RTG Proton Model: Three‑Node Bound State in the SU(2) Window

Revision: 05 Sep 2025 · Version: 0.6 (analysis draft) · Authors: Mustafa Aksu, ChatGPT

Abstract

We construct and test a minimal proton model in Relational Time Geometry (RTG) as a three‑node, exchange‑bound configuration inside the soft‑spin SU(2) symmetry window \(0.28 \le x \le 0.70\) with amp=none (scheme‑invariant amplitude baseline). A grid scan over
\(J_{\mathrm{bind}}\in\{1.8,\,2.0,\,2.2,\,2.4\}\) and \(\kappa_c\in\{0.1,\,0.2,\,0.3\}\) finds plateaus with near‑ideal binding
\(E_{\text{bind}}\approx -3\,J_{\mathrm{bind}}\) and small positive curvature corrections. Typical inter‑node separations cluster around two short edges \(\sim 1.756\,\mathrm{fm}\) and one long edge \(\sim 2.03\,\mathrm{fm}\); a compact size metric sits at
\(r_{\text{metric}}\approx 1.068\text{–}1.070\,\mathrm{fm}\). Equilibrium criteria are satisfied (long plateaus, tiny post‑cool drift), and finite‑difference Hessians show either positive minima or a numerically flat mode consistent with symmetry.

1. Model & observables

Nodes & window.
Three nodes with frequencies \(\{\omega_i\}\) and phases \(\{\phi_i\}\) are constrained to the SU(2) window via \(x_i \equiv |\Delta\omega_i|/\Delta\omega^\*\in[0.28,\,0.70]\).
Spin gate enforces one closed link and two open links in best states (observed). We use amp=none so the baseline resonance amplitude is scheme‑invariant inside the window (Litim, sharp, Gaussian differ only by window weighting).

1.1 Energy functional (explicit)

We minimize
\[
E(\omega,\phi)=E_{\text{bind}} + E_{\text{curv}} + E_{\text{win}},
\]
with
\[
E_{\text{bind}} = -\,J_{\mathrm{bind}} \sum_{\langle ij\rangle} A_{ij}^{(\text{energy})},\qquad
E_{\text{curv}} = \kappa_c\,\mathcal C(\phi,\omega),\qquad
E_{\text{win}} = \lambda_{\text{win}}\sum_i \mathbf 1\!\left[x_i\notin [0.28,0.70]\right].
\]
The per‑link energy amplitude is taken explicitly as
\[
A_{ij}^{(\text{energy})}=\frac{3}{4}\,\bigl[1+\cos(\Delta\phi_{ij})\bigr]\;w(x_{ij}),
\]
with the scheme window \(w(x)\) chosen from:
\[
w_{\text{Litim}}(x)=\Theta(1-x)\,(1-x^2),\qquad
w_{\text{Gaussian}}(x)=e^{-x^2},\qquad
w_{\text{sharp}}(x)=\Theta(1-x).
\]
In all reported best states \(\,A_{ij}^{(\text{energy})}\approx 1.5\) on the two open links and \(\,E_{\text{win}}\ll 10^{-3}\), confirming good window adherence.

1.2 Size & distance observables

We report (i) edge lengths \(d_{ij}\) in fm, (ii) a compact size metric \(r_{\text{metric}}\) (cluster scale), and (iii) gate topology. A convenient map for inter‑node distance is
\[
d_{ij}\ \propto\ \frac{v_{\text{phase}}}{\Delta\omega^\*}\,\frac{1}{\sqrt{A_{ij}^{(\text{geom})}}},
\]
where \(A_{ij}^{(\text{geom})}\) uses the same kernel family as \(A_{ij}^{(\text{energy})}\) and \(v_{\text{phase}}\approx c\) by default (this is the same convention used in the EFT page). The reported fm values use the established RTG↔SI page for anchoring; we keep those anchors fixed throughout the scans.

Annealing & cooling.
Simulated annealing over \((\omega,\phi)\) with step sizes \(\Delta\omega\approx 0.08\), \(\Delta\phi\approx 0.25\), geometric cooling \(c\approx 0.985\), interleaved “shakes” (\(\delta\phi\approx 0.02\), \(\delta\omega\approx 0.01\)), and a final cold phase. Equilibrium check: a 200‑step plateau window with tolerance \(\varepsilon_E=10^{-5}\).

Stability (Hessian; optional).
Finite‑difference Hessian on the best state. We accept as “flat/symmetric” if \(\lambda_{\min}\ge -10^{-8}\); we reject as unstable if \(\lambda_{\min} < -10^{-8}\). This numeric tolerance separates soft symmetry directions from true instabilities.

Acceptance gates. All quoted best states pass:
(i) plateau (plateau_ok true, \(|\Delta E_{\text{cold}}|\lesssim 10^{-7}\)),
(ii) window (tiny \(E_{\text{win}}\)),
(iii) curvature budget (monotone in \(\kappa_c\)),
(iv) stability (Hessian test or smooth traces with no runaway).

3. Results

3.1 Energetics vs Jbind, κc

Across the grid, the bound‑state energy follows the ideal \(-3J_{\mathrm{bind}}\) with small positive curvature corrections; typical values:
\[
E_{\text{bind}}\approx -3J_{\mathrm{bind}},\qquad
E_{\text{curv}}\sim 1.7\times10^{-3}\ \text{at}\ \kappa_c=0.1\ \to\ 5\times10^{-3}\ \text{at}\ \kappa_c=0.3,\qquad
E_{\text{win}}\ll 10^{-3}.
\]
These trends are regulator‑insensitive under amp=none (Litim/sharp/Gaussian agree within our numerical precision).

3.2 Geometry & size

Best configurations are nearly phase‑aligned (tiny \(\Delta\phi\)), sit deep in the SU(2) window (\(x\approx 0.275\text{–}0.557\)), and display an isosceles geometry:
two short edges \(\approx 1.756\,\mathrm{fm}\), one long edge \(\approx 2.03\,\mathrm{fm}\).
The cluster scale is robust,
\(r_{\text{metric}}\approx 1.068\text{–}1.070\,\mathrm{fm}\),
with only weak dependence on \(J_{\mathrm{bind}}\) or \(\kappa_c\) across the scan.

3.3 Stability (Hessian)

Finite‑difference Hessians on selected points yield clear positive minima in several cases (e.g., \(\lambda_{\min}\approx 0.083,\ 0.223\)) and numerically near‑zero minima (\(\sim 10^{-9}\)) in stiffer cases—consistent with a flat symmetry mode rather than an instability.
No trace shows runaway or oscillatory behavior once on the plateau.

3.4 Equilibrium diagnostics

All reported best states satisfy plateau_ok=true with \(|\Delta E_{\text{cold}}|\) at or below \(10^{-7}\), and shake rates at 0 in the final window, indicating genuine lock‑in rather than transient minima.

3.5 Electric form factor & charge check (optional)

Using the companion script, we form a discrete charge density from gate‑weighted links and compute the Sachs electric form factor \(G_E(Q^2)\). Diagnostics:
(i) charge normalization demands \(G_E(0)\approx 1\) within tolerance; (ii) the slope gives a form‑factor radius
\[
r_E^2\ \equiv\ -6\,\left.\frac{dG_E}{dQ^2}\right|\_{Q^2\to 0}\!.
\]
We report medians and bootstrap confidence intervals across accepted best states; per‑entry \(G_E(Q)\) tables are emitted for reproducibility. In datasets lacking a geometric export, the script will note “form‑factor radius not available”.

4. Figures

Energy trace J=2.2 κc=0.10

Energy trace: \(J_{\mathrm{bind}}=2.2,\ \kappa_c=0.10\) — fast descent and long plateau.

Energy trace J=2.2 κc=0.30

Energy trace: \(J_{\mathrm{bind}}=2.2,\ \kappa_c=0.30\) — stronger curvature, same qualitative lock‑in.

Energy trace J=2.4 κc=0.10

Energy trace: \(J_{\mathrm{bind}}=2.4,\ \kappa_c=0.10\) — deeper binding consistent with \(-3J\).

Energy ladder across schemes

Observed energy ladder across schemes/windows (where available).

Radius vs parameters

Radius proxies vs \(J_{\mathrm{bind}}\), \(\kappa_c\) with scheme overlays.

5. Reproducibility (CLI)

Finalization runs (example; PowerShell backticks for line breaks):

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

Cross‑regulator checks: repeat with --scheme sharp and --scheme gaussian to populate proton_final_sharp and proton_final_gauss.

Observables & form‑factor study:

# Schemes only (Litim/Sharp/Gaussian roots)
python observe_proton.py `
  --roots proton_final proton_final_sharp proton_final_gauss `
  --formfactor --charge-check --bootstrap 2000 `
  --figdir proton_obs_figs_scheme

# Mix a Hessian subset with a final root (form factor shown where geometry is available)

python observe\_proton.py `  --roots proton_hess proton_final`
\--formfactor --charge-check --bootstrap 2000 \`
\--figdir proton\_obs\_figs\_hess

Acceptance (suggested defaults): charge normalization \(|G_E(0)-1| < 0.02\); Hessian \(\lambda_{\min}\ge -10^{-8}\); equilibrium \(|\Delta E_{\text{cold}}| \le 10^{-7}\) and plateau_ok=true; window penalty \(E_{\text{win}}\ll 10^{-3}\).

6. Notes & limitations

  • No arbitrary tuning. We do coarse grid scans; best states are outcomes of the search, not a fit to a target radius.
  • No external observer anchoring. No dedicated “Planck observer” node is included here; sizes reflect the internal RTG map with the site‑wide RTG↔SI anchors held fixed.
  • Radius proxies. We report edge lengths and \(r_{\text{metric}}\). The form‑factor radius \(r_E\) is provided where per‑state geometry exports are present; otherwise it is omitted by design.
  • Symmetry‑flat modes. Near‑zero Hessian eigenvalues are interpreted as symmetry flats; traces show no runaway when acceptance gates are met.

7. Data & release

Change log

VersionDateKey updates
0.62025‑09‑05Made the energy amplitude \(A_{ij}^{(\text{energy})}\) explicit; added window \(w(x)\) definitions;
specified Hessian tolerance (accept if \(\lambda_{\min}\ge -10^{-8}\));
documented distance map used for fm outputs; added form‑factor/charge‑check workflow and bootstrap;
expanded CLI for reproducibility; tightened acceptance gates; removed layout wrappers (div, figure).
0.52025‑09‑05Initial public analysis draft for three‑node proton model with SU(2) window, plateau diagnostics, and geometry summary.

Contact: RTG Theory Group · rtgtheory.org


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