A New Way to Shrink Empty Space Energy

In the paper's words

The observed dark‑energy density is smaller than the naïve Planck‑scale vacuum energy by a factor of order 10^-120, a discrepancy that challenges the predictive power of quantum field theory and hinders the design of a unified quantum‑gravity framework. Existing approaches have not demonstrated a mechanism that naturally yields such a suppression without fine‑tuning. We adopt a non‑Archimedean spacetime model in which the underlying geometry is an ultrametric tree associated with a prime p.

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

Numbers it reports

1. Using the phase‑transition condition $p=3$ from the $p$‑adic Potts model [4] and the observed suppression factor $10^{-120}$, we solve $3^{-N}=10^{-120}$ to obtain a hierarchical depth $N=252$.

   from Abstract

2. The resulting suppression factor $3^{-252}\approx10^{-120}$ reproduces the observed dark‑energy scale without adjustable continuous parameters.

   from Abstract

3. Quantitative demonstration that the suppression factor $p^{-N}$ reproduces the observed $10^{-120}$ vacuum‑energy ratio (Section 5).

   from 1. Introduction

4. The depth $N$ is determined by requiring that the $p$‑adic vacuum energy be reduced by the observed factor $10^{-120}$: $$ p^{-N}=10^{-120}. $$ Solving for $N$ yields the hierarchical depth needed to match observations.

   from 3. Methods

5. Equation to solve. $$ 3^{-N}=10^{-120}. $$ Take logarithms (base 10). $$ -N\log_{10}3=-120. $$ Isolate $N$. $$ N=\frac{120}{\log_{10}3}. $$ Insert the numerical value $\log_{10}3\approx0.4771212547$ (standard logarithm). $$ N=\frac{120}{0.47712125472}\approx251.5. $$ Round to the nearest integer depth, because a tree must have an integer number of levels. $$ N=252. $$ Compute the suppression factor with $N=252$. $$ 3^{-252}=10^{-\;252\log_{10}3}=10^{-120.2346}. $$ For verification, we evaluate $3^{-252}$ directly: $$ 3^{-252}=e^{-252\ln3}=e^{-252\times1.09861228868}\approx e^{-276.85}\approx5.83\times10^{-121}.

   from 4. Analysis

How it is done

1. Identification of a minimal lattice‑simulation experiment that can falsify the suppression mechanism (Section 9).

   from 1. Introduction

2. Our goal is to compute the vacuum‑energy density $\rho_{\text{vac}}$ for a free scalar field when the underlying spacetime is the $p$‑adic field $\mathbb{Q}_{p}$ equipped with the Vladimirov operator $D^{\alpha}$, which serves as the $p$‑adic analogue of the Laplacian.

   from 3. Methods

3. We now solve the equation $p^{-N}=10^{-120}$ explicitly, using the prime $p$ identified in the Potts model.

   from 4. Analysis

4. (1) this inequality translates to $$ 119\le\frac{120}{\log_{10}p}\le121\quad\Longrightarrow\quad \frac{120}{121}\le\log_{10}p\le\frac{120}{119}. $$ Numerically, $$ 0.9917\le\log_{10}p\le1.0084\;\Longrightarrow\;9.8\lesssim p\lesssim10.2. $$ Since $p$ must be prime and no prime lies in the interval $9.8\le p\le10.2$ (the integer $10$ is composite and $11$ exceeds the upper bound), no prime achieves a suppression within a factor of ten of $10^{-120}$ when the depth is chosen as the exact real solution $N=120/\log_{10}p$ of Eq.

   from 6. Parameter‑space projection

What would falsify it

1. A direct calculation of the vacuum‑energy density on a $p$‑adic lattice (see Section 9(c)) that yields a suppression factor significantly larger than $10^{-120}$ (e.g., $\gt 10^{-100}$) would falsify the main claim because the hierarchical depth would then be insufficient to match observations.

   from Falsification Criterion

2. Can the hierarchical depth be linked to observable signatures in the cosmic microwave background via adelic factorization [3]?

   from Open Questions

3. Identification of a minimal lattice‑simulation experiment that can falsify the suppression mechanism (Section 9).

   from 1. Introduction

Where the claim stops

1. The observed suppression factor $10^{-120}$ is taken as exact; cosmological measurements have uncertainties that could shift the required depth by a few levels.

   from Limitations

2. Our analysis builds on three concrete observations drawn from the bibliography: The $p$‑adic Potts model exhibits a phase transition only when $p=3$ [4].

   from 1. Introduction

3. We postulate, as a model assumption not derived from the shell sum above, that each hierarchical level contributes a factor $p^{-1}$, so that the total suppression factor is $p^{-N}$.

   from 3. Methods

Where to read

1. Abstract177 words · 1 min
2. 1. Introduction331 words · 1 min
3. 2. Background and Related Work283 words · 1 min
4. 3. Methods275 words · 1 min
5. 4. Analysis154 words · 1 min
6. 5. Results79 words · 1 min
7. 6. Extended Derivations and Worked Examples963 words · 4 min
8. 7. Extended Derivations and Worked Examples615 words · 3 min
9. 8. Discussion217 words · 1 min
10. 9. Practical Implications and Engineering Pathways240 words · 1 min
11. 10. Conclusion104 words · 1 min
12. Provenance and AI Involvement225 words · 1 min