The Eleven Foundational Equations of HQR
Mathematical Framework of Holonomic Quantum Reality
1. Holonomic Action Principle
\[S_{HQR} = \int d^{11}x\sqrt{-G}\left(\frac{R_{11}}{16\pi G_{11}} + \mathcal{L}_{\mathcal{QI}}\right)\]
Unifies gravity and quantum information
2. Quantum Cosmos Field Equation
\[R_{MN} - \frac{1}{2}G_{MN}R_{11} = 8\pi G_{11}\langle T_{MN}^{QI}\rangle\]
Quantum information stress-energy coupling
3. Quantum Information Metric Tensor
\[Q_{\mu\nu} = \frac{\delta^2 S_{QI}}{\delta g^{\mu\nu}\delta g^{\rho\sigma}}\rho_{\rho\sigma}\]
Geometric role of quantum entropy
4. Temporal Projection Equation
\[t^{\mu}(x) = \frac{\delta S_{QI}}{\delta I_{\mu}(x)}\]
Emergent informational arrow of time
5. Infogeometric Curvature Identity
\[R_{\mu\nu}^{QI} = \frac{\partial_{\mu}I\partial_{\nu}I}{I^2}\]
Information-based spacetime curvature
6. Fine-Structure Informational Ratio
\[\alpha^{-1} \approx 137 = \frac{\rho_{QI}}{R_{QI}}\]
Information-based fine-structure constant
7. HQR Entanglement Entropy
\[S_{HQR} = \frac{A_{proj}}{4G\hbar} + S_{hidden}\]
Holographic entropy bound
8. Higher-Dimensional Flux Quantization
\[\oint_{C^2} F = \frac{2\pi n\hbar}{q}, \quad n \in \mathbb{Z}\]
Kaluza-Klein particle predictions
9. Dark Energy Informational Density
\[\Lambda_{QI}^{(4D)} = 8\pi G_4\left(\rho_{QI}^{(4D)} - \rho_{ST}^{(4D)}\right)\]
Quantum informational dark energy origin
10. Entangled Echoes Equation
\[\frac{d}{dl}\left(-\text{Tr}\left(\rho_{11D}\log\rho_{11D}\right)\right) \approx -\frac{1}{\tau}\left(\frac{A_{proj}}{4G^{11D}\hbar} - \text{Tr}\left(\rho_{11D}\log\rho_{11D}\right)\right) + \xi\left(-\text{Tr}\left(\rho_{11D}\log\rho_{11D}\right)\right)\]
Black hole information paradox resolution
11. Evolving Enigma Entropy Equation
\[\frac{dS}{dt} = -\frac{S}{\tau} + \xi S_{hidden}\]
Black hole entropy time evolution