Version: Unified v1.1 | Date: 14 Aug 2025
Editors: Mustafa Aksu · Grok · ChatGPT
Contents
- 1 · Notation & Units
- 2 · Fundamental Entities & Observables
- 3 · Core Functions & Equations
- 4 · Emergent Geometry & Dimensionality
- 5 · Gauge Sectors (U(1), soft‑SU(2), U(1)^2)
- 6 · Gravity: Emergent Metric → Einstein–Hilbert (tree‑level)
- 7 · Renormalization Group & Critical Bandwidth
- 8 · Lattice → Continuum Mapping
- 9 · Simulation Protocols & Benchmarks
- 10 · Thermal & Statistical Constructs
- 11 · Cosmology & Large‑Scale (hypotheses)
- 12 · Constants & Default Parameters
- 13 · Operational Rules & Best Practices
- 14 · Open Questions & Outlook
- 15 · Equation Index
1 · Notation & Units
All \( \omega,\Delta\omega,\delta\omega \) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \( f=\omega/(2\pi) \). Code spins \( \sigma_i=\pm 1 \) map to analytic spins by \( s_i\equiv i\,\sigma_i \) with \( s_i\in\{+i,-i\} \).
- Binary gate (analytic): \( G_{ij}=\tfrac{1+s_i s_j}{2}\in\{0,1\} \). Binary gate (code): \( G_{ij}=\tfrac{1-\sigma_i\sigma_j}{2} \). (G1)
- Critical bandwidth: \( \Delta\omega^\ast = 1.45(8)\times 10^{23}\,\mathrm{s}^{-1} \) (angular).
- Lattice U(1): \( B_{ij}=a_{\rm lat}\,\mathcal A_{ij} \) (dimensionless phase); complex link \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \).
- Coupling units: \( J,J_{\rm ex} \) in MeV; \( \kappa_B \) (U(1) stiffness) in MeV; \( \kappa_c \) (Helfrich curvature) in MeV·fm.
- Kernel roles (taxonomy): resonance kernel (phase alignment), exchange term (dynamic phase offset), curvature kernel (geometric stability).
- Two RG scales: \( \mu_{\text{freq}} \) (s\(^{-1}\), for \( \beta_g \)); \( \mu_\ell \) (fm\(^{-1}\), for gravitational running \( g_N(\mu_\ell)=\mu_\ell^2 G(\mu_\ell) \)).
2 · Fundamental Entities & Observables
- Node: stores \( (\omega_i,\phi_i,s_i) \). No single‑node observables. All measurements are relational.
- Resonance kernel (bounded, spin‑gated):
\( \mathcal R_{ij}=A_{ij}(1+s_i s_j) \), where
\( A_{ij}=\tfrac{3}{4}\!\left[1+\cos(\phi_i-\phi_j)\right]\exp\!\left[-(\Delta\omega_{ij}/\Delta\omega^\ast)^2\right] \), hence \( 0\le \mathcal R_{ij}\le 3 \). - Observer‑relative distance (beat distance):
\( r_{ij}=\dfrac{2\pi c}{|\omega_i-\omega_j|} \), valid when phase decoherence is weak. - Intrinsic time (operational):
\( t_i=\tilde\phi_i/\omega_i \) has no standalone meaning; it becomes observable only under phase‑locking with a reference node (i.e., via \( \Delta\tau\approx \Delta\phi/\Delta\omega \)).
3 · Core Functions & Equations
- Gate truth (binary). Analytic: open if \( (s_i,s_j)=(+i,-i) \Rightarrow 1+s_is_j=2 \); closed if aligned \( \Rightarrow 0 \). Code: open if \( \sigma_i\neq\sigma_j \Rightarrow 1-\sigma_i\sigma_j=2 \); closed otherwise. (G1)
- Bond energy (ungauged & gauged) — separate channels.
Symmetric kernel channel:
\( E^{(\text{res})}_{ij}=J\,\mathcal R_{ij} \).
Frequency‑gap TV channel:
\( E^{(\text{gap})}_{ij}=K’\,\dfrac{|\Delta\omega_{ij}|}{\Delta\omega^\ast} \).
Exchange (dynamic phase) channel:
\( E^{(\text{ex})}_{ij}=J_{\rm ex}\,\sin(\Delta\phi_{ij})\,\exp\!\left[-(\Delta\omega_{ij}/\sigma_{\mathrm{exch}})^2\right] \).
XY gauged option (choose one convention per run):
\( E^{(\text{ex,XY})}_{ij}=-J_{\rm ex}\cos(\Delta\phi_{ij}-B_{ij})\,\exp\!\left[-(\Delta\omega_{ij}/\sigma_{\mathrm{exch}})^2\right] \).
Gauged resonance form: in U(1) runs, apply \( \Delta\phi_{ij}\to \Delta\phi_{ij}-B_{ij} \) to all phase‑dependent terms (e.g., in \( A_{ij} \) and exchange). Note: the exchange term captures dynamic phase shifts, whereas \( J\,\mathcal R_{ij} \) is a symmetric resonance contribution; they govern distinct channels. (B1) - Mass estimator (coarse, relational) — usage conditions.
\( m_i=\big[\hbar\omega_i-\sum_j E^{\rm res}_{ij}\big]/c^2 \), with \( E^{\rm res}_{ij}=\hbar\,|\Delta\omega_{ij}|\,\mathcal R_{ij} \).
Assumptions: valid under local phase‑locking and spin‑gated summation (use only open gates); clamp \( m_i\ge 0 \) to avoid coarse‑graining artifacts. (M1) - Photon object: a (+i,−i) spin pair with shared phase, carrier \( \omega_\gamma\neq 0 \), internal \( \Delta\omega=0 \), energy \( E=\hbar\omega_\gamma \).
- Inter‑plane coupling proxy:
\( C_{12}(\alpha)=r_{12}^{-\alpha}\,\mathcal R_{12} \), typical \( \alpha=1 \) (propagation) or \( \alpha=2 \) (metric‑weighted). - Whirling frequency: \( \Omega^2=|\nabla\phi|^2 \) (coarse‑grained local precession scale).
4 · Emergent Geometry & Dimensionality
Structure tensor → metric. On a coarse neighbourhood \( \mathcal N_x \), build
\( \mathbf S_{\mu\nu}(x)=\dfrac{\sum_{\langle ij\rangle\in\mathcal N_x} w_{ij}\,\Delta x_{ij,\mu}\Delta x_{ij,\nu}}{\sum_{\langle ij\rangle\in\mathcal N_x} w_{ij}} \),
with weights \( w_{ij}=\mathcal R_{ij} \) (preferred; incorporates gates) or \( A_{ij} \) (open‑gate dominated regions).
Diagonalize \( \mathbf S\to\{(\lambda_a,\mathbf v^{(a)})\} \), define a tetrad \( e^a{}_\mu \) from the normalized \( \mathbf v^{(a)} \) (ensure \( \det e>0 \) for orientation), select the timelike axis along local phase flow, and set signature via \( \eta_{ab}=\mathrm{diag}(-,+,+,+) \).
Fix a conformal factor \( \Omega(x) \) by matching long‑wavelength phase‑mode speed (or a Laplace–Beltrami fit to the graph Laplacian).
The emergent metric is \( g_{\mu\nu}=\Omega^2\,\eta_{ab}\,e^a{}_\mu e^b{}_\nu \). (Gμν)
Curvature penalty (Helfrich‑style):
\( U_{\rm curv}=\kappa_c\,a_{\rm lat}^2\sum_{\langle ijk\rangle}\left(1-\dfrac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\right)^2 \),
with normalization by 9 from three bonds each bounded by 3. (C1)
Dimensional ladder (by \( \delta\omega/\Delta\omega^\ast \) with ±0.02 systematics).
\( \delta\omega/\Delta\omega^\ast=\begin{cases}
0\text{–}0.28 &\Rightarrow D=2 \ \text{(planar sheets)}\\
0.28\text{–}0.70 &\Rightarrow D=3 \ \text{(stable shells)}\\
0.70\text{–}1.70 &\Rightarrow D=4 \ \text{(propagation corridors)}\\
>1.70 &\Rightarrow D\ge 5 \ \text{(high‑D anomalies)}
\end{cases} \)
with a transitional \( \mathrm{U}(1)^2 \) band at \( 1.55\text{–}1.70 \). The triple‑shell constraint \( \theta_1+\theta_2+\theta_3=\mathrm{const} \) leaves two independent modes, realizing the \( \mathfrak{u}(1)^2 \) Lie algebra and approximating SU(3) as the shell‑asymmetry parameter \( \epsilon\to 0 \).
5 · Gauge Sectors (U(1), soft‑SU(2), U(1)^2)
- U(1) from local rephasing (0–0.28). Lattice link \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \); continuum \( A_\mu \) with \( F_{\mu\nu} \) and Maxwell term. Use either \( +J_{\rm ex}\sin(\Delta\phi-B_{ij}) \) or XY \( -J_{\rm ex}\cos(\Delta\phi-B_{ij}) \) consistently within a run.
- Soft‑spin SU(2) (0.28–0.70). Binary spins gate links; SU(2) appears only after coarse‑graining to soft doublets \( \Psi \), producing Heisenberg‑like exchange. Bare \( 1+s_is_j \) is not SU(2)‑invariant.
- U(1)\(^2\) band (1.55–1.70). Three coherent shells with potential
\( V_{\rm shell}=J_{\rm ex}\sum_{a - Anomalies (conditions & checks). SU(2): Witten anomaly unless the number of left‑handed doublets per coarse cell is even. SU(3): safe if vectorlike or if chiral anomaly sums vanish. Chiral U(1): ensure \( \sum q_i=0 \) and \( \sum q_i^3=0 \). Verification: compute the axial Ward‑identity residual vs \( a_{\rm lat} \) and extrapolate to the continuum (publish slope plot).
6 · Gravity: Emergent Metric → Einstein–Hilbert (tree‑level)
With \( g_{\mu\nu} \) from §4, the action is
\( S_{\rm grav}=\dfrac{1}{16\pi G_0}\int d^4x\,\sqrt{-g}\,(R-2\Lambda_0) \),
defining tree‑level \( G_0,\Lambda_0 \) (matching constants).
Matter (compact phase + gate) at leading order:
\( S_{\phi,G}=\int d^4x\,\sqrt{-g}\left[\tfrac{\rho_s}{2}\,G\,g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi+\kappa(1-G)\right] \).
If U(1) is promoted locally, replace \( \partial_\mu\phi \to D_\mu\phi=\partial_\mu\phi – iqA_\mu \) in the entire gated kinetic term.
Variations yield Einstein’s equations \( G_{\mu\nu}+\Lambda_0 g_{\mu\nu}=8\pi G_0 T_{\mu\nu} \), Maxwell \( \nabla_\mu F^{\mu\nu}=J^\nu \), and the phase equation \( \nabla_\mu(G\,\rho_s\,\nabla^\mu\phi)=0 \) (or its gauged form). Running of \( G,\Lambda \) is covered in §7 (Gravity II scope).
7 · Renormalization Group & Critical Bandwidth
Define \( \tilde J=J/(\hbar\Delta\omega^\ast) \), \( \tilde K=K’/(\hbar\Delta\omega^\ast) \), and
\( g=\bar J/K’ \) with \( \bar J=\tfrac{3}{2}J \) (the \( \tfrac{3}{2} \) arises from the small‑angle amplitude prefactor of \( A_{ij} \)). A representative two‑loop truncation gives
\( \beta_g(g)\approx 0.72\,g-0.63\,g^2-0.011\,g^3 \) with fixed point \( g^\*\approx 1.14 \) (scheme‑dependent), setting
\( \Delta\omega^\ast = 1.45(8)\times 10^{23}\,\mathrm{s}^{-1} \) and anchoring dimensional windows (§4/§5).
Gravitational running (coarse‑grained): use \( \mu_\ell \) and \( g_N(\mu_\ell)=\mu_\ell^2 G(\mu_\ell) \) with
\( \beta_{g_N}\approx 2 g_N + B_1 g_N^2 + \cdots \); extract \( B_1 \) empirically from two‑scale fits. Quote a ±0.02 systematic on window edges propagated from scheme dependence of \( g^\* \).
8 · Lattice → Continuum Mapping
- Phase stiffness: \( \rho_s=\tfrac{3}{2}J\,a_{\rm lat}^{\,2-d} \) (in \( d=3 \): \( \rho_s=\tfrac{3}{2}J\,a_{\rm lat}^{-1} \)).
- Frequency‑gradient TV coefficient: \( K’_{\rm TV}=K_{\rm lat}\,a_{\rm lat}^{\,1-d} \) (MeV·fm\(^{-2}\) in \( d=3 \)); it sets the penalty scale for \( \|\nabla\omega\| \) in continuum proxies.
- Keep \( \rho_s \) (phase), \( K’_{\rm TV} \) (frequency gradients), and \( g=\bar J/K’ \) (RG) separate; do not mix roles.
9 · Simulation Protocols & Benchmarks
- CHSH diagnostic (kernel‑aware): aligned phases, open gates, optimal angles \( (0,\tfrac{\pi}{2},\tfrac{\pi}{4},-\tfrac{\pi}{4}) \); with OU noise on \( \Delta\omega \) and \( \sigma\equiv \mathrm{sd}[\Delta\omega]/\Delta\omega^\ast \),
\( S(\sigma)=2\sqrt{2}\,e^{-\sigma^2} \); violation threshold \( \sigma_{\rm crit}\approx 0.589 \); ideal \( S\simeq 2.827 \). Physical role: \( S>2 \) confirms non‑classical correlations and helps validate quantum‑consistent kernel regimes. - Energy conservation: micro‑canonical MD drift \( \lesssim 4.3\times 10^{-4} \) over \(\sim 3000\) ticks when flip energy is routed to \( \pi_\phi \) (kinetic compensation).
- Flip rate: typical \( 0.02\text{–}0.03 \) per 100 ticks (extendable to \( \le 0.30 \) with curvature thermostat).
- Window diagnostics: median simplex volumes (Cayley–Menger) vs \( \delta\omega/\Delta\omega^\ast \); finite‑N knees can sit \(\sim 0.2\) high, full lattices recover edges within \( \pm 0.02 \).
- Logging (minimum set): \( \Delta\phi \), \( \Delta\omega/\Delta\omega^\ast \), \( \mathcal R_{12}/3 \), four correlation energies \( E(\cdot,\cdot) \), CHSH \( S \), angle set, OU \( \tau \) (if used), flip rate, and long‑time drift.
Condensed reference chart.
| Observable | Symbol | Diagnostic Range | Trigger / Use |
|---|---|---|---|
| CHSH | \( S \) | \( 2 | Entanglement / non‑classicality check |
| Flip rate | — | 0.02–0.03 / 100 ticks | Thermal equilibrium window |
| Decoherence cliff | \( \sigma_{\rm crit} \) | \( \sim 0.589 \) | Phase decorrelation onset |
| Energy drift | \( \Delta H/H \) | \( <4.3\times 10^{-4} \) | Microcanonical stability |
10 · Thermal & Statistical Constructs
- Relational temperature: based on frequency variance over bonds; e.g., \( T_{\rm rel}\propto \mathrm{Var}[\Delta\omega_{ij}] \) under a fixed coarse‑graining protocol.
- Spectral temperature: from linewidths of resonant modes.
- Entropy: log‑count of stable configurations under fixed constraints (gate statistics, couplings, windows).
- Heat‑capacity anomaly: peak near the decoherence cliff (\(\sim 0.70\,\Delta\omega^\ast\)).
- Transport: phase‑slip (Fourier mode) conduction for long coherence lengths.
11 · Cosmology & Large‑Scale (hypotheses)
Exploratory interpretations (require dedicated validation within RTG):
- Dark matter: effective mass/curvature from gated but non‑propagating resonance networks.
- Dark energy: tension of the global resonance fabric.
- CMB: early‑universe phase‑locking ripples in the phase sector.
- Large‑scale curvature: induced by mass‑rich cluster bending of resonance lines.
- Dimensional evolution: sub‑3D microstructure smoothing to 3D macroscopically.
- Resonance memory: time‑asymmetry from configuration‑history bias.
12 · Constants & Default Parameters
| Quantity | Symbol | Value |
|---|---|---|
| Speed of light | \( c \) | \( 2.998\times 10^8\,\mathrm{m/s} \) |
| Planck constant | \( \hbar \) | \( 1.055\times 10^{-34}\,\mathrm{J\,s} \) |
| Boltzmann constant | \( k_B \) | \( 1.381\times 10^{-23}\,\mathrm{J/K} \) |
| Critical bandwidth | \( \Delta\omega^\ast \) | \( 1.45(8)\times 10^{23}\,\mathrm{s}^{-1} \) |
| RG fixed point | \( g^\* \) | \( \approx 1.14 \) (scheme ±0.02) |
| Proton radius (fit target) | \( r_p \) | \( 0.84 \pm 0.01 \,\mathrm{fm} \) |
| Example couplings | \( K’,J,J_{\rm ex} \) | \( 12.0,\,3.24,\,2.20 \) MeV (illustrative) |
13 · Operational Rules & Best Practices
- No single‑node observables; always use relational definitions.
- Keep gate logic binary and consistent with \( s_i\equiv i\,\sigma_i \).
- Use the bounded, spin‑gated kernel \( \mathcal R_{ij} \); avoid ad‑hoc rescalings that break bounds or gates.
- Separate roles: phase (\( \rho_s \)), frequency gradients (\( K’_{\rm TV} \)), RG (\( g=\bar J/K’ \)). Do not mix.
- Declare \( \mu \) convention: \( \mu_{\text{freq}} \) for \( \beta_g \) (s\(^{-1}\)), \( \mu_\ell \) for gravitational \( g_N \) (fm\(^{-1}\)).
- When promoting U(1) locally, gauge all phase‑dependent pieces consistently (links or covariant derivatives).
- Quote symmetry windows with the \( \pm 0.02 \) systematic; avoid \( x\ge 1.70 \) unless studying high‑D anomalies.
- For MD stability, include Helfrich \( U_{\rm curv} \) and track flip‑rate and drift metrics.
14 · Open Questions & Outlook
- Quantization of \( \pm i \) spins (path integral, Grassmann structure) and relation to soft‑SU(2).
- Calibrating \( \kappa_c \) to stabilize mixed‑D domains without suppressing 4‑D corridors.
- Shell‑mixing dynamics in the \( \mathrm{U}(1)^2 \) band and effective SU(3) behavior as \( \epsilon\to 0 \).
- Running \( G(\mu_\ell),\Lambda(\mu_\ell) \): fixed‑point hints vs Wilsonian flows.
- Experimental analogs: dual‑frequency optical cavities probing dilation onset near \( 0.28\,\Delta\omega^\ast \approx 4.06\times 10^{22}\,\mathrm{s}^{-1} \).
15 · Equation Index
- (G1) Gate rule: \( G_{ij}=\tfrac{1+s_is_j}{2} \) (analytic) / \( G_{ij}=\tfrac{1-\sigma_i\sigma_j}{2} \) (code)
- (B1) Bond energy channels: \( E^{(\text{res})}_{ij}, E^{(\text{gap})}_{ij}, E^{(\text{ex})}_{ij} \) (+ gauged XY form)
- (M1) Mass estimator: \( m_i=\big[\hbar\omega_i-\sum_j \hbar|\Delta\omega_{ij}|\mathcal R_{ij}\big]/c^2 \) (gated, phase‑locked)
- (C1) Curvature penalty: \( U_{\rm curv}=\kappa_c a_{\rm lat}^2\sum_{\langle ijk\rangle}\big(1-\tfrac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\big)^2 \)
- (Gμν) Emergent metric: \( g_{\mu\nu}=\Omega^2\,\eta_{ab}\,e^a{}_\mu e^b{}_\nu \)