physical relevance requires matching to actual black‑hole data.

   from 6.1 Limitations

3. Future work will focus on numerical simulations using the toolkit of [8] and on resolving the exponential causal‑cone growth to achieve exact Minkowski‑like causality.

   from 8. Conclusion

Where to read

1. Abstract143 words · 1 min
2. 1. Introduction292 words · 1 min
3. 2. Background and Related Work333 words · 1 min
4. 3. Methods347 words · 2 min
5. 4. Analysis424 words · 2 min
6. 5. Results95 words · 1 min
7. 6. Discussion235 words · 1 min
8. 7. Practical Implications and Engineering Pathways246 words · 1 min
9. 8. Conclusion116 words · 1 min
10. Correction Notice417 words · 2 min
11. Provenance and AI Involvement241 words · 1 min
12. Appendix A. Divergence report81 words · 1 min