Branching Fractal p‑adic Reality as a Replacement for Minkowski Spacetime and Hamiltonian Dynamics

In the paper's words

The incompatibility between relativistic spacetime geometry and Hamiltonian quantum dynamics imposes a persistent barrier to a unified physical theory, consuming substantial research resources and limiting predictive power in high‑energy and cosmological regimes. Existing frameworks assume an Archimedean continuum and linear time evolution, leaving the structural origin of quantum‑gravitational tension unexplained. We formulate a Wheeler–DeWitt‑type equation on an infinite rooted p‑adic tree, derive its branching solutions, and demonstrate that a coarse‑grained limit reproduces approximate Lorentzian geometry and Schrödinger‑type evolution.

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

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. Using the $p$‑adic fractal‑string tube formula of Lapidus et al. [2] we compute the Minkowski dimension of the 3‑adic Cantor string as $D=\log 2/\log 3\approx0.63093$, and from the $p$‑adic ball volume $V(p^{-n})=p^{-n}$ we obtain explicit inner‑tube volumes.

   from Abstract

2. Building on this, Lapidus et al. in [2] derived exact tube formulas for $p$‑adic self‑similar strings, showing that the volume of an inner $\varepsilon$‑neighbourhood can be expressed as a sum over complex dimensions; their Section 1 provides the concrete example of the 3‑adic Cantor string; we compute its Minkowski dimension in (7)–(8) from the similarity relation.

   from 2. Background and Related Work

3. To connect with emergent spacetime, we define a coarse‑grained field $\Phi(t,\mathbf{x})$ by averaging $\Psi$ over all vertices within a ball of radius $p^{-N}$ at depth $N$.

   from 3.4 Coarse‑Graining

4. (c) Minimal de‑risking experiment.

   from 7. Practical Implications and Engineering Pathways

What would falsify it

1. A single observation that would falsify the main result is the detection of a dispersion relation in ultrafast interferometry that deviates from the $p$‑dependent quadratic form predicted by (13).

   from 6.2 Falsifiability

2. Identification of falsifiable experimental signatures and a concrete next‑step prototype (Section 7).

   from 1. Introduction

3. Observation of a deviation exceeding $5\%$ from the predicted scaling for $p=2$ would falsify the model.

   from 7. Practical Implications and Engineering Pathways

Where the claim stops

1. The analysis assumes a perfectly regular $p$‑ary tree; physical configurations may involve defects or varying branching factors.

   from 6.1 Limitations

2. The moment formulas rely on the standard Haar measure of $\mathbb{Z}_{p}$ and on the annulus measure $(1-1/p)p^{-k}$; deviations from these assumptions would alter the results.

   from 6.1 Limitations

3. If any of these assumptions fail, the derived dimension $D$, inner‑tube volumes, or emergent Schrödinger equation could be altered.

   from 6.1 Limitations

Where to read

1. Abstract176 words · 1 min
2. 1. Introduction252 words · 1 min
3. 2. Background and Related Work803 words · 3 min
4. 3. Methods245 words · 1 min
5. 4. Analysis234 words · 1 min
6. 5. Results163 words · 1 min
7. 6. Discussion211 words · 1 min
8. 7. Practical Implications and Engineering Pathways225 words · 1 min
9. 8. Conclusion109 words · 1 min
10. Correction Notice386 words · 2 min
11. Provenance and AI Involvement243 words · 1 min
12. Appendix A. Divergence report13 words · 1 min