Summary: Mathematical Structure of Stage 40 and Completion Theorem
1. Metric from Substrate Density
$$\boxed{ g_{\mu\nu} = \eta_{\mu\nu} \left( 1 + \frac{P}{K}\rho \right) }$$
2. Connection (Christoffel Symbols)
$$\boxed{ \Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} \left( \partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta} \right) }$$
3. Curvature from Density Variation
$$\boxed{ R_{\mu\nu} \sim \partial_\mu \partial_\nu \rho - g_{\mu\nu} \Box \rho }$$
4. Einstein Tensor
$$\boxed{ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R }$$
5. Stress–Energy from Density
$$\boxed{ T_{\mu\nu}^{(\text{eff})} = \partial_\mu \partial_\nu \rho - g_{\mu\nu} \Box \rho }$$
6. Gravitational Coupling
$$\boxed{ 8\pi G = \frac{P}{K} }$$
7. Einstein-Type Equation
$$\boxed{ G_{\mu\nu} = 8\pi G\, T_{\mu\nu}^{(\text{eff})} }$$
8. Stochastic Dynamics (Microscopic Level)
$$\boxed{ dx^\mu = v^\mu dt + \sigma\, dW^\mu_t }$$
$$\boxed{ dv^\mu = -\Gamma^\mu_{\alpha\beta} v^\alpha v^\beta\,dt + \sigma_v\, dW^\mu_t }$$
9. Geodesic Emergence (Macroscopic Limit)
$$\boxed{ \mathbb{E}[x^\mu(\tau)] \;\rightarrow\; \text{Geodesic} }$$
$$\boxed{ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0 }$$
10. Stationary Density (Equilibrium)
$$\boxed{ \rho_{\text{eq}}(P) \propto P }$$
11. Equation of State
$$\boxed{ G_{\mu\nu} = 8\pi G \left( \nabla_\mu \nabla_\nu \rho_{\text{eq}} - g_{\mu\nu} \Box \rho_{\text{eq}} \right) }$$
12. Structural Flow
$$\boxed{ \text{Stochastic Dynamics} \;\longrightarrow\; \rho \;\longrightarrow\; g_{\mu\nu} \;\longrightarrow\; G_{\mu\nu} }$$
Final Mathematical Statement
$$\boxed{ \text{Spacetime geometry is the stationary macroscopic limit of stochastic substrate dynamics} }$$
Conclusion:
$$\boxed{ \text{General relativity emerges as the equilibrium equation of state of the substrate} }$$
Completion Theorem — Stage 40
Statement
$$\boxed{ \text{A stochastic substrate with multiplicative dynamics admits a unique stationary density } \rho, \text{ which induces a metric } g_{\mu\nu}, \text{ whose curvature and geodesic motion reproduce gravitational structure as an equilibrium state.} }$$
Expanded Form
$$\boxed{ \begin{aligned} &\text{Stochastic dynamics } (dA \sim A\,dW) \\ &\Longrightarrow\; \text{Stationary density } \rho(P) \propto P \\ &\Longrightarrow\; g_{\mu\nu} = \eta_{\mu\nu}\left(1 + \frac{P}{K}\rho\right) \\ &\Longrightarrow\; R_{\mu\nu} \sim \partial_\mu \partial_\nu \rho \\ &\Longrightarrow\; G_{\mu\nu} = 8\pi G\, T_{\mu\nu}^{(\text{eff})} \\ &\Longrightarrow\; \mathbb{E}[x^\mu(\tau)] \rightarrow \text{Geodesic} \\ &\Longrightarrow\; \text{Stable spacetime geometry} \end{aligned} }$$
Uniqueness Clause (Important)
The resulting geometric structure is uniquely fixed by:
- multiplicative stochastic scaling
- normalization constraint
- phase invariance
- second-order curvature structure
Closure Statement
$$\boxed{ \text{The framework is structurally closed: geometry, curvature, and motion arise from a single stochastic substrate without external postulates.} }$$
Final One-Line Form:
$$\boxed{ \text{Spacetime geometry is the unique stationary realization of stochastic substrate dynamics.} }$$