Distant Worlds Equations

"ACROSS SPACE & TIME TOWARDS DISTANT WORLDS"

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Dedicated page to Distant Worlds Equations. Note: I am not qualified Mathematician or Physicist, Almost all equations are either based on my notes or brainstormed through LLM's, Equations may contain Mistakes or make no sense at all, Done with love.

MATHEMATICSGRAPHS

Refer to respective articles for detailed rundown, D-Particles, for AdS Subspace Anti-De-Sitter Drive, for Bridges The Bridges Bridge Operator Box, for Library The Library, for Grav Well Gravitational Well, For Ang Metal Angelic Metal, Q-Language, For MT Metric Tensors.

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]

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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}.

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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)

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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}

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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}

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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}.

xychart-beta
    title "AdS Drive Stress Over Proper Time, Δτ is normalized to 1.0"
    x-axis ["0.00","0.09","0.18","0.27","0.36","0.45","0.55","0.64","0.73","0.82","0.91","1.00"]
    y-axis "Stress Intensity S(τ)" 0 --> 100
    line [5,12,18,26,35,45,55,65,75,82,90,95]

---

xychart-beta
    title "AdS Drive Energy Consumption Over Proper Time"
    x-axis ["0","τ₁","τ₂","τ₃","Δτ"]
    y-axis "Energy Consumed E(τ) in PW" 0 --> 10000
    line [2500, 4600, 6000, 8600, 10000]