Drogidi Theory — Overview

1. Philosophical Core

The Drogidi theory treats reality not as a single spacetime, but as a synergistic multiplicity of curved manifolds that interact.

• Reality: synergistic multiplicity of manifolds, not a single spacetime.
• Cosmology: network-based, not monolithic.
• Big Bang: local high-expansion episode, not a universal origin.
• Dark sector: projection phenomena of the coupling, not new particles or fields.

2. Basic Mathematical Structures

2.1 Family of Manifolds

$$\mathcal{M}={\{(M_a,g_{\mu \nu }^{(a)})\}}_{a\in \mathcal{A}}$$

• Each $M_a$: four-dimensional Lorentzian manifold.
• No privileged choice: full symmetry with respect to the index $a$.

2.2 Local General Relativity

$$G_{\mu \nu }^{(a)}+{\mathrm{\Lambda }}_a{g}_{\mu \nu }^{(a)}=8\pi G{T}_{\mu \nu }^{(a)}$$

The Drogidi theory does not modify GR; it extends it to multiple manifolds.

2.3 Mapping Field

$${\mathrm{\Phi }}_{ab}:M_a\to M_b$$

• Pushforward: ${({\mathrm{\Phi }}_{ab})}_{\ast }:T_x{M}_a\to T_{{\mathrm{\Phi }}_{ab}(x)}{M}_b$
• Pullback: ${({\mathrm{\Phi }}_{ab})}^{\ast }:T_{{\mathrm{\Phi }}_{ab}(x)}^{\ast }{M}_b\to T_{\mathrm{x\ast }}^{}{M}_a$
• Weak metric compatibility: $g^{(a)}\approx {({\mathrm{\Phi }}_{ab})}^{\ast }{g}^{(b)}$ on large scales.
• Kinematic compatibility: $u^{(a)}\approx {({\mathrm{\Phi }}_{ab})}^{\ast }{u}^{(b)}$.

2.4 Phase-space Structure

$$f_a(x,p),p^{\mu }{\mathrm{\nabla }}_{\mu }^{(a)}{f}_a=0$$

3. The Synergy Tensor $J_{\mu \nu }^{(ab)}$

3.1 General Definition

$$J_{\mu \nu }^{(ab)}=F(g^{(a)},g^{(b)},{\mathrm{\Phi }}_{ab},R^{(a)},R^{(b)},{\theta }_a,{\theta }_b,f_a,f_b)$$

3.2 Decomposition into Components

$$J_{\mu \nu }^{(ab)}=A_{ab}\mathrm{\Delta }{R}_{ab}{g}_{\mu \nu }^{(a)}+B_{ab}\mathrm{\Delta }{\theta }_{ab}{u}_{\mu }^{(a)}{u}_{\nu }^{(a)}+C_{ab}{\mathrm{\Xi }}_{\mu \nu }^{(ab)}$$

• Curvature difference: $$\mathrm{\Delta }{R}_{ab}=R^{(a)}-{({\mathrm{\Phi }}_{ab})}^{\ast }{R}^{(b)}$$
• Expansion difference:$$\mathrm{\Delta }{\theta }_{ab}={\theta }_a-{({\mathrm{\Phi }}_{ab})}^{\ast }{\theta }_b$$
• Kinetic mismatch tensor:

$${\mathrm{\Xi }}_{\mu \nu }^{(ab)}(x)=\int p_{\mu }{p}_{\nu }(f_b({\mathrm{\Phi }}_{ab}(x),{({\mathrm{\Phi }}_{ab})}_{\ast }p)-f_a(x,p))d^{4}p$$

4. Central Equations

4.1 Effective Energy-Momentum Tensor

$$T_{\mu \nu }^{\text{eff}(a)}=T_{\mu \nu }^{(a)}+\sum _{b\ne a}{K}_{ab}{J}_{\mu \nu }^{(ab)}$$

• Curvature projection: effective gravity → “dark matter”.
• Expansion projection: effective negative pressure → “dark energy”.
• Kinetic projection: enhancement of structure growth → cosmic web.

4.2 Cosmological Feedback Equation

Local Raychaudhuri Equation

$$\frac{d{\theta }_a}{d\tau }=-\frac{1}{3}{\theta }_a^2-{\sigma }_{\mu \nu }^{(a)}{\sigma }_{(a)}^{\mu \nu }+{\omega }_{\mu \nu }^{(a)}{\omega }_{(a)}^{\mu \nu }-R_{\mu \nu }^{(a)}{u}^{(a)\mu }{u}^{(a)\nu }$$

Synergistic Term

$${\mathcal{F}}_{ab}={\alpha }_{ab}\mathrm{\Delta }{R}_{ab}+{\beta }_{ab}\mathrm{\Delta }{\theta }_{ab}+{\gamma }_{ab}{\mathrm{\Xi }}_{ab}$$

$${\mathcal{S}}_a=\sum _{b\ne a}{K}_{ab}{\mathcal{F}}_{ab}$$

Full Equation

$$\boxed{\frac{d{\theta }_a}{d\tau }=-\frac{1}{3}{\theta }_a^2-{\sigma }_{\mu \nu }^{(a)}{\sigma }_{(a)}^{\mu \nu }+{\omega }_{\mu \nu }^{(a)}{\omega }_{(a)}^{\mu \nu }-R_{\mu \nu }^{(a)}{u}^{(a)\mu }{u}^{(a)\nu }+{\mathcal{S}}_a}$$

5. Observables & Predictions

5.1 Dark Matter

• Nature: effective curvature, not a particle.
• Structure: smooth distribution, without subhalos/cusps.
• Scales: constant DM ratio at all scales.
• Dependence:$${\rho }_{\text{DM}}^{(a)}\propto \sum _{b\ne a}{K}_{ab}\mathrm{\Delta }{R}_{ab}$$

5.2 Dark Energy

• Emergent: from expansion mismatch $\mathrm{\Delta }{\theta }_{ab}$.
• Equation of state: $w_{\text{eff}}\approx -1$, without fine-tuning.
• Stability: $\frac{dw}{dz}\approx 0$.
• Non-clustering: geometric projection, not a field.

5.3 Cosmic Web & Structures

• Characteristic scale: $k_{\ast }\approx 0.1h/\text{Mpc}$.
• Transfer function: Gaussian bump around $k\approx 0.1$, exponential cutoff at small scales.
• Structure growth: without CDM halos, via kinetic projection.

5.4 CMB & Big Bang Episodes

• CMB homogeneity: from inter-domain synchronization, not inflation.
• Primordial B-modes: prediction of absence.
• Big Bang: local episodes when $\sum _{b\ne a}{K}_{ab}{\mathcal{F}}_{ab}\gg 0$.
• Big Bang cascades: through synergistic symmetry ${\mathrm{\Phi }}_{ba}\circ {\mathrm{\Phi }}_{ab}$.

6. Quantum Structure

6.1 Multiple Hilbert Spaces

$$|\mathrm{\Psi }⟩\in \mathcal{F}\left(\underset{a\in \mathcal{A}}{⨂}{\mathcal{H}}^{(a)}\right)$$

• Local operators: ${\hat{O}}^{(a)}:{\mathcal{H}}^{(a)}\to {\mathcal{H}}^{(a)}$
• Synergistic operators: ${\hat{S}}^{(ab)}:{\mathcal{H}}^{(a)}\otimes {\mathcal{H}}^{(b)}\to {\mathcal{H}}^{(a)}\otimes {\mathcal{H}}^{(b)}$
• Dark matter / dark energy / Big Bangs: expectation values of synergistic operators.

7. Core Equations

7.1 Core Field Equation

$$G_{\mu \nu }^{(a)}+{\mathrm{\Lambda }}_a{g}_{\mu \nu }^{(a)}=8\pi G\left(T_{\mu \nu }^{(a)}+\sum _{b\ne a}{K}_{ab}{J}_{\mu \nu }^{(ab)}\right)$$

7.2 Interaction Tensor

$$J_{\mu \nu }^{(ab)}=\alpha (R^{(a)}-{\mathrm{\Phi }}_{ab}^{\ast }{R}^{(b)})g_{\mu \nu }^{(a)}+\beta (R_{\mu \nu }^{(a)}-{\mathrm{\Phi }}_{ab}^{\ast }{R}_{\mu \nu }^{(b)})+\gamma ({\theta }_a-{\mathrm{\Phi }}_{ab}^{\ast }{\theta }_b)$$

8. Diagrams (conceptual layout)