1. High-Dimensional Fundamental Actions
1.1 M-Theory (11D)
S_{\text{M-Theory}} = \int d^{11}x \, \sqrt{-g} \left( -\frac{1}{2\kappa^2} R + \frac{1}{12} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}} + \mathcal{L}_{\text{M2}} + \mathcal{L}_{\text{M5}}
1.2 11D Supergravity
S_{\text{SUGRA}} = \frac{1}{2 \kappa^2} \int d^{11}x \, \sqrt{-g} \left( R - \frac{1}{48} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}} + \text{fermionic terms}
1.3 Causal Dynamical Triangulations (CDT)
S_{\text{CDT}} = \sum_{\Delta_2} \left( \frac{1}{2 \kappa^2} \text{Vol}(\Delta_2) R(\Delta_2) \right) - \sum_{\Delta_3} \text{Vol}(\Delta_3)
1.4 Combined 11D + 4D + 2D + CDT Action
S_{\text{Total}} = \int d^{11}x \, \sqrt{-g} \left( -\frac{1}{2\kappa^2} R + \frac{1}{12} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}}
+ \frac{1}{2 \kappa^2} \int d^{11}x \, \sqrt{-g} \left( R - \frac{1}{48} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) + \text{fermionic terms}
+ \int d^2 \sigma \left( -\frac{1}{2} \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} - i \bar{\psi} \gamma^a \partial_a \psi \right)
+ \int d^4x \left( \mathcal{L}_{\text{SM}} + \sum_{\text{superpartners}} \left( \bar{\psi} i \gamma^\mu D_\mu \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) + \mathcal{L}_{\text{Yukawa}} + \mathcal{L}_{\text{Higgs}} \right)
+ \int d^4x \left( \bar{\psi} \gamma^\mu D_\mu \psi + \frac{1}{2} F_{\mu\nu} F^{\mu\nu} + \text{superpotential terms} \right)
+ \sum_{\Delta_2} \left( \frac{1}{2 \kappa^2} \text{Vol}(\Delta_2) R(\Delta_2) \right) - \sum_{\Delta_3} \text{Vol}(\Delta_3)
2. 4D Field-Theory Actions
S_{\text{Superstrings}} = \int d^2 \sigma \left( -\frac{1}{2} \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} - i \bar{\psi} \gamma^a \partial_a \psi \right)
2.2 (Minimal) Supersymmetric Standard Model (MSSM)
S_{\text{MSSM}} = \int d^4x \left( \mathcal{L}_{\text{SM}} + \sum_{\text{superpartners}} \left( \bar{\psi} i \gamma^\mu D_\mu \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) + \mathcal{L}_{\text{Yukawa}} + \mathcal{L}_{\text{Higgs}} \right)
2.3 General SUSY Action
S_{\text{SUSY}} = \int d^4x \left( \bar{\psi} \gamma^\mu D_\mu \psi + \frac{1}{2} F_{\mu\nu} F^{\mu\nu} + \text{superpotential terms} \right)
2.4 Cosmological ReConstant ჴ(\Lambda)
ჴ = \frac{1}{T}\int_{0}^{T} H_{\mathcal R}(t)\,dt = \frac{1}{T}\ln\!\biggl(\frac{V(T)}{V(0)}\biggr) = \frac{1}{T}\ln\bigl[1 + D(T)\bigr]
3. D-Particles Field Theories
3.1 Fracton Lagrangian
\mathcal{L}_{\text{Fracton}} = \overline{\Psi}_f \left( i \gamma^\mu D_\mu - m_f - \frac{S_b}{2} \Phi_{\text{string}} \right) \Psi_f - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2} \left( \nabla^2 \Phi_{\text{string}} \right)^2
3.2 Boreon Lagrangian
\mathcal{L}_{\text{Boreon}} = \frac{1}{2} (\partial_\mu \Phi_B)(\partial^\mu \Phi_B) - \frac{1}{2} m_B^2 \Phi_B^2 - \frac{\lambda_B}{4} \Phi_B^4 + g_B \Phi_B \overline{\Psi} \gamma^\mu \Psi
3.3 Mashtakov Metric
A = \frac{1}{1 + \alpha_B \Phi_B^2} \left( 1 + \frac{g_B \Phi_B}{M} \right)
3.4 Aurora Lagrangian
\mathcal{L}_{\text{Aurora}} = \frac{1}{2} (\partial_\mu \Phi_A)(\partial^\mu \Phi_A) - \frac{1}{2} m_A^2 \Phi_A^2 - \frac{\kappa}{4} \Phi_A^4 - \xi \Phi_A^2 \Phi_B^2 + ჱ_{փ ֆ^A}
3.5 Axion Lagrangian
xxx
4. Graviton/Gravity Related
4.1 Graviton Lagrangian
\mathcal{L}_{\text{Graviton}} = -\frac{1}{2} h^{\mu\nu} \mathcal{E}_{\mu\nu}^{\alpha\beta} h_{\alpha\beta} + \frac{\lambda}{2} \left( \partial_\lambda h_{\mu\nu} \partial^\lambda h^{\mu\nu} - \partial_\lambda h \partial^\lambda h \right) + \frac{\xi}{2} \Phi_p^2 h_{\mu\nu} h^{\mu\nu} + ჱ_{փ ֆ^g}
4.2 Dual Graviton
\mathcal{L}_{\text{Dual-Graviton}} = \lambda_D (h_{\mu\nu} h^{\mu\nu})^2
4.3 Gravitino Lagrangian
\mathcal{L}_{\text{Gravitino}} = \bar{\psi}_\mu \left( i \gamma^{\mu\nu\lambda} \partial_\lambda - m \gamma^{\mu\nu} \right) \psi_\nu + \frac{1}{2} \kappa \, \bar{\psi}_\mu \gamma^{\mu\nu} \psi_\nu h_{\alpha\beta} + \frac{\lambda_S}{2} S_G^2 \, \bar{\psi}_\mu \gamma^\mu \psi^\mu + \mathcal{L}_{\text{int}}^{\text{dim}}
4.4 Gravitational Well
Schwarzschild GW: \dot{G}\underset{\cdot}{G}(r) = 1 - \frac{2 G}{c^2\,r}\,\Biggl( M_c + \int_{0}^{r} 4\pi\,r'^{2}\,\bigl(\rho_{g0}e^{-\beta r'} + \rho_{q}(r')\bigr)\,dr' \Biggr) - \int_{0}^{r} \frac{G\,\bigl(\rho_{g0}e^{-\beta r'} + \rho_{q}(r')\bigr)}{r'^{2}} \,e^{-\alpha r'}\,dr' - \frac{k_{B}\,\Phi_{k}}{\lambda_{s}^{2}} \int_{0}^{r} \rho_{s}(r')\,4\pi\,r'^{2}\,dr'
second Kerr version: \vec{G}\underset{\vec{}}{G}(r,\theta) = \dot{G}\underset{\cdot}{G}(r) \;-\;\frac{2 G\, a\, \sin^2\!\theta}{c^3\,r^2}\, \Bigl(\,L_c + \!\int_0^r\! \ell(r')\,dr'\Bigr)
Third Charged, Stationary version: \begin{aligned} \dot{G}_Q(r) &= 1 - \frac{2 G}{c^2\,r}\Bigl( M_c + \int_{0}^{r}4\pi\,r'^{2}\bigl(\rho_{g0}e^{-\beta r'}+\rho_{q}(r')\bigr)\,dr' \Bigr) + \frac{G\,Q^2}{c^4\,r^2}\\ &\quad - \int_{0}^{r}\frac{G\,\bigl(\rho_{g0}e^{-\beta r'}+\rho_{q}(r')\bigr)}{r'^{2}} \,e^{-\alpha r'}\,dr' - \frac{k_{B}\,\Phi_{k}}{\lambda_{s}^{2}} \int_{0}^{r}\rho_{s}(r')\,4\pi\,r'^{2}\,dr'. \end{aligned}
Fourth Charged, Spinning, and Cosmological Constant version: \begin{aligned} \dot{G}_{\Lambda,Q}(r) &= 1 - \frac{2 G}{c^2\,r}\Bigl( M_c + \int_{0}^{r}4\pi\,r'^{2}\bigl(\rho_{g0}e^{-\beta r'}+\rho_{q}(r')\bigr)\,dr' \Bigr) + \frac{G\,Q^2}{c^4\,r^2} - \frac{\Lambda\,r^2}{3}\\ &\quad - \int_{0}^{r}\frac{G\,\bigl(\rho_{g0}e^{-\beta r'}+\rho_{q}(r')\bigr)}{r'^{2}} \,e^{-\alpha r'}\,dr' - \frac{k_{B}\,\Phi_{k}}{\lambda_{s}^{2}} \int_{0}^{r}\rho_{s}(r')\,4\pi\,r'^{2}\,dr',\\[6pt] \vec{G}_{\Lambda,Q}(r,\theta) &= \dot{G}_{\Lambda,Q}(r) \;-\;\frac{2\,G\,a\,\sin^2\!\theta}{c^3\,r^2} \Bigl( L_c + \int_{0}^{r}\ell(r')\,dr' \Bigr). \end{aligned}
4.5 Graviton Cloud Antsatz
\delta\beta_{\mu\nu}(r,\theta,\phi) = \varepsilon(r)\,\Xi_{\mu\nu}(r,\theta,\phi), \quad \varepsilon(r) = \varepsilon_0\,e^{-(r/r_q)^\gamma}, \\ \Xi_{\mu\nu} = \sum_{\ell,m} \Phi_{\ell m}(r)\,Y_{\ell m}(\theta,\phi)\,e_{\mu\nu}.
Assembled: \beta_{\mu\nu}(r,\theta) = g^{(\mathrm{classic})}_{\mu\nu}(r,\theta) + \varepsilon_0\,e^{-(r/r_q)^\gamma} \sum_{\ell,m}\Phi_{\ell m}(r)\,Y_{\ell m}(\theta,\phi)\,e_{\mu\nu}.
4.6 Piker-Baltzov Metric
Linear element: ds^2 = \beta_{\mu\nu}(x)\,dx^\mu dx^\nu = g^{(\mathrm{cl})}_{\mu\nu}(x)\,dx^\mu dx^\nu + \delta\beta_{\mu\nu}(x)\,dx^\mu dx^\nu
Schwarzschild
g^{(\mathrm{Schw})}_{tt} = -\bigl(1 - \tfrac{2GM}{c^2r}\bigr), \quad g^{(\mathrm{Schw})}_{rr} = \bigl(1 - \tfrac{2GM}{c^2r}\bigr)^{-1}, \quad d\Omega^2 = r^2(d\theta^2 + \sin^2\theta\,d\phi^2).
Kerr
\Sigma = r^2 + a^2\cos^2\theta, \qquad g^{(\mathrm{Kerr})}_{tt} = -\Bigl(1 - \tfrac{2GMr}{\Sigma c^2}\Bigr), \quad g^{(\mathrm{Kerr})}_{t\phi} = -\frac{2GMar\sin^2\theta}{\Sigma c^2}, \quad g^{(\mathrm{Kerr})}_{\phi\phi} = \sin^2\theta\,\frac{r^2 + a^2 + \tfrac{2GMa^2r\sin^2\theta}{\Sigma c^2}}{}.
Reissner–Nordström
g^{(\mathrm{RN})}_{tt} = -\Bigl(1 - \tfrac{2GM}{c^2r} + \tfrac{GQ^2}{c^4r^2}\Bigr), \quad g^{(\mathrm{RN})}_{rr} = \bigl(g^{(\mathrm{RN})}_{tt}\bigr)^{-1}.
Kerr-Newman-de Sitter
\Sigma = r^2 + a^2\cos^2\theta,\\ g^{(\mathrm{KNdS})}_{tt} = -\Bigl(1 - \tfrac{2GMr}{\Sigma c^2} + \tfrac{GQ^2}{\Sigma c^4} - \tfrac{\Lambda r^2}{3}\Bigr), \quad g^{(\mathrm{KNdS})}_{t\phi} = -\frac{2GMar\sin^2\theta}{\Sigma c^2}.
5. Interaction Terms
5.1 Fracton–Boreon Coupling
The potential interaction between the Fracton and Boreon fields, including the reduced interdimensional coupling constant \Phi_k:
\mathcal{L}_{\text{Fracton-Boreon}} = K_{\text{FB}} \cdot \bar{\psi}_{\mu\nu} \varphi \psi^{\mu\nu} + M \cdot \partial^\mu \varphi \partial_\mu \psi
5.2 Dual-Graviton with Matter
\mathcal{L}_{\text{Dual Graviton Interaction}} = \frac{1}{2} \left( h_{\mu \nu} h^{\mu \nu} \right) \left( \bar{\psi}_\mu h^{\mu \nu} \psi_\nu \right) + \Phi_{11}(h_{\mu \nu}, h_{\rho \sigma})
5.3 Topological Entanglement
\mathcal{L}_{\text{entanglement}} = \mathcal{L}_{\text{Fracton}} + \mathcal{L}_{\text{Boreon}} + \frac{k}{4\pi} \epsilon^{\mu\nu\alpha} A_{\mu} \partial_\nu \varphi \partial_\alpha \bar{\psi} + g \cdot \bar{\psi} \gamma^\mu (\partial_\mu \varphi) \psi + \xi \cdot |\nabla \varphi|^2 + \lambda \cdot \varphi^4
5.4 Topological Entanglement within Bridges
V_{\text{total}} = \sum_{i=1}^{N_B} V_{\text{wormhole}, i} - \beta \sum_{i=1}^{N_B} \sum_{j > i}^{N_B} \left( E_B^{(i)} \cdot E_B^{(j)} \right) \cdot \Phi_{ij} - \gamma \sum_{i=1}^{N_B} \lambda_i \left( \frac{1}{r_{\text{throat}, i}} \right)^{2}
5.5 Lagrangian for Gravioli Interaction with wormhole potential
\mathcal{L}_{\text{Gravioli}} = \frac{1}{2} (\partial_\mu \varphi)(\partial^\mu \varphi) - \frac{1}{2} m_\varphi^2 \varphi^2 - g_{\varphi h} \varphi \, h^{\mu\nu} R_{\mu\nu} - g_{\varphi \psi} \varphi \, \bar{\psi_\mu} \gamma^\mu \psi_\nu - \left( \alpha_1 \varphi^2 R + \alpha_2 (\bar{\psi_\mu} \gamma^\mu \psi_\nu) h^{\mu\nu} + \alpha_3 \varphi^4 - \alpha_4 \frac{1}{M_{\text{Pl}}^2} (\bar{\psi_\mu} \gamma^\mu \psi_\nu)^2 \right)
6. Wormhole Geometry & Potentials
6.1 5D Cylindrical Metric
V_{\text{wormhole}}(\varphi, h_{\mu\nu}, \psi_\mu, g_{AB}) = \alpha_1 \varphi^2 R_4 + \alpha_2 (\bar{\psi}_A \Gamma^A \psi_B) h^{AB} + \alpha_3 \varphi^4 + \alpha_5 \, \frac{1}{M_{\text{Pl}}^2} (\bar{\psi}_A \Gamma^A \psi_B)^2 + \alpha_6 \, R_5
6.2 5D Cylindrical Wormhole Metric
ds^2 = - c^2 dt^2 + d\rho^2 + r^2(\rho) \, d\Omega_3^2 + \lambda^2(\rho) \, d\psi^2
6.3 Traversability & Stability
The equation for throat radius stability is: \frac{d^2r(\rho)}{d\rho^2} + 4 \frac{r(\rho)}{\lambda(\rho)} \frac{d\lambda(\rho)}{d\rho} = 0
The effective gravitational force on a 3D object passing through the wormhole is: F_{\text{effective}} = - \frac{d}{d\rho} \left( r(\rho) \right) + \frac{\lambda(\rho)}{r(\rho)^2}
7. Bridge Dynamics
7.1 Stability Lagrangian
\mathcal{L} = \int d^5x \left( -\frac{1}{2} \left( \partial_\mu h_{\nu \rho} \partial^\mu h^{\nu \rho} - \frac{1}{2} \partial_\mu h \partial^\mu h \right) + \frac{i}{2} \left( \bar{\psi}_\mu \gamma^\mu D_\nu \psi^\nu - \bar{\psi}_\nu \gamma^\mu D^\nu \psi_\mu \right) + \frac{1}{2} \left( \partial_\mu \phi_F \partial^\mu \phi_F - m_F^2 \phi_F^2 \right) + \frac{1}{2} \left( \partial_\mu \phi_A \partial^\mu \phi_A - m_A^2 \phi_A^2 \right) + g_{\text{FA}} \phi_F \phi_A + g_{\text{GrF}} h_{\mu \nu} \partial^\mu \phi_F \partial^\nu \phi_F + \frac{1}{2} \epsilon^{\mu \nu \rho \sigma \tau} \partial_\mu A_\nu \partial_\rho B_{\sigma \tau} \right)
7.2 Configuration Action
\mathcal{S}_{\text{Bridge}} = \int d^5x \sqrt{-g} \left[ \frac{1}{2} R - \Lambda - \frac{1}{2} \left(\alpha_F |\nabla \phi_F|^2 + \alpha_B |\nabla \phi_B|^2 \right) + \beta \sum_{n} \frac{|\phi_{F,n}|^2}{\left(1 + r^2 \right)^{n}} + \gamma \epsilon^{\mu \nu \rho \sigma \tau} \partial_\mu A_\nu \partial_\rho B_{\sigma \tau} + \delta_{FB} \phi_F \phi_B + \eta_{GG} h_{\mu \nu} h^{\mu \nu} + \zeta \mathcal{A}_{\text{Boreon}} \right]
7.3 Triple‑Modular Redundancy Constants
\text{TMR Constants: } \left\{ \begin{aligned} \beta &\rightarrow \left( \beta_1,\, \beta_2,\, \beta_3 \right) \\ \gamma &\rightarrow \left( \gamma_1,\, \gamma_2,\, \gamma_3 \right) \\ \delta_{FB} &\rightarrow \left( \delta_{FB,1},\, \delta_{FB,2},\, \delta_{FB,3} \right) \\ \eta_{GG} &\rightarrow \left( \eta_{GG,1},\, \eta_{GG,2},\, \eta_{GG,3} \right) \\ \zeta &\rightarrow \left( \zeta_1,\, \zeta_2,\, \zeta_3 \right) \end{aligned} \right.
7.4 Evaporation Rate
\frac{dM_{\text{Bridge}}}{dt} = -\alpha \frac{\hbar c^2}{G} \frac{\Delta D}{r_{\text{Horizon}}^2} \exp\left(-\frac{\Phi_k}{k_B T}\right)
7.5 Bridge Stability safeguard
\mathcal{L}_{\text{Stability}} = \lambda_{\text{GF}} \bar{\psi}_\mu \gamma^\mu \Psi
7.6 Aurora Borealis Aymptotic Safety
\mathcal{L}_{\text{Aurora-Boreon}} = g_{\text{AB}} \phi_B \partial_\mu \phi_A + \lambda_{\text{AB}} \phi_A \phi_B^2
V_{\text{total}}(d) = \frac{1}{d^\beta} - \frac{1}{d^\gamma}, \quad \gamma > \beta
7.7 Electromagnetic Spike Decoherence
\lvert\psi\rangle = \alpha\lvert0\rangle + \beta\lvert1\rangle \;\xrightarrow{\text{EM spike}}\; \rho(\tau) = \begin{pmatrix} |\alpha|^2 & \alpha\beta^*\,e^{-\Gamma \tau} \\ \beta\alpha^*\,e^{-\Gamma \tau} & |\beta|^2 \end{pmatrix}
8. AdS-Drive & Subspace
8.1 AdS-Drive Action & EOM
S = \int d\tau \, \mathcal{L}_{\text{AdS}}(x^\mu, \dot{x}^\mu, \phi, g_{\mu\nu}, \epsilon, E)
\frac{d}{d\tau} \left( \frac{\partial \mathcal{L}_{\text{AdS}}}{\partial \dot{x}^\mu} \right) - \frac{\partial \mathcal{L}_{\text{AdS}}}{\partial x^\mu} = 0
8.2 Lagrangian for AdS Dynamics
\mathcal{L}_{\text{AdS}} = \frac{1}{2} m \left( g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu - 1 \right) - \epsilon \, g_{\mu\nu} \, \phi \, F(\dot{x}^\mu, \partial_\mu \phi) + \frac{\alpha}{2} \dot{x}^\mu \frac{d}{d\tau} \dot{x}_\mu - \frac{\beta}{r^2 + \gamma} + \mathcal{L}_{\text{AdS Transition}}
8.3 Equation of Transition
t_{\text{transition}} = \frac{2d}{\sqrt{\frac{2P}{m}} + \sqrt{\frac{2Q}{m}}}
8.4 Stress & Energy Build-Up
\tau = S_0 + \kappa_1 \,\epsilon\,\tau + \kappa_2 \,\epsilon^2\,\tau^2 - \delta\,e^{-\lambda\,\tau}
E(\tau) = E_0 + \eta\,\epsilon\,\tau + \zeta\,\epsilon^2\,\tau^2
8.5 Subspace Projection
V_{3D} = P_{4 \to 3} \cdot V_{4D}
V_{4D} = \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}, \quad V_{3D} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}
P_{4 \to 3} = \begin{bmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{bmatrix}
9. Linear Algebra & Metrics
9.1 (A)dS & dS Metrics
M = [v_1 \quad v_2 \quad \dots \quad v_k] \in \mathbb{R}^{n \times k}
Conditions:
\forall \alpha, \beta \in \mathbb{R}, \quad \alpha v_1 + \beta v_2 \in V
ANTI DE SITTER:
g_{\mu\nu}^{(AdS)} = \eta_{\mu\nu} + h_{\mu\nu}
DE SITTER:
g_{\mu\nu}^{(dS)} = \eta_{\mu\nu} - h_{\mu\nu}
9.2 Tensorial Pressure & Anti-de Sitter Tensors
T^{\mu\nu}_{AdS} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}
9.3 5x5 AdS Subspace Metric Matrice
G_{MN}(t,w,x,y,z) =\begin{pmatrix} G_{00} & G_{01} & G_{02} & G_{03} & G_{04} \\ G_{10} & G_{11} & G_{12} & G_{13} & G_{14} \\ G_{20} & G_{21} & G_{22} & G_{23} & G_{24} \\ G_{30} & G_{31} & G_{32} & G_{33} & G_{34} \\ G_{40} & G_{41} & G_{42} & G_{43} & G_{44} \end{pmatrix},\quad M,N=0,\dots,4
9.4 (AdS) Energy Requirement per Dimension
E_d = E_3 \cdot k^{(d - 3)} \quad \text{with } k \in [2, 5]
9.5 (AdS) Estimated Travel Time Scaling
T_d = \frac{T_3}{(d - 2)^{1.5}}
9.6 (AdS) Symmetry Preservation Function
S_d = \begin{cases} 1 & \text{if } d \leq 4 \\\\ e^{-(d - 4)} & \text{if } d > 4 \end{cases}
9.7 Christoffel & Geodesic
\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right)
\frac{d^2 x^\rho}{d\tau^2} + \Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = 0
9.8 Non-Zero Christoffel Symbol Derivation (Kerr)
ds^2 = -\Bigl(1 - \frac{2 G M r}{\Sigma c^2}\Bigr) c^2 dt^2 - \frac{4 G M a r \sin^2\theta}{\Sigma c}\,dt\,d\phi + \frac{\Sigma}{\Delta}\,dr^2 + \Sigma\, d\theta^2 + \Bigl(r^2 + \frac{a^2 G^2}{c^4} + \frac{2 G M a^2 r \sin^2\theta}{\Sigma c^2}\Bigr)\sin^2\theta\,d\phi^2
\Sigma = r^2 + \frac{a^2 G^2}{c^4}\cos^2\theta, \quad \Delta = r^2 - \frac{2 G M r}{c^2} + \frac{a^2 G^2}{c^4}.
10. Metric Tensors
10.1 Dimensional Metric Tensor
D_{\mu\nu} =\begin{bmatrix}g_{11} & g_{12} & g_{13} & \cdots & g_{1d} \\g_{21} & g_{22} & g_{23} & \cdots & g_{2d} \\g_{31} & g_{32} & g_{33} & \cdots & g_{3d} \\\vdots & \vdots & \vdots & \ddots & \vdots \\g_{d1} & g_{d2} & g_{d3} & \cdots & g_{dd}\end{bmatrix}
10.2 Dimensional Particles Stress-Tensor
S_{\mu\nu} =\begin{bmatrix}\sigma_{11} & \tau_{12} & \tau_{13} & \cdots & \tau_{1d} \\\tau_{21} & \sigma_{22} & \tau_{23} & \cdots & \tau_{2d} \\\tau_{31} & \tau_{32} & \sigma_{33} & \cdots & \tau_{3d} \\\vdots & \vdots & \vdots & \ddots & \vdots \\\tau_{d1} & \tau_{d2} & \tau_{d3} & \cdots & \sigma_{dd}\end{bmatrix}
10.3 Expanded Einstein Field Metric Tensor (Rank-2 MT Expanded)
\mathcal{G}_{\mu\nu} =\begin{bmatrix}g_{11} & g_{12} & \cdots & g_{1,10} \\g_{21} & g_{22} & \cdots & g_{2,10} \\\vdots & \vdots & \ddots & \vdots \\g_{10,1} & g_{10,2} & \cdots & g_{10,10}\end{bmatrix}
11. Miscellaneous Formulas
Example of Six Dimensional De Sitter Matrice Embedding Curvature
g_{\mu\nu}^{(6D)} = \begin{bmatrix} -1 & 0 & 0 & 0 & 0 & \epsilon \\ 0 & 1 & 0 & 0 & \epsilon & 0 \\ 0 & 0 & 1 & \delta & 0 & 0 \\ 0 & 0 & \delta & 1 & 0 & 0 \\ 0 & \epsilon & 0 & 0 & 1 & 0 \\ \epsilon & 0 & 0 & 0 & 0 & 1 \end{bmatrix}
11.1 Electroweak Leakage Representation
\Psi_{Bino}(x,t) = A \cdot e^{-(\gamma + i\omega)t} \cdot \psi(x)
11.2 Galactic Numerical System
\text{S}_r = \text{hex}\left( \left\lfloor \frac{r}{R_{\text{max}}} \times 0xFFFF \right\rfloor \right), \quad
\text{S}_\theta = \text{hex}\left( \left\lfloor \frac{\theta}{360^\circ} \times 0xFFFF \right\rfloor \right), \quad
\text{S}_\phi = \text{hex}\left( \left\lfloor \frac{\phi + \phi_{\text{offset}}}{\Phi_{\text{max}}} \times 0xFFFF \right\rfloor \right)
11.3 Berkenstein Bound
I_{\text{max}} \leq \frac{D R^\prime E^\prime}{\hbar c \ln 2},
11.4 Angelic Metal Resonance
\mathcal{R}_{\text{AM}} = \lambda_\text{res} \int d^4 x \left( \bar{\psi}_\mu h^{\mu \nu} \right))
11.5 Chemical Notation for Angelic Metal
\mathcal{L}_{\text{res}}^{(5D)} = \alpha \left( \bar{\psi}_{\mu} \gamma^{\nu} \psi^{\mu} \right) \left( h_{\mu\nu}^{(5D)} h^{\mu\nu(5D)} \right)
\ce{^{{314}}_{{126}}{AM}\ }
\ce{^{{318}}_{{126}}\text{AM}\ }
\ce{^{{325}}_{{140}}\text{AM}^{+}\ }
\ce{^{{314}}_{{126}}\text{AM}^{+}\ }
\ce{^{{314}}_{{126}}\text{AM}^{-}\ }
\ce{^{{318}}_{{126}}\text{Am} + \gamma^ \triangledown -> \text{Am}^* + \phi_b + E_{\text{burst}} + \gamma^ \triangledown_{\text{secondary}}}
\ce{\text{Am}^* \rightarrow \text{Am} + \phi_b + E_{\text{burst}} + \gamma^ \triangledown_{\text{secondary}} + e^\triangledown}
\ce{ e^\triangledown \xrightarrow{\text{Absorption}} E_{\text{usable}} }
\ce{E_{\text{reaction}} = \phi_b + E_{\text{burst}}}
\ce{E_{\text{SSR}} = 500 \, \text{MeV/reaction}}
12. Remus and Romulus
\tilde g_{\mu\nu}(r) = \Lambda\bigl(r;C_f\bigr)\;g_{\mu\nu}
N_{\max} = \frac{\ln C_f}{\ln s}, \quad \Lambda(r\to0;C_f)\to s^{N_{\max}} = C_f.
\mathcal M(\mathcal E) = \frac{1}{1 + e^{-\,k(\mathcal E - \mathcal E_0)}},
\begin{cases} \mathcal M\approx1 & (\text{Absorption}),\\ \mathcal M\approx0 & (\text{Deflection}). \end{cases}
E_{\rm abs}(t) = \int_0^t \Lambda\bigl(r(t');C_f\bigr)\; K\bigl(v(t')\bigr)\; \mathcal M\bigl(\mathcal E(t')\bigr)\; \Phi_{\rm in}(t')\;dt'.
E_{\rm abs}(t) \ge E_{\max} \quad\Longrightarrow\quad \text{Field collapse \& expulsion of }E_{\rm abs}.
N_{\max} = \frac{\ln C_f}{\ln s}
ds^2 = \bigl[\Lambda(r;C_f)\bigr]\, g_{\mu\nu}\,dx^\mu\,dx^\nu.
\Box\,h_{\mu\nu}(x) = -\,\frac{16\pi G}{c^4}\,T_{\mu\nu}(x) \;+\;S_{\mu\nu}(x),
\Phi(r) = \pm\,\alpha\,\frac{G\,M_{\rm usr}}{r}, \quad \begin{cases} +\;\text{“Push”} \\[4pt] -\;\text{“Pull”} \end{cases}
h_{\mu\nu}(x,t) = A\,e_{\mu\nu}\, \exp\bigl[i(\mathbf{k}\cdot\mathbf{x}-\omega t)\bigr], \quad \omega = c|\mathbf{k}|.
\kappa(v) = 1 + \gamma\bigl(\tfrac{v}{c}\bigr)^n, \quad \mathbf{a}(r,v) = -\kappa(v)\,\alpha\,G\,M_{\rm usr}\,\frac{\mathbf{\hat r}}{r^2}.
\mathbf{a}(r) = -\,\nabla\Phi(r) = \mp\,\alpha\,G\,M_{\rm usr}\,\frac{\mathbf{\hat r}}{r^2}.
g_{tt}(r) \approx -\Bigl(1 + \tfrac{2\,\Phi(r)}{c^2}\Bigr), \quad g_{ij}(r) \approx \delta_{ij}\Bigl(1 - \tfrac{2\,\Phi(r)}{c^2}\Bigr).
\chi_B(\mathbf{x}) = \Theta\bigl(R_B - \|\mathbf{x}-\mathbf{x}_0\|\bigr)
\mathbf{x}' \;=\; T_{\rm Tess}^{(4)}(\boldsymbol\phi)\;\mathbf{x}, \qquad \bigl[T_{\rm Tess}^{(4)}\bigr]^{i}{}_{j} = \exp\!\bigl(\phi_{ab}\,M^{ab}\bigr)^{i}{}_{j},
g''_{\mu\nu}(\mathbf{x}) = \bigl[1-\chi_B(\mathbf{x})\bigr]\,g_{\mu\nu} \;+\; \chi_B(\mathbf{x})\, \bigl(T_{\rm Tess}^{(4)}\,g\,T_{\rm Tess}^{(4)\,T}\bigr)_{\mu\nu}.
M_{\rm eff} = \frac{N_g\,\hbar\,\omega}{c^2}.
r_s = \frac{2\,G\,M_{\rm eff}}{c^2} = \frac{2\,G\,N_g\,\hbar\,\omega}{c^4}.
ds^2 = -\Bigl(1 - \frac{r_s}{r}\Bigr)c^2\,dt^2 + \Bigl(1 - \frac{r_s}{r}\Bigr)^{-1}dr^2 + r^2\,d\Omega^2.
K(r) = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = \frac{48\,G^2\,M_{\rm eff}^2}{c^4\,r^6}
\tau \sim \frac{5120\pi\,G^2\,M_{\rm eff}^3}{\hbar\,c^4}.