Basic Axioms (v0.9)

Axioms

Basic Axioms (v0.9)

Qaether 2025. 6. 8. 14:32

A1. Fundamental Reality: Qaether and Void

Qaether is the smallest unit-cell of space that composes the universe. (Quantum Aether)
- Placed as spherical cells of Planck-scale radius $l_p$ on the lattice sites of an FCC lattice.
- Each cell can bond in up to 12 directions to other cells; bonding both releases energy and provides the condition for space to emerge.
Qaether sphere surface area
- If the radius of a Qaether sphere is $r_p = l_p$, then the total surface area of a single cell is $V_s := 4\pi\,r_p^2 = 4\pi\,l_p^2$.
Void: Non-space boundary condition
- A Void is not a physical substance but a mathematical boundary condition that defines the limit of the region in which the Qaether system can exist.
- Qaether inherently holds phase energy. This energy drives expansion, but since expansion beyond the Void is impossible, Qaether’s expansion energy is converted into internal stress or outward pressure, producing an effect equivalent to 100% reflection at the boundary.
- In other words, the Void does not exert a force; it merely blocks Qaether’s expansion.

A2. FCC Lattice Structure

Assume that Qaether is packed in a Face-Centered Cubic lattice structure.
Therefore, each Qaether can bond with up to 12 nearest neighbors, giving it the 12 unit vectors of an FCC lattice.
Why the FCC lattice structure is chosen:
- Minimum-energy arrangement
  - Since Qaether is modeled as a spherical cell of Planck-radius scale, the phase-interaction potential grows stronger as cells get closer.
  - In an FCC arrangement, each cell has twelve nearest-neighbor bond vectors, all separated by angles of 60° or 90°, so the phase-difference potential is distributed uniformly.
  - As a result, the total phase-potential energy of the lattice is minimized, making it energetically very stable.
- Restoration of isotropy
  - In the long-wavelength approximation (considering dynamics or wave-propagation speed), FCC is a discrete lattice microscopically but best restores isotropy at long distances. $$\lim_{\lambda \gg l_p} \to \text{Lorentz effective symmetry}$$
  - This means that physical quantities (propagation speed, spin-interaction energy, etc.) do not vary with direction; the bulk energy distribution is uniform.

A3. Mathematical Definition of Qaether

Each Qaether $i$ is defined by the state vector \[Q_i = \bigl(\phi_i,\;\{\hat{b}_{ij}\},\;k_i,\;\hat{z}_i\bigr)\]

Phase $\phi_i$
- A periodic cyclic variable representing the internal oscillation state of each Qaether cell, as if each cell carries a phase $\phi_i$.
- Discrete quantization: the phase difference $\Delta\phi_{ij} = \phi_i - \phi_j$ is quantized in units of $\pi/3$.
- Physical mediator: quantization of phase differences underlies electromagnetism, color charge, spin, and topological defects and serves as a relational time standard.
- Discrete phase derivatives: \[\ddot\phi_i = \frac{\phi_i^{N+1}-2\phi_i^N+\phi_i^{N-1}}{t_p^2}, \quad \dot\phi_i = \frac{\phi_i^N-\phi_i^{N-1}}{t_p}\]
  - Here $t_p$ is the Planck time. One can define an effective Qaether time $t_q$ which under synchronization becomes equal to $t_p$.
Set of bond-direction vectors $\{\hat{b}_{ij}\}$
- Bond-vector sum: $B_i = \sum_{j\in\mathbb{N}(i)} \hat{b}_{ij}$.
- Number of bonds $m_i = |\{\hat{b}_{ij}\}|$, with $0 \le m_i \le 12$.
- From this, one can define the effective bonding pressure $P_i(m_i)$.
Frequency constant $k_i$
- Each frequency $\Omega_i(k_i) = \tfrac{2\pi c}{k_i\,l_p}$.
- This follows from the condition that the minimum wavelength is $l_p$ and increases in integer multiples $k_i$.
Spin axis $\hat{z}_i$
- Chosen perpendicular to maximize bonding stability and degrees of freedom: \[\hat{z}_i = \frac{D_i \times B_i}{\|D_i \times B_i\|} \quad\bigl(\text{if }D_i \times B_i=0,\ \hat{z}_i=\tfrac{D_i}{\|D_i\|}\bigr), \quad D_i = \sum_{j\in\mathbb{N}(i)} \Delta\phi_{ij},\,\hat{b}_{ij}.\]

A4. Bonding Phase Conditions

\[
\Theta_i := \sum_{j\in\mathcal{N}(i)} \Delta\phi_{ij},
\quad \Theta_i = 2\pi\,n_i,\;n_i\in\{-1,0,1\}
\]
Here $\mathcal{N}(i)$ is the set of nearest-neighbor Qaether cells to cell $i$.

Phase quantization: $$\Delta\phi_{ij} \in \mathbb{Z}_6\cdot\tfrac{\pi}{3} = \{0,\pm\tfrac{\pi}{3},\pm\tfrac{2\pi}{3},\pm\pi\}$$
If $n_i=0$, there is no local phase mismatch (no charge).
If $n_i\neq0$, a local phase mismatch occurs (charge or topological defect).
Projection plane and ordering of bonds:
- The reference plane for ordering bond vectors is the plane orthogonal to the spin axis $\hat{z}_i$; if $\hat{z}_i = 0$, use the local (111) plane.
- Bond directions are taken clockwise as “+” and counterclockwise as “–"

A5. Bonding Patterns and Phase-Difference Constraints

Basic loop definitions
- Links that form loops must satisfy the phase-quantization condition of A4, but cases with $\Delta\phi=0$ and $\pi$ are predefined like below:
  - $\Delta\phi=0$ (link value 0): a perfectly synchronized, stable bond with no energy mismatch.
  - $\Delta\phi=\pi$ (link value 1/2): the circulating phase reverses, producing a standing wave at the bond point with no reflection; the energy packet can appear as a particle.
- Triangular loop ($\Delta$, Trianglet, $\ell_3$):
  - Consists of 3 closed links.
  - Phase-closure condition: $$\sum_{(a,b)\in\ell_3}\Delta\phi_{ab} = 2\pi\,n_{\Delta},\;n_{\Delta}\in\{-1,0,1\}$$
  - $n_{\Delta}$ is the triangular-loop index.
  - Patterns for $n_{\Delta}=0$ and $n_{\Delta}=\pm1$ correspond to specific phase-difference assignments.
- Quadrilateral loop($\Box$, Plaquette, $\ell_4$):
  - Consists of 4 closed links.
  - Phase-closure condition: $$\sum_{(a,b)\in\ell_4}\Delta\phi_{ab} = 4\pi\,n_{\Box},\;n_{\Box}\in\{-1,0,1\}$$
  - $n_{\Box}=\pm1$ ⇒ phase sum $\pm4\pi$ (spin-½ and color-charge patch); $n_{\Box}=0$ ⇒ no spin or color charge (locally flat).
  - For $n_{\Box}=\pm1$, one simultaneously assigns conjugate spin angular momentum $\pm\tfrac{\hbar}{2}$ and color-charge Cartan components $C^3=\pm\tfrac12$, $C^8=\pm\tfrac{1}{2\sqrt3}$ (see A9, A11).
“Surface-flux conservation” for closed polyhedra
- Flux conservation means that the sum of loop indices over all faces of a closed polyhedron must be zero. Shared edge phase differences cancel, leaving only the sum over external faces; a nonzero result would indicate a topological defect (monopole).
- Tiara (regular tetrahedron):
  - 4 trianglet faces $\ell_{3,1}\ldots\ell_{3,4}$, $$\sum_{k=1}^4 n_{\Delta_k}=0$$
  - Examples: (1,1,–1,–1), (1,–1,0,0), (0,0,0,0). Combinations like (1,1,1,1) are forbidden.
- Pyramid (square pyramid):
  - 4 trianglet faces $\ell_{3,1}\ldots\ell_{3,4}$ + 1 plaquette base $\ell_4$,
    $$\sum_{k=1}^4 n_{\Delta_k} + 2\,n_{\Box} = 0$$ When $n_{\Box}=\pm1$, $$\sum n_{\Delta} = -2\,n_{\Box}$$
  - Examples given for spin/color generation and flat cases.
- Diamond (regular octahedron):
  - 8 trianglet faces $\ell_{3,1}\ldots\ell_{3,8}$, $$\sum_{k=1}^8 n_{\Delta_k}=0$$
  - Examples: four faces +1 and four faces –1; or all zero.
Examples by pattern

Pattern	Flux Formulas	Allowed loops	Recommendation	Physical config.
Tiara (Regular tetrahedron)	$$\sum_{k=1}^4 n_{Δ_k}=0$$	• (1,1,–1,–1), (1,–1,0,0), (0,0,0,0)	• Most links $\Delta\phi=0,\pm\pi/3$ • $\pm2\pi/3·\pm\pi$ possible but high-energy	•No plaquette → No Spin·Color charge
Pyramid (Square pyramid)	$$\sum_{k=1}^4 n_{Δ_k} + 2\,n_{□}=0$$	• $n_{□}=1$→ $\sum n_{Δ}=-2$ ex: (–1, –1, 0, 0 ∥ □:+1) • $n_{□}=–1→ \sum n_{Δ}=+2$ 예: (1, 1, 0, 0 ∥ □:–1) • Flat: $n_{□}=0,\;\sum n_{Δ}=0$ ex: (1, –1, 1, –1 ∥ □:0)	• Bottom □ link: $\Delta\phi=\pm\pi$ (Patching Spin½·Color charge) • Trianglet face link: $0,\pm\pi/3$ (low-energy) • $\pm2\pi/3·\pm\pi$ possible but high-energy	•Quark Cell → plaquette •If $n_{□}=\pm1$, Emerging Spin ½ & Color charge
Diamond (Regular octahedron)	$$\sum_{k=1}^8 n_{Δ_k}=0$$	• (+1 × 4 times, –1 × 4 times) • (0,0,0,0,0,0,0,0) • ETC (+1 × n, –1 × n, remain 0) But $n_{+}=n_{–}$	• Most links $\Delta\phi=0,\pm\pi/3$ • $\pm2\pi/3·\pm\pi$ possible but high-energy	•No plaquette → No Spin·Color charge •Applicable as Balyon & Neutral Cell

A6. Definition of Effective Time and Synchronization

The smallest meaningful physical event ($\Delta\phi=\pi/3$) occurs over one Planck time $t_p$. $$\boxed{t_q = t_p}$$ Therefore $c_\phi = \tfrac{l_p}{t_p} = c$.

A7. Global Time Definition (Einstein Synchronization Protocol)

Phase-pulse send and reflect
- A reference cell $r$ sends a phase pulse of size $\Delta\phi=\pi/3$ to cell $i$, and $i$ immediately reflects it. The round-trip time is $t_{r\leftrightarrow i}$.
Synchronization offset
- Since the round-trip time is measured at finite speed $c_\phi = c$, define the time offset $\Delta t_i := \tfrac12\,t_{r\leftrightarrow i}$.
Assigning global time coordinate
- The global time at cell $i$ is $t_i := t_r + \Delta t_i$, where $t_r$ is the send time at cell $r$.
- Applying this over the lattice synchronizes global Qaether time causally.

A8. Effective Bonding (Boundary) Pressure of Qaether

Blocked area per bond
- When two spherical cells bond, their contact area is approximated as a constant $V_b$.
- Although the true shape is a spherical cap, for Planck-scale simplicity assume each bond blocks the same area $V_b$.
Unbonded boundary area
- If a cell has $m_i$ bonds, then $m_i V_b$ of its surface is blocked. The remaining unblocked area is
  $$V_i(m_i) = V_s - m_i\,V_b = 4\pi\,l_p^2 - m_i\,V_b$$
- Since $\alpha := \tfrac{V_b}{V_s} \, \text{and} \, V_s \gg 12 \, V_b,$ then $$\alpha \ll 1/12$$
Reflection-pressure model
- Define $p_0$ as the pressure experienced per unit area when a phase wave is 100% reflected. If the phase-wave energy density is $u_\phi$, then $p_0=2\,u_\phi$ (assuming speed $c$).
- $p_0$ may be treated as constant or vary with local $\phi$ distribution.
- The baseline boundary pressure on cell $i$ is proportional to the fraction of reflective area:
  \[
  P_i(m_i) := p_0\,\frac{V_i(m_i)}{V_s}
  = p_0\bigl(1 - \alpha\,m_i\bigr)
  \]
- Pressure Aanlysis:
  - Maximum: $m_i=0$ ⇒ $P_i(0)=p_0$
  - As $m_i$ increases, $P_i(m_i)$ decreases linearly, but remains like below $$p_0(1-12\alpha) \le P_i(m_i) \le p_0$$
In summary, the Void’s boundary effect gives Qaether a constant baseline pressure that locally curves space, creating effective curvature and thus a “baseline-mass condition” (see A9).
At Planck scale, $\alpha$ may be approximated to 0, so $P_i(m_i)=p_0$.

A9. Definition of Spin

Canonical angular momentum
- Each Qaether cell $i$ has canonical angular momentum
  \[L_i = I_i(m_i)\,\dot\phi_i, \quad \text{with} \,\, I_i(m_i)=I_0\bigl(1-\eta\,\tfrac{m_i}{12}\bigr)\]
- Dimension: $[L_i]=\hbar$
Partial spin
- Cell $i$ must traverse one plaquette (4 links) so that $\sum\Delta\phi_{ij}=4\pi$ in order to acquire spin-½.
- Each time $\Delta\phi_{ij}=\pm\pi$ occurs on a link, it generates a local partial angular momentum (“partial spin”):
  \[
  \Delta S_{ij} := \tfrac{\hbar}{8}\,\operatorname{sgn}(\Delta\phi_{ij})
  \quad(\Delta\phi_{ij}=\pm\pi),
  \]
  where $\tfrac{\hbar}{8}$ is chosen so that four such events sum to $\hbar/2$.
- Going around the plaquette accumulates total $\sum\Delta\phi=4\pi$, and
  $\sum_{\text{4 links}}\Delta S_{ij}=4\times\tfrac{\hbar}{8}=\tfrac{\hbar}{2}$, yielding spin-½.
- Label the four contributing partial spins as $\psi_i^{(k)}$ ($k=1,2,3,4$), corresponding to the four links.
Spin axis
- As defined in A3; choose the axis based on the plaquette orientation around cell $i$.
Construction of a four-component spinner
- Collect the four partial-spin modes into the column vector
  \[\Psi_i \equiv \begin{pmatrix}\psi_i^{(1)} \\ \psi_i^{(2)} \\ \psi_i^{(3)} \\ \psi_i^{(4)}\end{pmatrix} \quad \text{with} \,\, \psi_i^{(k)}=\pm\tfrac{\hbar}{8}\,\vec z_i\quad(\Delta\phi^k=\pm\pi)\]
Spin vector
- Only after all four components accumulate does one have
  $\sum_{k=1}^4\psi_i^{(k)} = \pm\tfrac{\hbar}{2}\,\hat z_i$, forming a complete spin-½.
Emergent Dirac-spinner structure (4×4 gamma-matrix correspondence)
- Divide the four components of $\Psi_i$ into left- and right-handed parts:
  - Left-handed: $(\psi_i^{(1)},\psi_i^{(2)})$
  - Right-handed: $(\psi_i^{(3)},\psi_i^{(4)})$
- In the continuum limit, $\Psi_i$ can transform under Lorentz via 4×4 gamma matrices $\gamma^\mu$, leading to the Dirac equation $$\bigl(i\gamma^\mu\partial_\mu - m\bigr)\Psi(x)=0$$
Phase closure on a plaquette loop
- For each plaquette, the below condition should be satisfied
  $$\sum_{k=1}^4\Delta\phi^k = 4\pi\,n_i,\ n_i\in\{-1,0,1\}$$
  so that $$\sum_{k=1}^4\psi_i^{(k)} = n_i\,\tfrac{\hbar}{2}\,\hat z_i$$
  - Only $n_i=\pm1$ yields half-integer spin.
  - $n_i=0$ is spinless.
Emergent minimal action
- Per single loop: $$\oint L_i\,d\phi_i = 2\pi\,\hbar_q \;\Longrightarrow\;L_{\min}=\tfrac{\hbar_q}{2}$$
Quantization of canonical angular momentum

$$L_i = n_i\,L_{\min} = n_i\,\tfrac{\hbar_q}{2}$$

A10. Definition of the Charge Operator

Charge is not on links but belongs to closed faces.
Trianglet and plaquette faces combine to form particles, so the charge of a face is the physical charge.
Simply put, fractional charges are assigned per link unit, and then the particle’s charge is defined as the sum of charges on closed faces. This reflects the physical intuition in the Qaether theory that faces close space and form particles.
Fractional charge assignment per link
- Trianglet bond link:
  \[
  q_{ij} = \frac{e}{3} \operatorname{sgn}(\Delta\phi_{ij}), \quad \operatorname{sgn}(x) = \begin{cases} +1 & (x>0) \\ 0 & (x=0) \\ -1 & (x<0) \end{cases}
  \]
- Plaquette bond link $(ij) \in \ell_4$:
  \[
  q_{ij} = \frac{e}{4} \operatorname{sgn}(\Delta\phi_{ij})
  \]
- Other links: $q_{ij} = 0$.
Charge of closed faces
- Trianglet face $\ell_3$ charge:
  \[
  Q_{\ell_3} = \sum_{(ij) \in \ell_3} q_{ij} \in \{-e, -\tfrac{e}{3}, 0, \tfrac{e}{3}, e\}
  \]
- Plaquette face $\ell_4$ charge:
  \[
  Q_{\ell_4} = \sum_{(ij) \in \ell_4} q_{ij} \in \{-e, -\tfrac{e}{2}, 0, \tfrac{e}{2}, e\}
  \]
Cell charge operator
- The physical charge of cell $i$ is defined as the sum of charges of all adjacent closed faces:
  \[
  Q_i := \sum_{\substack{\ell \ni i \\ \ell = \ell_3, \ell_4}} Q_{\ell}
  \]
- Open bonds and faces other than trianglets or plaquettes carry no charge.
- These accumulated face charges close space and form particles.
Comparison with traditional charge concepts
- Conventional charge operators are point-like or link-centered,
- but in the Qaether model, charge emerges and accumulates on the face level, enabling spatial closure and indicating particles.

A11. Definition of the Color Charge Operator

Basic pattern definition
- The square-pyramid cell pattern (4 trianglets + 1 plaquette) corresponds to a quark.
- When forming closed loops polyhedrally, a base curvature patch effect due to the Void arises.
- That is, the plaquette plus the four surrounding gluon links must form one closed loop to realize an $F_{\mu\nu}$-like field strength, completing the scale-dependent mass-curvature correlation.
- The ‘I-shaped’ 3-link structure corresponds to gluons.
- The plaquette corresponds to local color and curvature patches (only curvature due to bonding).
- All phase differences are quantized as
  \[
  \Delta \phi_{ij} = m_{ij} \cdot \frac{\pi}{3}, \quad (m_{ij} \in \mathbb{Z}_6).
  \]
Quark cell (square pyramid) bonding pattern - fundamental 3 colors
- $C^3, C^8$ are diagonal components of SU(3), distinguishing the directions of the basic colors (R,G,B) in color-charge space.
  \[
  \begin{array}{|c|c|c|}
  \hline
  \text{Cell color} & \text{Plaquette phase sum } S_{\mathcal{C}} & \text{Cartan }(C^3, C^8) \\
  \hline
  R & +4\pi & \left(+\tfrac{1}{2}, +\tfrac{1}{2\sqrt{3}}\right) \\
  G & -4\pi & \left(-\tfrac{1}{2}, +\tfrac{1}{2\sqrt{3}}\right) \\
  B & 0 & \left(0, -\tfrac{1}{\sqrt{3}}\right) \\
  \hline
  \end{array}
  \]
  \[
  S_{\mathcal{C}} := \sum_{\ell^4 \subset \mathcal{C}} \sum_{(ij) \in \ell^4} \Delta\phi_{ij}, \quad S_{\mathcal{C}} \in \{+4\pi, 0, -4\pi\}
  \]
Plaquette ($\Box$) $\leftrightarrow$ color $C^3$
- Independent plaquettes
- When a plaquette exists alone without bonding to other cells or links, it follows the standard phase quantization condition.
- Plaquettes bonded to gluons
- Each plaquette’s color charge must be connected to at least one gluon link to transfer color charge externally.
- When a plaquette bonds with gluons to form a closed bonding loop, it satisfies:
  \[
  \sum_{(ij) \in \ell_4} \Delta\phi_{ij} = 4\pi n_{\ell_4}, \quad n_{\ell_4} \in \{-1, 0, +1\},
  \]
  which leads to $C^3 = \pm \frac{1}{2}, 0$.
Gluon = ‘I-shaped 3-link’ phase pattern
- Three links arranged in a straight line, with phase difference set
  \[
  \{\pm\pi, \pm \tfrac{2\pi}{3}, \pm \tfrac{\pi}{3} \}
  \]
  each used once.
- The sum of these phase differences must always be zero:
  $\Delta\phi_1 + \Delta\phi_2 + \Delta\phi_3 = 0$.
Eight gluon states
\[
\begin{array}{|c|c|}
\hline
\text{Gluon state} & (\Delta\phi_1, \Delta\phi_2, \Delta\phi_3) \\
\hline
G^{R\bar G} & (+\pi, -\tfrac{2\pi}{3}, -\tfrac{\pi}{3}) \\
G^{G\bar R} & (-\pi, +\tfrac{2\pi}{3}, +\tfrac{\pi}{3}) \\
G^{G\bar B} & (+\tfrac{2\pi}{3}, -\pi, -\tfrac{\pi}{3}) \\
G^{B\bar G} & (-\tfrac{2\pi}{3}, +\pi, +\tfrac{\pi}{3}) \\
G^{R\bar B} & (+\tfrac{2\pi}{3}, +\tfrac{\pi}{3}, -\pi) \\
G^{B\bar R} & (-\tfrac{2\pi}{3}, -\tfrac{\pi}{3}, +\pi) \\
\lambda_7 (\text{diag}) & (+\pi, -\pi, 0) \\
\lambda_8 (\text{diag}) & (+\tfrac{2\pi}{3}, +\tfrac{2\pi}{3}, -\tfrac{4\pi}{3}) \\
\hline
\end{array}
\]
- Color transfer mechanism
- Flipping a link phase $m_{ij} = +3 \leftrightarrow -3$ changes $\Delta\phi = \pm \pi$ to $\pm 2\pi$, causing the central cell’s $S_C$ to vary by $\pm 4\pi$.
- Changing link phases $m_{ij} \to m_{ij} \pm 6$ changes the plaquette phase sum by $\pm 4\pi$, which corresponds exactly to transferring one unit of color charge $\Delta C^3 = \pm \tfrac{1}{2}$ to adjacent cells.
- Thus, one of the eight gluon patterns “emerges,” moving one unit of color charge across cells.
Local Gauss’s law (color flux conservation)
\[
\sum_{(ij) \in \partial \mathcal{C}} E_{ij}^a = C_{\mathcal{C}}^a, \quad E_{ij}^a = m_{ij} \varepsilon^a, \quad a=3,8
\]
Bonding hierarchy & color neutrality
\[
\begin{array}{|c|c|c|}
\hline
\text{Structure} & \text{Phase condition} & \text{Meaning} \\
\hline
\text{Link} & \Delta m = \pm 6 & \text{Gluon flux tube} \\
\text{Plaquette} & \sum m = \pm 12, 0 & F_{\mu\nu} \text{ patch} \\
\text{Cell (square pyramid)} & S = 4\pi n & \text{Quark color charge} \\
Y \text{ coupling} & S_R + S_G + S_B = 0 & \text{Baryon} \\
q\bar{q} & S + \bar S = 0 & \text{Meson} \\
\text{Link loop} & \text{no external cell} & \text{Glueball} \\
\hline
\end{array}
\]

A12. Phase-Spin Dynamical Equation

Gauge-covariant phase difference
\[
\Delta \phi_{ij}^{\rm tot} = (\phi_j - \phi_i) - q_e A_{ij} - g\,\vec C_i \cdot \vec A_{ij}
\]
- $A_{ij}$: U(1) electromagnetic link potential
- The phase-bond difference $\Delta\phi_{ij}^{U(1)} = (\phi_j-\phi_i) - \Delta\phi_{ij}^{SU(3)}$ can be used to define $A_{ij} = \frac{1}{q_e}\Delta\phi_{ij}^{U(1)}$. In the continuum limit, $A_{ij} \approx a\, \hat{e}_{ij}^\mu A_\mu(x)$.
- The 4-potential is $A_\mu = (A_0, A_1, A_2, A_3) \iff (\phi_{\rm EM}, \mathbf{A})$, where $A_0 = \phi_{\rm EM}$ is the electric potential, and $\mathbf{A} = (A_1, A_2, A_3)$ is the magnetic potential.
$\vec A_{ij}$: SU(3) gauge link
\[
U_{ij}^{(3)} = \exp(i \Delta\phi_{ij}^{SU(3)}), \quad A_{ij}^a = \frac{1}{g} \operatorname{Tr}[T^a \log U_{ij}^{(3)}]
\]
- $\vec C_i$: color charge vector of cell $i$
- Corresponding to adjacent plaquettes $\ell_4$ with phase sum $S_C^\ell = \sum_{(jk) \in \ell_4} \Delta\phi_{jk} \in \{\pm4\pi, 0\}$ is
  \[
  (C_3, C_8) = \left\{ \left(+\tfrac{1}{2}, +\tfrac{1}{2\sqrt{3}}\right), \left(-\tfrac{1}{2}, +\tfrac{1}{2\sqrt{3}}\right), \left(0, -\tfrac{1}{\sqrt{3}}\right) \right\},
  \quad \mathbf{C}_i = \sum_{\ell_4 \ni i} (C_3^\ell, C_8^\ell).
  \]
- $q_e, g$: coupling constants of U(1) and SU(3), respectively.
- See A10 for U(1) charge operator reference.
- See A11 for SU(3) color charge operator reference.
Complex link field
\[
\chi_{ij} := e^{i \Delta\phi_{ij}^{\rm tot}}, \quad \chi_{ji} = \chi_{ij}^{-1}
\]
Dynamical equation
The phase $\phi_i$ of each Qaether cell $i$ evolves according to
\[
I_i(m_i) \ddot{\phi}_i = \sum_{j \in \mathcal{N}(i)} \Bigl[ K_{ij} \Im \chi_{ij} - U_0 \Im(\chi_{ij}^6) - q_e A_{ij} \Im \chi_{ij} - g \vec C_i \cdot \vec A_{ij} \Im \chi_{ij} \Bigr] - P_i(m_i) V_s l_p \sin\phi_i,
\]
where $\Im$ denotes taking the imaginary part of $\exp(i \Delta\phi_{ij}^{\rm tot})$.
Coupling constant
\[
K_{ij} = K_0 \exp\left[-\lambda \frac{V_i(m_i) + V_j(m_j)}{2 V_s}\right] \left| \hat b_{ij} \cdot \hat n_{ij} \right|
\]
Differentiation follows the definition given in A7.

Symbol & Form	Description
$$I_i(m_i)\ddot\phi_i$$	$\text{Inertia term: } I_i(m_i) = I_0 \left(1 - \eta \frac{m_i}{12}\right), \text{ local moment of inertia depending on bond number } m_i$
$$K_{ij} \Im \chi_{ij}$$	$\text{Phase-coupling term: basic oscillation coupling depending on } \Delta\phi_{ij}^{\rm tot}$
$$- U_0 \Im(\chi_{ij}^6)$$	$\text{Phase-quantization potential enforcing } \pi/3 \text{ phase difference steps}$
$$- q_e A_{ij} \Im \chi_{ij}$$	$\text{Electromagnetic gauge coupling (U(1) minimal coupling)}$
$$- g \vec C_i \cdot \vec A_{ij} \Im \chi_{ij}$$	$\text{Color charge–gluon coupling maintaining SU(3) gauge covariance}$
$$- p_0 V_s l_p \sin \phi_i$$	$\text{Void effective pressure restoring force (approximate } P_i \simeq p_0,\, V_s=4\pi l_p^2\text{)}$

==> Deleting the void gravitational term from the dynamical term makes it equivalent to gauge lattice theory. To be precise, the Qaether theory encompasses the gauge lattice theory.