ERRT Summary

Externally Rendered Reality Theory (ERRT) proposes that reality as we perceive it is a rendered construct produced by an external source with unlimited computational capacity. The theory aims to unify quantum mechanics, general relativity, consciousness studies, and information theory within a single framework.

Key principles of ERRT include:

1. An external fundamental source rendering reality
2. Logical-mathematical primacy as reality's substrate
3. Informational ontology as the basis of existence
4. Rendered physicality of space, time, matter, and energy
5. Scale-dependent rendering from quantum to cosmic scales
6. Observer-dependent rendering
7. Integration of consciousness as fundamental
8. Dynamic and adaptive rendering processes
9. Fractal geometry as a core aspect of the rendering process

Key formulas in ERRT:

1. Universal State Function: $$\Psi(x, t) = R(x, t) \cdot \Phi(x, t)$$ Where $\Psi(x, t)$ is the universal state function, $R(x, t)$ is the rendering operator, and $\Phi(x, t)$ is the underlying quantum state.
2. Rendering Operator: $$R(x, t) = \mathcal{F}(S(s), O(o), \Lambda, \kappa)$$ Where $S(s)$ is the scale-dependent operator, $O(o)$ is the observer-dependent operator, $\Lambda$ is a large-scale structure parameter, and $\kappa$ adjusts for potential feedback.
3. Scale Operator: $$S(s) = \int_{0}^{\infty} \frac{\sigma(s')}{(s - s')^2 + \epsilon^2} \, ds'$$
4. Observer Effect: $$O(o) = \int \rho(o') \, f(x, o') \, do'$$
5. Modified Hubble Parameter: $$H_{\text{ERRT}}(z) = H_0 \cdot \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda + \delta H(z, s)}$$
6. Rendering Dynamics: $$\frac{\partial \Psi(x, t)}{\partial t} = \mathcal{H}_R \Psi(x, t)$$
7. ERRT Entropy: $$S_{\text{ERRT}} = -k_B \int \rho(x, t) \ln \rho(x, t) \, dx$$
8. Modified Friedmann Equation: $$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho_{\text{total}} + \frac{\Lambda_{\text{ERRT}}}{3} - \frac{k}{a^2}$$

These equations form the mathematical foundation of ERRT, describing how reality is rendered across scales, incorporating observer effects, and modifying standard cosmological models. The theory provides a framework for understanding quantum phenomena, cosmic structure, consciousness, and the nature of information in physical systems.