Python Code for Fig. 1 of "Testing Information-Gauge Entanglement Couplings with Hydrogenic Spectroscopy"

This repository provides the Python source code used to generate Fig. 1 of the manuscript “Testing Information-Gauge Entanglement Couplings with Hydrogenic Spectroscopy.”

The figure presents a benchmark sensitivity map for information-gauge couplings in the ((m_\Lambda,g_\Lambda)) plane. The plotted curves use the illustrative conversion
[
\epsilon_{\rm IG}=\left(\frac{g_\Lambda E_0}{m_\Lambda}\right)^2,
]
with (E_0=h\nu_{21}=5.87\times10^{-6},\mathrm{eV}), corresponding to the hydrogen 21-cm hyperfine energy scale. The normalization is fixed to the illustrative benchmark (C_O\Delta I=1). Therefore, the curves should be interpreted as benchmark reaches or sensitivity mappings, not as model-independent exclusions.

The script includes benchmark inputs for hydrogen 21-cm spectroscopy, atomic neutrality, muonium hyperfine spectroscopy, positronium hyperfine spectroscopy, and projected hydrogen-maser/Ramsey and leptonic-atom improvements. It also marks the (m_\Lambda\lesssim E_0) nonlocal regime and the (g_\Lambda>1) nonperturbative guide region. Atomic neutrality is explicitly indicated as applying only to charge-screening realizations.

No experimental data were generated in this work. All numerical values used to reproduce the plotted curves are included directly in the Python script and are taken from the benchmark inputs discussed in the manuscript and Supplemental Material.

Files included

• Fig1_constraints.py: Python script for generating Fig. 1.
• fig1_constraints.pdf: Vector PDF output generated by the script.
• fig1_constraints.png: High-resolution PNG output generated by the script.

Notes
The code uses (C_O\Delta I=1) as an illustrative normalization. Model-specific bounds require replacing this factor with the appropriate IG atomic matrix element and comparing with residuals relative to the Standard Model prediction.

Keywords
information gauge theory; IG-RUEQFT; hydrogen spectroscopy; 21-cm line; muonium; positronium; atomic neutrality; benchmark sensitivity; Stueckelberg field; quantum information; entanglement; precision spectroscopy