Python code for Figure 2: (nS)-state contact scaling of the information-gauge shift in hydrogenic spectroscopy

This repository contains the Python script used to generate Figure 2 of the manuscript Prepared-State Hydrogenic Spectroscopy as a Null Test of Information-Gauge Entanglement Couplings. The figure illustrates the benchmark (nS)-state contact scaling of an information-gauge spin-entanglement response in hydrogenic (2S-nS) transitions.

The script computes and plots the benchmark frequency shift

A_{\rm IG}
\left(
2^{-3}-n^{-3}
\right),
]

using (A_{\rm IG}=2.0,{\rm mHz}), corresponding to the hydrogen 21-cm benchmark sensitivity scale discussed in the manuscript. It also converts the shift into an energy scale using (h=4.135667696\times10^{-15},{\rm eV,s}), showing that the asymptotic (n\rightarrow\infty) value corresponds to (\delta\nu_\infty=0.250,{\rm mHz}) or (\delta E_\infty\simeq1.03\times10^{-18},{\rm eV}).

The code generates publication-ready PDF and PNG outputs:

• fig2_nS_contact_scaling_PLA.pdf
• fig2_nS_contact_scaling_PLA.png

The plot is intended as a benchmark sensitivity visualization, not as a model-independent exclusion. The robust feature illustrated by the figure is the (n^{-3}) contact-scaling pattern for (S)-states and the absence of a contact shift for (\ell\neq0) states.

Keywords
hydrogenic spectroscopy; hyperfine structure; information-gauge field; entanglement; Ramsey spectroscopy; null test; Python; matplotlib; benchmark sensitivity; effective field theory