Author: Mustafa Aksu, ChatGPT, Grok | Date: 12 Aug 2025 | Version: 1.3.3
Preamble: This workbook maps the RTG lattice Hamiltonian to a continuum action with a Z₂ gate, typically in emergent 3+1D (Core Principles §4). All constants align with Two-Loop RG Derivation v1.3.3.
Notation guard: All \(\omega,\Delta\omega,\delta\omega\) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \(f=\omega/(2\pi)\). \(\sigma_{\mathrm{exch}}\) (default \(\sigma_{\mathrm{exch}} \simeq \Delta\omega^*\)) is an independent UV regulator for the exchange term, distinct from CHSH noise \(\sigma_{\mathrm{noise}}\).
Contents
1 Overview
This workbook shows how the discrete RTG lattice Hamiltonian used in Monte Carlo work translates into a continuum compact-phase action with a Z₂ gate. All constants match the Two-Loop RG Derivation v1.3.3.
2 Lattice Hamiltonian
The nearest-neighbour bond energy is
\[ H_{\text{lat}} = \sum_{\langle ij\rangle} \left[ K_{\text{lat}}\,\frac{|\Delta\omega_{ij}|}{\Delta\omega^*} + J_{\text{lat}}\,\mathcal{R}_{ij} + J_{\text{lat,ex}}\,\sin(\Delta\phi_{ij})\,e^{-(\Delta\omega_{ij}/\sigma_{\rm exch})^{2}} \right] \]
where \(\mathcal R_{ij} \equiv A_{ij}(1+s_i s_j)\), \(A_{ij}=\frac{3}{4}[1+\cos(\Delta\phi_{ij})]\,e^{-(\Delta\omega_{ij}/\Delta\omega^*)^{2}}\) (dimensionless, max \(3/2\) for \(\cos=1\)), \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \mathrm{s}^{-1}\) is the critical bandwidth, and \(\sigma_{\mathrm{exch}} \simeq \Delta\omega^*\) is the independent UV regulator for the exchange term.
Gauge note: A U(1) gauge term (e.g., \(\cos(\Delta\phi_{ij} – 2\pi a_{\rm lat} A_{ij})\)) is omitted unless explicitly coupling to a gauge sector—use it for gauge-coupled simulations or U(1) symmetry studies (Gauge Symmetries §2; see Mathematical Foundations §4 for the coupled form). Default runs drop it to focus on pure resonance/phase effects.
Ranges: With \(\Delta\phi=0\) and open gate (analytic \(s_i s_j=+1\), anti-aligned \(\pm i\)), \(A_{ij}^{\max}=3/2\), \(\mathcal R_{ij}^{\max}=3\). With aligned spins (gate closed), \(\mathcal R_{ij}=0\). Thus \(\mathcal R_{ij} \in [0,3]\).
Gaussian suppression note: Dropping \(e^{-(\Delta\omega/\Delta\omega^*)^{2}}\) in \(A_{ij}\) changes stiffness by \(\leq 1\%\) for \(|\Delta\omega| \lesssim 0.1 \Delta\omega^*\), and 5–10% at \(|\Delta\omega| \sim 0.3 \Delta\omega^*\) (see simulation summaries, Mathematical Foundations §11).
Spin-gate truth table
| Convention | Spin Pair | Gate Value |
|---|---|---|
| Analytic \(s = \pm i\) | (+i, −i) | \(1+s_i s_j=2\) (open) |
| Analytic \(s = \pm i\) | (+i, +i) or (−i, −i) | \(1+s_i s_j=0\) (closed) |
| Code \(\sigma_i = \pm 1\) | (+1, −1) or (−1, +1) | \(1-\sigma_i\sigma_j=2\) (open) |
| Code \(\sigma_i = \pm 1\) | (+1, +1) or (−1, −1) | \(1-\sigma_i\sigma_j=0\) (closed) |
Open for anti-aligned analytic spins \((+i,-i)\) or opposite code spins \((+1,-1)\) or \((-1,+1)\); matches Core Principles §2.
3 Field identification (compact phase + Z₂ gate)
Introduce the composite link \(\mathcal U_{ij} = G_{ij} U_{ij}\) with \(U_{ij} = e^{i(\phi_i-\phi_j)}\) and \(G_{ij} = (1-\sigma_i\sigma_j)/2 \in \{0,1\}\) (open/closed gate). Small-angle expansion gives \(|1-U_{ij}|^2 \approx (\Delta\phi_{ij})^2\). The bond term expands as
\[ J\,\mathcal R_{ij} = J\,A_{ij}(1+s_i s_j) \Rightarrow J\left[3-\frac{3}{4}(\Delta\phi_{ij})^2\right]\,G_{ij} \quad (\text{to } \mathcal{O}(\Delta\phi^2), \text{ open gate}) \]
Up to an additive constant, this yields a phase-stiffness term \(\frac{3}{4}J\,G_{ij}(\Delta\phi_{ij})^2\). On a hypercubic lattice (undirected links counted once),
\[ \sum_{\langle ij\rangle} G_{ij}(\Delta\phi_{ij})^2 \longrightarrow a_{\rm lat}^{2-d} \int d^d x \, G(x) \, |\nabla\phi|^2 \]
Thus, \[\rho_s = \frac{3}{2} J a_{\rm lat}^{2-d}\] In \(d=3\), \(\rho_s \propto a_{\rm lat}^{-1}\), so halving \(a_{\rm lat}\) doubles the stiffness (RTG Gravity I §4.2).
Sign convention: The lattice bond energy uses a \(+J\) prefactor, so the small-angle quadratic piece has a negative sign relative to \(\Delta\phi=0\). Phase stiffness is defined from positive curvature (energy increase with angle). For a binding convention, use \(-J \mathcal{R}_{ij}\) with \(J > 0\); \(\rho_s\) is unchanged (Gauge Symmetries §6).
Open-gate fraction: For fluctuating gates, define a coarse-grained gate field \(0 \leq G(x) \leq 1\), so \(\rho_{s,\text{eff}}(x) = G(x) \rho_s\). For homogeneous statistics, \(\rho_{s,\text{eff}} = \langle G \rangle \rho_s\) (RTG Gravity I §4.2).
Sanity check: For \(\phi(\mathbf x) = \mathbf q \cdot \mathbf x\), \(\sum_{\langle ij\rangle} (\Delta\phi)^2 = a_{\rm lat}^{2-d} V |\mathbf q|^2\), so lattice quadratic energy \(\frac{3}{4}J \sum (\Delta\phi)^2\) matches continuum \(\frac{\rho_s}{2} \int |\nabla\phi|^2\) iff \(\rho_s = \frac{3}{2} J a_{\rm lat}^{2-d}\).
4 Frequency scale vs. lattice spacing
Define the spectral length \(\ell_* \equiv c/\Delta\omega^* \approx 2.07 \pm 0.11 \, \mathrm{fm}\) and simulation lattice spacing \(a_{\rm lat} = 0.08 \, \mathrm{fm}\). These are distinct. The long-wavelength beat length is \(r_* = 2\pi \ell_* \approx 13.0 \pm 0.7 \, \mathrm{fm}\). Proton radius \(r_p = 0.84 \pm 0.01 \, \mathrm{fm}\) is recovered by retuning \(J\) by +8% at fixed \(\Delta\omega^*\) (Mathematical Foundations §9). The 1.55–1.70 window in \(\Delta\omega/\Delta\omega^*\) hosts U(1)\(^2\) phase shells (Gauge Symmetries §4).
5 Coupling normalization and mapping
For RG, use dimensionless couplings at scale \(\Lambda\): \(\tilde J(\Lambda) = J(\Lambda)/(\hbar \Lambda)\), \(\tilde K'(\Lambda) = K'(\Lambda)/(\hbar \Lambda)\), \(g(\Lambda) = \frac{\bar J(\Lambda)}{K'(\Lambda)}\), \(\bar J \equiv \frac{3}{2}J\). Matches Two-Loop RG Derivation v1.3.3 (\(g^* \approx 1.14\)). Do not mix \(\rho_s\) with \(g\).
Phase stiffness: \(\rho_s = \frac{3}{2} J a_{\rm lat}^{2-d}\) (§3).
6 Continuum action (compact phase with Z₂ gate)
A continuum proxy is \[ S = \int d^{d}x \left[ \frac{\rho_s}{2}\,G(x)\,|\nabla\phi|^{2} + \kappa\,(1-G(x)) \right] + K’_{\rm TV} \int d^{d}x \,\frac{\|\nabla\omega(x)\|_1}{\Delta\omega^*} + \int d^{d}x \,\frac{J_{\rm ex}}{a_{\rm lat}^{d}} \,\sin(\nabla\phi)\, \exp\!\left[-\left(\frac{\nabla\omega}{\sigma_{\rm exch}}\right)^{2}\right], \] where \(\rho_s = \frac{3}{2} J a_{\rm lat}^{2-d}\), \(\kappa\) (MeV·fm\(^{-3}\)) penalises closed gates (RTG Gravity I §4.2), and \(K’_{\rm TV} = K_{\rm lat} a_{\rm lat}^{1-d}\) (MeV·fm\(^{-2}\) in \(d=3\)). The last term is a continuum surrogate for the lattice \(J_{\rm ex}\) contribution.
Continuum coefficient for \(K’\) (TV form): Using \(\sum_{\langle ij\rangle} \mapsto a_{\rm lat}^{-d} \int d^d x \sum_{\mu=1}^d\), \(\Delta\omega_{i,i+\hat\mu} \approx a_{\rm lat} \partial_\mu \omega\), \[ \sum_{\langle ij\rangle} \frac{|\Delta\omega_{ij}|}{\Delta\omega^*} \longrightarrow a_{\rm lat}^{1-d} \int d^d x \,\frac{\|\nabla\omega(x)\|_1}{\Delta\omega^*}. \] Thus, \(\int K’_{\rm TV} \|\nabla\omega\|_1 \Delta\omega^{*-1} d^3 x\) has energy units.
L2 option: Replace \(\|\cdot\|_1\) by \(\|\cdot\|_2\) with an \(O(1)\) calibration factor for smooth fields.
Numerical check: For \(\omega(\mathbf x) = \mathbf k \cdot \mathbf x\), \(\sum_{\langle ij\rangle} |\Delta\omega_{ij}| = a_{\rm lat}^{1-d} V \|\mathbf k\|_1\), matching continuum iff \(K’_{\rm TV} = K_{\rm lat} a_{\rm lat}^{1-d}\).
Units: In \(d=3\), \(\rho_s\): MeV·fm\(^{-1}\); \(\kappa\): MeV·fm\(^{-3}\); \(K’_{\rm TV}\): MeV·fm\(^{-2}\).
7 Numerical example
Phase stiffness: With \(a_{\rm lat} = 0.08 \, \mathrm{fm}\), \(d=3\), \(J_{\rm lat} = 0.85 \, \text{MeV}\): \(\rho_s = \frac{3}{2} J_{\rm lat} a_{\rm lat}^{2-d} = \frac{3}{2} \times 0.85 \times (0.08)^{-1} \, \text{MeV·fm}^{-1} \approx 15.9 \, \text{MeV·fm}^{-1}\). Note: \(J_{\rm lat}\) is a lattice coupling; continuum \(J = 3.24 \, \text{MeV}\) (Mathematical Foundations §10) is defined after continuum mapping (e.g., \(J \propto J_{\rm lat}\,a_{\rm lat}^{1-d}\)).
RG coupling: For \(K’ = 12.0 \, \text{MeV}\), \(J = 3.24 \, \text{MeV}\), \(\bar J = \frac{3}{2} J = 4.86 \, \text{MeV}\), \(g = \bar J / K’ \approx 0.405\) (dimensionless, Two-Loop RG Derivation §5).
8 Uncertainty budget
| Source | \(\Delta g\) impact (relative) |
|---|---|
| Lattice spacing \(a_{\rm lat}\) (±3 %) | ±3 % |
| \(J_{\rm lat}\) statistical error (1 %) | ±1 % |
| Phase-expansion truncation | ≤ 2 % up to \(|\Delta\phi|=\pi/4\) |
| Two-loop truncation beyond sunset | ±1 % |
| RG scheme spread | ±1 % (Two-Loop RG Derivation §5) |
| RG fixed point | ±2 % from \(g^* \pm 0.02\) |
9 Conclusion & cross-link
This workbook matches Two-Loop RG Derivation v1.3.3, using the compact-phase + Z₂-gate continuum. The \(J\)-derived phase stiffness is separate from the \(K’\)-controlled frequency-gap term, with \(K’_{\rm TV} = K_{\rm lat} a_{\rm lat}^{1-d}\). The U(1)\(^2\) window (1.55–1.70) relates to shell rigidity (Enriched Geometric Concepts §6).
Scales: \(r_* = 2\pi c / \Delta\omega^* \approx 13.0 \pm 0.7 \, \mathrm{fm}\) is a long-wavelength beat length, not a hadronic size. Proton fits retune \(J\) by +8% at fixed \(\Delta\omega^*\) to recover \(r_p = 0.84 \pm 0.01 \, \mathrm{fm}\) (Mathematical Foundations §9).
Related work
RTG Gravity I | RTG Gravity II | RTG Glossary | Core Principles | Mathematical Foundations | Enriched Geometric Concepts | Gauge Symmetries | Two-Loop RG Derivation | Quantum Behaviours | Thermodynamics
Change Log
| Version | Date (UTC) | Main updates |
|---|---|---|
| 1.3.3 | 2025-08-12 | Added preamble with 3+1D context; moved notation guard to §1; clarified gauge term omission and typical use-cases; updated Core Principles reference; added proton radius uncertainty; defined exchange term in continuum action; clarified \(J_{\rm lat}\) vs. \(J\) conversion; removed redundant notation in §8; aligned with Two-Loop RG v1.3.3. |
| 1.3.2 | 2025-08-12 | Added \(\sigma_{\rm exch}\) vs \(\sigma_{\rm noise}\) distinction; inserted U(1)\(^2\) window cross-links; added gauge-term omission note; clarified units for \(K’_{\rm TV}\); noted \(g\) dimensionless; added related-work section; compact-uncertainty notation was referenced to RG note. |
| 1.3.1 | 2025-08-12 | Defined \(\mathcal R_{ij}\); added gate-table caption; extended notation guard; fixed \(\rho_s\) prefactor; added sign-convention, open-gate fraction, plane-wave sanity-check; renamed \(\sigma\) to \(\sigma_{\rm exch}\); clarified Gaussian suppression; rewrote \(K’\) mapping; updated numerical example; minor edits. |
| 1.3 | 2025-08-08 | Rewritten continuum mapping; corrected \(\rho_s\) exponent; removed z-factor; separated spectral length; clarified \(K’\) mapping; qualified Gaussian suppression; added exchange-term role; added RG uncertainties; removed proton \(\delta\omega\) claim. |