RTG Hydrogen: Proton Triad + Single Electron (U(1)×SU(2))

Revision: 11 Sep 2025 · Version: 0.1 (first draft) · Authors: Mustafa Aksu, ChatGPT

Summary. We model the hydrogen atom in Relational Time Geometry (RTG) as a fixed
three‑node proton triad (inside the SU(2) window) that binds a single electron node
(inside the U(1) window). With the scheme‑invariant baseline amp=none, the binding
amplitude becomes regulator‑independent; differences between Litim / sharp / Gaussian
enter only through the window weight w(x). Simulated‑annealing searches over a small
grid of electron couplings yield stable plateaus with dimensionless total energies in the
band E ≈ −2.7 … −3.3 and radii from the beat‑distance map in the
10⁴–10⁵ fm range once near‑resonant outliers are trimmed—consistent with a
Bohr‑scale envelope when interpreted through the RTG↔SI calibration.

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1) Model & observables

1.1 Proton seed (fixed)

The proton is a three‑node bound state in the SU(2) window, obtained from the
proton search script. For hydrogen runs we load a canonical seed file:

proton_seed.json  // emitted by rtg_hydrogen.py
{
  "proton_nodes": [
    {"omega": &omega1, "phi": &phi1, "spin": +1},
    {"omega": &omega2, "phi": &phi2, "spin": -1},
    {"omega": &omega3, "phi": &phi3, "spin": +1}
  ]
}

This file mirrors the best proton triad (ω, φ, spin) exported by the
updated proton script (see CLI below). No proton parameters are tuned for hydrogen;
the proton is treated as a rigid internal cluster during the electron search.

1.2 Energy functional (explicit)

Windows and weights follow the EFT convention with
\(x \equiv |\Delta \omega|/\Delta\omega^\*\) and window weight \(w(x)\):

w_litim(x)   = Θ(1-x)·(1 − x²)
w_sharp(x)   = Θ(1-x)
w_gaussian(x)= exp(−x²)

The per‑link energy amplitude is explicit:

A_ij^(energy) = (3/4)·[1 + cos(Δφ_ij)] · w(x_ij)         (angular phases)

Hydrogen total energy (proton triad fixed, one electron “e”):

E_total = E_bind + E_curv + E_win.

E_bind  = − J_e · Σ_{p ∈ proton} A_{ep}^(energy)         // 3 ep links

E_curv  = (κ_e/2) · Σ_{p} (wrap(φ_e − φ_p))²             // quadratic phase smoothness

E_win   = λ_win · 1[x_e ∉ U(1)],  with  x_e = |ω_e|/Δω*

Scheme‑invariant baseline: amp=none is used everywhere, so binding is
regulator‑independent; Litim/sharp/Gaussian differ only through w(x) inside
A_ij^(energy).

1.3 Size & distance metrics

We report two radius proxies:

• Beat‑distance radius \(r_\star\): from the electron–proton beat,
  
  using the page‑standard RTG↔SI map (phase velocity \(v_{\text{phase}}\), critical
  
  bandwidth \(\Delta\omega^\*\)). This is the “raw” kinematic radius and is
  
  sensitive to near‑resonant \(x \to 0\) outliers.
• Bohr‑band estimate \(r_{\mathrm{Bohr}}\): a trimmed metric (median or
  
  10–90% trimmed mean over accepted cases) meant to capture the envelope scale,
  
  robust to occasional near‑resonant spikes.

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2) Search algorithm & acceptance

• Annealing. Geometric cooling with local proposals
  
  \(Δ\omega \sim 0.08\cdot(Δ\omega^\*)·10^{-2}\) and \(Δ\phi \sim 0.25\), early lock‑in
  
  if a 200‑step plateau satisfies \(|ΔE|<10^{-7}\).
• Acceptance gates.
  • Plateau window 200 steps with tolerance \(ε_E=10^{-7}\).
  • Electron inside U(1) window (small \(E_{\text{win}}\)).
  • (Optional) Form‑factor check on future runs: charge normalization \(G_E(0)=1\pm 0.02\).

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3) Results

Energetics. Over
\(J_e \in \{0.8,1.0,1.2\}\), \(κ_e \in \{0.05,0.10\}\) (and refined grids),
best plateaus fall in three clear bands, monotone in \(J_e\) and only weakly dependent
on \(κ_e\):

• \(J_e=0.8\): E ≈ −2.0 … −2.2
• \(J_e=1.0\): E ≈ −2.5 … −2.8
• \(J_e=1.2\): E ≈ −3.1 … −3.4

Best‑case energy per grid point. Monotone decrease with \(J_e\) and mild \(κ_e\) dependence.

Radii. The raw beat‑distance radius \(r_\star\) shows a wide dynamic range because
electron–proton \(x\) can approach the U(1) boundary; trimming near‑resonant outliers gives a
mid‑band in the 10⁴–10⁵ fm range, compatible with a Bohr‑scale envelope
(0.5 Å = 5.29×10⁴ fm) under the site’s RTG↔SI map. A representative plot is shown below;
the single large spike corresponds to a near‑resonant case (kept here for transparency but
excluded from trimmed summaries).

Radius policy \(r_\star\) vs case index. The tall spike is a near‑resonant outlier;
trimmed statistics (10–90%) yield O(10⁴–10⁵) fm.

Scale sanity check (relative). The proton’s internal radius metric from the
triad is ≈1.07 fm (SU(2) window), while the hydrogen envelope (trimmed \(r_{\mathrm{Bohr}}\))
is O(10⁴–10⁵) fm. The ratio is therefore O(10⁴–10⁵),
consistent with atomic vs hadronic scales without introducing extra anchors.

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4) Reproducibility (CLI)

4.1 Export a proton triad with (ω, φ, spin)

# Re‑run proton finalization to emit proton_nodes.json alongside index.json
python rtg_proton.py `
  --outdir proton_final `
  --window SU2 --scheme litim --amp none `
  --Jbind 2.0 2.2 2.4 `
  --kappa_c 0.10 0.30 `
  --lambda-win 2.0 `
  --trials 512 --iters 6000 `
  --workers 8 --seed 20250905 `
  --save-traces --plots --hess

# Files of interest (all in proton_final/):
#   - index.json
#   - summary.csv
#   - proton_nodes.json   <— explicit [ω, φ, spin] for the best triad

4.2 Hydrogen (electron search around fixed triad)

python rtg_hydrogen.py `
  --outdir hydrogen_final `
  --scheme litim `
  --pwindow SU2 `
  --ewindow U1 `
  --amp none `
  --Je 0.8 1.0 1.2 `
  --Ke 0.05 0.10 `
  --lambda-win 2.0 `
  --trials 256 `
  --iters 4000 `
  --save-traces --plots `
  --proton-root proton_final
# Emits:
#   hydrogen_final/index.json
#   hydrogen_final/summary.csv
#   hydrogen_final/proton_seed.json  (the triad actually used)

4.3 Observables & summaries

# Quick summary + plots (energy ladder, radius policy)
python observe_hydrogen.py `
  --root hydrogen_final `
  --outcsv H_observables.csv `
  --summary H_observables_summary.csv `
  --figdir H_refine_figs `
  --radius-policy trimmed
#   - H_energy_ladder.png
#   - H_radius_vs_case.png
#   - hydrogen_nodes.json   <— canonical [proton triad + electron] JSON

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5) Data & artifacts

• Hydrogen summary: H_observables_summary.csv
• Per‑case table: H_observables.csv
• Nodes (for reuse): hydrogen_nodes.json
• Figures: H_energy_ladder.png, H_radius_vs_case.png
• Run index: hydrogen_final/index.json, hydrogen_final/summary.csv

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6) Notes, caveats, next steps

• No extra tuning. The proton is fixed; hydrogen scans only vary \((J_e, κ_e)\)
  
  and random seeds for the electron.
• Near‑resonant outliers. Raw \(r_\star\) can spike when \(x \to 0\). We therefore
  
  report a trimmed radius \(r_{\mathrm{Bohr}}\) for scale comparisons.
• Charge/form‑factor. The electron charge normalization and Sachs \(G_E(Q^2)\)
  
  are slated for the next iteration (script hooks are in place).

Planned next steps.

1. Form factor: implement and publish \(G_E(Q^2)\) for hydrogen, verify
   
   \(G_E(0)=1\) and extract a small‑\(Q^2\) slope radius \(r_E\) with bootstrap errors.
2. Spin configurations: document which proton triad spins favor the deepest hydrogen
   
   plates; sweep electron spin and record degeneracies.
3. Calibration pass: apply the two‑anchor RTG↔SI map (Ω, Λ) from the unified page to
   
   quote hydrogen energies and radii with uncertainties.
4. Excited shells: repeat the electron search with different U(1) sub‑windows to
   
   probe excited states (compare \(r_\star\) bands and energy gaps).

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7) Change log

Version	Date	Key updates
0.1	2025‑09‑11	First public draft. Explicit energy model for hydrogen (3 proton links), plateau diagnostics, energy ladder plot, radius policy with trimmed Bohr band, reproducibility CLI, and export of hydrogen_nodes.json.

Contact: RTG Theory Group · rtgtheory.org