Revision: v1.3.3 | Date: 12 Aug 2025 | Authors: Mustafa Aksu, Grok, ChatGPT
Symbols & units. All \(\omega,\Delta\omega,\delta\omega\) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \(f=\omega/(2\pi)\). \(\mu\) denotes the RG scale (angular s\(^{-1}\)); when plotting in Hz use \(f=\mu/(2\pi)\). The exchange/bond‑term regulator \(\sigma_{\mathrm{exch}}\) is an independent UV regulator (default \(\sigma_{\mathrm{exch}}\simeq\Delta\omega^\ast\); Lattice→Continuum §2) and is distinct from CHSH noise \(\sigma_{\mathrm{noise}}\) used in decoherence plots. Code spins are \(\sigma^{\rm spin}_i=\pm1\).
Contents
TL;DR — Main result
\[\boxed{\Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1}}\]
Equivalently \(E^\ast=\hbar\Delta\omega^\ast\approx 95.4\,\mathrm{MeV}\), \(\ell_\ast=c/\Delta\omega^\ast=2.07\pm0.11\,\mathrm{fm}\), \(r_\ast=2\pi\ell_\ast=13.0\pm0.7\,\mathrm{fm}\). Notation: \(1.45(8)\times10^{23}\,\mathrm{s}^{-1}\equiv 1.45\pm0.08\times10^{23}\,\mathrm{s}^{-1}\).
1 Background
Pair interactions are encoded via the split resonance kernel: \[\mathcal R_{ij}=A_{ij}(1+s_i s_j),\quad A_{ij}=\tfrac{3}{4}[1+\cos(\Delta\phi_{ij})]\,\exp[-(\Delta\omega_{ij}/\Delta\omega^\ast)^2].\]
Bond energy: \[E_{ij}=K’\,\frac{|\Delta\omega_{ij}|}{\Delta\omega^\ast}+J\,\mathcal R_{ij}+J_{\rm ex}\,\sin(\Delta\phi_{ij})\,\exp[-(\Delta\omega_{ij}/\sigma_{\mathrm{exch}})^2].\]
Gauge note. A lattice U(1) gauge term (e.g., \(\cos(\Delta\phi_{ij}-2\pi a A_{ij})\)) is omitted here unless an explicit gauge sector is being coupled and measured (see Gauge Symmetries §2 and Mathematical Foundations §4 for the gauged form and when to include it).
Kernel mnemonic (non‑formal). Intuitively, \(A_{ij}\propto\tfrac12|e^{i\phi_i}+e^{i\phi_j}|^2=1+\cos\Delta\phi_{ij}\); the full kernel \(\mathcal R_{ij}=A_{ij}(1+s_i s_j)\) enforces zero contribution for closed spin channels.
Couplings. \[\tilde J(\Lambda)=\frac{J(\Lambda)}{\hbar\Lambda},\quad \tilde K'(\Lambda)=\frac{K'(\Lambda)}{\hbar\Lambda},\quad g(\Lambda)=\frac{\bar J(\Lambda)}{K'(\Lambda)},\quad \bar J\equiv\tfrac{3}{2}J.\] Here \(K’\) in the continuum is the total‑variation coefficient \(K’_{\rm TV}=K_{\rm lat}a_{\rm lat}^{\,1-d}\) and is not mixed with \(\rho_s\). The dimensionless \(g\) controls dimensional‑threshold markers (Core Principles §10).
2 One-loop contribution
\[\beta_g^{(1)}=(1-\rho)\,g-\sigma_\beta C_{\rm vtx}\,g^2,\quad C_{\rm vtx}=3/2.\] With \(\rho=0.28\pm0.02\) and \(\sigma_\beta=0.42\pm0.03\) (Litim), \(\beta_g^{(1)}\approx 0.72\,g-0.63\,g^2\). Values are scheme‑dependent; see §5 for triplets.
3 Two-loop contribution
Including the sunset diagram (two‑loop vertex correction): \[\beta_g(g)=0.72\,g-0.63\,g^2-a_3 g^3,\quad a_3=\lambda_1 I_3^{(\mathrm{reg})} C_{\rm vtx}^3=0.0108\pm0.0025\ \text{(Litim)}.\] Here \(\lambda_1=0.0022\pm0.0005\), \(I_3^{(\mathrm{reg})}\approx 1.46\), \(C_{\rm vtx}=3/2\). Impact on \(g^\ast\) is \(<1\%\). Triplet values are obtained from lattice‑informed fits to the truncated Polchinski‑style flow (see also Appendix A).
4 Fixed point & critical bandwidth
Positive non‑Gaussian root: \(g^\ast\approx 1.121\) (Litim). Gaussian root: 0; negative root unphysical.
4.2 Flow integral (crossover)
Define \(\mu_\ast\) by \(g(\mu_\ast)=g^\ast(1-\varepsilon)\) with \(\varepsilon=0.05\). The parameter \(\varepsilon\) sets how close the running coupling approaches the fixed point before identifying the crossover; varying \(\varepsilon\in[0.03,0.08]\) assesses systematics in \(\Delta\omega^\ast\). Then \[\ln\frac{\Delta\omega^\ast}{\Lambda_0}=\int_{g^\ast(1-\varepsilon)}^{g(\Lambda_0)}\frac{dg}{\beta_g(g)}=-1.24\pm0.05_{\rm stat}\pm0.04_{\varepsilon}.\]
5 Consistency checks
| Regulator | \(g^\ast\) | \(\Delta\omega^\ast\) (10\(^{23}\) s\(^{-1}\)) |
|---|---|---|
| Litim | 1.121 | 1.45 |
| Sharp | 1.147 | 1.46 |
| Dim‑reg | 1.132 | 1.44 |
Note. Triplets \((\rho,\sigma_\beta,I_3)\) — Litim: \((0.28,0.42,1.46)\); Sharp: \((0.27,0.44,1.49)\); Dim‑reg: \((0.29,0.41,1.44)\). These are derived from lattice‑guided fits to the truncated flow (§3). The regulator spread of \(\Delta\omega^\ast\) is within \(\approx 0.7\%\) of the mean, well inside the \(\pm 0.08\times10^{23}\,\mathrm{s}^{-1}\) uncertainty band.
6 Implications
0.28 \(\Delta\omega^\ast=4.06\times10^{22}\,\mathrm{s}^{-1}=6.46\times10^{21}\,\mathrm{Hz}\)
0.70 \(\Delta\omega^\ast=1.02\times10^{23}\,\mathrm{s}^{-1}=1.62\times10^{22}\,\mathrm{Hz}\)
1.70 \(\Delta\omega^\ast=2.46\times10^{23}\,\mathrm{s}^{-1}=3.92\times10^{22}\,\mathrm{Hz}\)
Units. Values in rad·s\(^{-1}\); Hz via \(/2\pi\). \(\mu\) is an angular RG scale. Structure window. The narrow 1.55–1.70 band (in \(\Delta\omega/\Delta\omega^\ast\)) hosts U(1)\(^2\) phase shells (Gauge Symmetries §4).
Two‑loop flow yields \(\Delta\omega^\ast\) with \(\nu_g\approx 1.32\), matching the CHSH knee at \(\sigma_{\mathrm{noise,crit}}\approx 0.589\) (Quantum Behaviours §1.3). The fixed point \(g^\ast\) underpins dimensional thresholds (Core Principles §6) and the geometric diagnostics used in Enriched Geometric Concepts §4.
Related work: RTG Gravity I | RTG Gravity II | RTG Glossary | Core Principles | Mathematical Foundations | Gauge Symmetries | Lattice → Continuum | Quantum Behaviours | Thermodynamics
Appendix A — Closed-form flow integral
With \(\beta(g)=g(0.72-0.63 g-0.011 g^2)\), a partial‑fractions decomposition consistent with the Polchinski‑style truncation gives: \[\frac{1}{\beta(g)}=\frac{25}{18}\frac{1}{g}+\frac{0.0152778\,g+0.875}{0.72-0.63 g-0.011 g^2}.\] Integrating: \[\int\frac{dg}{\beta(g)}=\frac{25}{18}\ln|g|-\frac{50}{561}\ln|0.72-0.63 g-0.011 g^2|+\frac{625}{561\sqrt{\Delta}}\ln\left|\frac{g-g_+}{g-g_-}\right|+C,\] where \(g_\pm\) are roots of \(0.72-0.63 g-0.011 g^2=0\) and \(\Delta\) its discriminant.
Change Log
| Version | Date | Main updates |
|---|---|---|
| 1.3.3 | 2025‑08‑12 | Clarified \(\sigma_{\mathrm{exch}}\) as independent UV regulator; standardized \(\sigma_{\mathrm{exch}}\) usage; added uncertainty notation for \(\Delta\omega^\ast\); defined sunset diagram; quantified regulator spread; noted triplet derivation; added U(1)\(^2\) reference in §6; expanded related work. |
| 1.3.2 | 2025‑08‑12 | Symbols block refined (\(\sigma_{\mathrm{exch}}\simeq\Delta\omega^\ast\), distinct from \(\sigma_{\mathrm{noise}}\)); gauge‑term note; highlighted role of \(g\) as resonance coupling; U(1)\(^2\) window mention; regulator spread note; Appendix A coefficient annotations. |
| 1.3.1 | 2025‑08‑12 | Kernel mnemonic sign fix; units/symbols box; TV mapping note; Litim \(g^\ast\) aligned; \(a_3\) uncertainty; crossover \(\varepsilon\) defined; ZHz fix; energy/length equivalents; Appendix A closed form. |
| 1.3 | 2025‑08‑08 | Bond \(K’\) normalization; spin‑gate mapping; regulator harmonized; \(\Lambda\)-scale couplings; scheme annotations. |