Anisotropic Fine‑Structure Constant from Non‑Isotropic $p$‑adic Embeddings

In the paper's words

Precision metrology assumes the fine‑structure constant α to be isotropic. Existing Lorentz‑violation analyses within the Standard‑Model Extension constrain any directional variation of α to a fractional level of 10^-20, but no theoretical framework links this bound to a deeper geometric origin. We develop a formal model in which the embedding of p‑adic completions into the Archimedean sector is direction‑dependent: each spatial direction i couples to a distinct prime p_i, modifying the electromagnetic coupling tensor.

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

Numbers it reports

1. (5): $$ \frac{\Delta K_{J}}{K_{J}}\bigg|_{i}= \varepsilon_{i},\qquad \frac{\Delta R_{K}}{R_{K}}\bigg|_{i}= -\varepsilon_{i}. \tag{12} $$ Numerically: $$ \begin{aligned} &\frac{\Delta K_{J}}{K_{J}}\big|_{x}=1.5\times10^{-20},\quad \frac{\Delta R_{K}}{R_{K}}\big|_{x}= -1.5\times10^{-20},\\[2mm] &\frac{\Delta K_{J}}{K_{J}}\big|_{y}=1.0\times10^{-20},\quad \frac{\Delta R_{K}}{R_{K}}\big|_{y}= -1.0\times10^{-20},\\[2mm] &\frac{\Delta K_{J}}{K_{J}}\big|_{z}=6.0\times10^{-21},\quad \frac{\Delta R_{K}}{R_{K}}\big|_{z}= -6.0\times10^{-21}. \end{aligned} \tag{13} $$ Rotation‑induced modulation amplitude.

   from 4. Analysis

2. Existing Lorentz‑violation analyses within the Standard‑Model Extension constrain any directional variation of $\alpha$ to a fractional level of $10^{-20}$, but no theoretical framework links this bound to a deeper geometric origin.

   from Abstract

3. Current Lorentz‑violation searches within the Standard‑Model Extension (SME) place an upper bound of $10^{-20}$ on fractional anisotropies of $\alpha$.

   from 1. Introduction

4. Quantitative bound $\Delta\alpha/\alpha\lt 10^{-20}$ obtained by enforcing the SME limit on the $p=3$ direction (Section 5).

   from 1. Introduction

5. The SME analysis provides an empirical upper bound on any directional fractional variation of $\alpha$: $$ |\varepsilon_{y}| \le 10^{-20}. \tag{6} $$ Since $p_{y}=3$, Eq.

   from 3.3 Relating $\lambda$ to SME bounds

6. (6) and $p_{y}=3$: $$ \varepsilon_{y}= \frac{\lambda}{3}\le10^{-20} \;\Longrightarrow\; \lambda \le 3\times10^{-20}. \tag{8} $$ We set $\lambda = 3\times10^{-20}$ (saturating the bound) for a worst‑case estimate.

   from 4. Analysis

How it is done

1. Existing Lorentz‑violation analyses within the Standard‑Model Extension constrain any directional variation of $\alpha$ to a fractional level of $10^{-20}$, but no theoretical framework links this bound to a deeper geometric origin.

   from Abstract

2. Current Lorentz‑violation searches within the Standard‑Model Extension (SME) place an upper bound of $10^{-20}$ on fractional anisotropies of $\alpha$.

   from 1. Introduction

3. The idea that number‑theoretic structures influence physical constants has appeared in several recent studies: [1] formulates a $p$‑adic analogue of Borel’s regulator theorem, relating $p$‑adic $K$‑groups to special values of $L$‑functions.

   from 2. Background and Related Work

4. Let $\mathbf{n}=(n_{x},n_{y},n_{z})$ be a unit vector indicating the orientation of a laboratory apparatus.

   from 3.1 Direction‑dependent embedding model

What would falsify it

1. Observation of a periodic modulation in $K_{J}$ or $R_{K}$ with amplitude exceeding $9\times10^{-21}$ under controlled rotation would falsify the present linear $p$‑adic embedding model.

   from 6.2 Falsifiability

2. Conversely, a null result at the $10^{-20}$ level would tighten the bound on $\lambda$ to below $3\times10^{-20}$, further constraining the hypothesis.

   from 6.2 Falsifiability

3. Can the model be embedded in a full quantum‑field‑theoretic framework that predicts additional observables (e.g., anisotropic photon dispersion)?

   from 6.3 Open Questions

Where the claim stops

1. Assumption Sensitivity Linear dependence $\varepsilon_{i}=\lambda/p_{i}$ (Eq.

   from 6.1 Limitations

2. Future work should explore higher‑order $p$‑adic corrections, embed the model in a full quantum‑field‑theoretic setting, and pursue the pilot rotation experiment outlined herein.

   from 8. Conclusion

3. Precision metrology assumes the fine‑structure constant $\alpha$ to be isotropic.

   from Abstract

Where to read

1. Abstract186 words · 1 min
2. 1. Introduction235 words · 1 min
3. 2. Background and Related Work277 words · 1 min
4. 3. Methods189 words · 1 min
5. 4. Analysis202 words · 1 min
6. 5. Results64 words · 1 min
7. 6. Discussion190 words · 1 min
8. 7. Practical Implications and Engineering Pathways330 words · 1 min
9. 8. Conclusion106 words · 1 min
10. Correction Notice275 words · 1 min
11. Provenance and AI Involvement233 words · 1 min
12. Appendix A. Divergence report247 words · 1 min