Functional Fuzziness Framework (FFF): Key Equations

Foundational Binary

The foundational binary describes the interplay between Being and Non-Being:

$$\mathcal{B}(t) = \begin{cases} 1 & \text{(Being)} \\ 0 & \text{(Non-Being)} \end{cases}$$

To smooth transitions, we define:

$$\mathcal{B}(t) = P(t) = \frac{1}{1 + e^{-k(t-t_0)}}$$

The flow of causality is given by:

$$\Psi(t) = \frac{d\mathcal{B}(t)}{dt}$$

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Planck Length

The Planck length is the smallest meaningful unit of spacetime:

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}}$$

It arises naturally from the interplay of the constants $\hbar$ (Planck's constant), $G$ (gravitational constant), and $c$ (speed of light).

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Speed of Causality

The speed of light, $c$, represents the maximum speed of causality:

$$c = 1 \quad \text{(in natural units)}$$

This universal constant ensures the unidirectional flow of causality across spacetime.

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Quantum Foam Dynamics

The energy density of the quantum foam is modeled as:

$$\rho_{\text{foam}} = \alpha \Psi(t) + \beta \Psi^2(t) + \xi(t)$$

where $\xi(t)$ is a stochastic term with correlations:

$$\langle \xi(t)\xi(t') \rangle = \exp\left(-\frac{|t-t'|}{\tau_P}\right)$$

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Dark Energy and Spacetime Expansion

The rate of spacetime creation is proportional to the energy density in the quantum foam:

$$\frac{dV}{dt} = \beta \rho_{\text{foam}}$$

Cosmic acceleration is derived as:

$$\ddot{a}(t) \propto \beta \alpha \frac{d\Psi(t)}{dt}$$

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Energy Recycling

The recycling of energy across process levels is expressed as:

$$\int_{t_0}^t E_{\text{upper}}(t') dt' + \int_{t_0}^t \rho_{\text{recycle}} dt' = E_{\text{total}}(t)$$

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Cosmological Constant Upper Limit

The cosmological constant $\Lambda$ is tied to the quantum foam’s energy density:

$$\Lambda_{\text{max}} \propto \rho_{\text{foam,max}}$$