Adelic Euler‑Product Cancellation of Vacuum Energy in Ultrametric Cosmology

In the paper's words

The observed dark‑energy density ρ_Λ^rm obs≈10^-47 GeV⁴ is smaller than the naïve quantum‑field‑theoretic vacuum estimate by roughly 10¹20, a discrepancy that dominates the cosmological constant problem. Existing proposals invoke new dynamical fields or fine‑tuned counterterms, yet they retain the Archimedean real number field as the sole arena for physics. We examine the hypothesis that the total vacuum energy should be evaluated adelically, i.e. as an Euler product over the real place and all p‑adic completions of Q.

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

Numbers it reports

1. The LSST Dark Energy Science Collaboration [2] describes how weak‑lensing shear $\gamma$ and galaxy clustering constrain the energy budget, noting that the observed vacuum energy is $10^{100}$–$10^{120}$ times smaller than naïve quantum estimates.

   from 2. Background and Related Work

2. The real‑sector vacuum density for a cutoff $\Lambda$ is $$ \rho_{\infty}= \frac{\Lambda^{4}}{16\pi^{2}}. \tag{6} $$ Equating $\rho_{\infty}$ to the $p$‑adic ground‑state energy density per site $E_{p}$ with $b=2$ gives $$ \frac{\Lambda^{4}}{16\pi^{2}} = \frac{\omega_{0}}{2(p-1)}. $$ Solving for $\omega_{0}$: $$ \omega_{0}= (p-1)\,\frac{\Lambda^{4}}{8\pi^{2}} = (p-1)\,\frac{\Lambda^{4}}{16\pi^{2}}\times 2. $$ Since the factor $2$ cancels with the $1/2$ in $E_{p}$, the compact matching condition is $$ \boxed{\omega_{0}= (p-1)\,\frac{\Lambda^{4}}{16\pi^{2}}}. \tag{7} $$ Finally we estimate the tree depth $n$ required to achieve a cancellation precision $\varepsilon=10^{-120}$.

   from 4. Analysis

How it is done

1. Using the product formula $\prod_{v}|q|_{v}=1$ for any non‑zero rational $q$ (derived in [9]), we compute the combined contribution of the test mode $q=2$ and show that the real‑sector factor $|2|_{\infty}=2$ is exactly cancelled by the $2$‑adic factor $|2|_{2}=2^{-1}=1/2$, while all other primes contribute unity, yielding a net adelic factor of $1$.

   from Abstract

2. Identification of falsifiable signatures and a minimal quantum‑simulation experiment to test the adelic cancellation (Sections 6 and 7).

   from 1. Introduction

3. Finally, Ostrowski’s theorem classifies all non‑trivial absolute values on $\mathbb{Q}$, showing that the Archimedean and $p$‑adic valuations exhaust the possibilities [10].

   from 2. Background and Related Work

4. Using the definitions above we compute $\mathcal{F}(2)$ step by step.

   from 4. Analysis

What would falsify it

1. A single observation that would falsify the main claim is the detection of a residual vacuum contribution that scales with the ultraviolet cutoff of the real sector, i.e. $\rho_{\rm vac}\propto\Lambda_{\rm UV}^{4}$, despite the presence of $p$‑adic degrees of freedom.

   from 6.2 Falsifiability

2. High‑precision measurements of Casimir forces or vacuum‑induced energy shifts that reveal such a scaling would refute the adelic cancellation hypothesis.

   from 6.2 Falsifiability

3. Identification of falsifiable signatures and a minimal quantum‑simulation experiment to test the adelic cancellation (Sections 6 and 7).

   from 1. Introduction

Where the claim stops

1. The derivation treats the momentum spectrum as a dense subset of rational numbers; a fully continuous spectrum is approximated by this rational discretisation.

   from 6.1 Limitations

2. The projection to the observed dark‑energy density assumes a simple regularisation of the countable sum; alternative regularisations could modify the magnitude.

   from 6.1 Limitations

3. Solving the eigenvalue problem on a regular tree of branching $b$ (the number of forward edges from any vertex) yields a ground‑state energy per site $$E_{p}= \frac{\omega_{0}}{2}\,\frac{b}{p-b+1},\tag{3}$$ where $\omega_{0}$ is the fundamental frequency of the oscillator and the expression is convergent only if $b\le p$ (the denominator must be positive).

   from 3.2 $p$‑adic harmonic oscillator on a Bethe lattice

Where to read

1. Abstract216 words · 1 min
2. 1. Introduction287 words · 1 min
3. 2. Background and Related Work342 words · 1 min
4. 3. Methods299 words · 1 min
5. 4. Analysis318 words · 1 min
6. 5. Results153 words · 1 min
7. 6. Discussion194 words · 1 min
8. 7. Practical Implications and Engineering Pathways288 words · 1 min
9. 8. Conclusion134 words · 1 min
10. Correction Notice298 words · 1 min
11. Provenance and AI Involvement231 words · 1 min
12. Appendix A. Divergence report128 words · 1 min