# **Mathematical Expressions for Information Dynamics** **1. Existence (\( X \))** \[ X \in \{0, 1\} \quad \text{(Binary predicate for existence)} \] --- **2. Information (\( \mathbf{I} \))** \[ \mathbf{I} \in \mathbb{R}^n \quad \text{(Continuous vector of informational axes)} \] **Example (Particle):** \[ \mathbf{I}*{\text{particle}} = \begin{pmatrix} I*{\text{position}} \\ I_{\text{spin}} \\ I_{\text{energy}} \end{pmatrix} \quad \text{(Components: } I_{\text{position}} \in \mathbb{R}^3, I_{\text{spin}} \in \mathbb{R}, I_{\text{energy}} \in \mathbb{R} \text{)} \] --- **3. Contrast (\( \kappa \))** \[ \kappa(\mathbf{I}_1, \mathbf{I}_2) = \frac{\|\mathbf{I}_1 - \mathbf{I}_2\|}{\epsilon} \quad \text{(Normalized vector difference)} \] --- **4. Sequence (\( \tau \))** \[ \tau = \{ \mathbf{I}_1, \mathbf{I}_2, \dots, \mathbf{I}_n \} \quad \text{(Ordered progression of states)} \] **Time Emergence:** \[ t \propto \frac{\tau}{\epsilon} \quad \text{(Time scales with resolution)} \] --- **5. Resolution (\( \epsilon \))** \[ \epsilon > 0 \quad \text{(Measurement discretization parameter)} \] **Discretization Formula:** \[ \hat{\mathbf{I}} = \text{round}\left( \frac{\mathbf{I}_{\text{continuous}}}{\epsilon} \right) \cdot \epsilon \quad \text{(Observed data)} \] --- **6. Information Density (\( \rho_{\mathbf{I}} \))** \[ \rho_{\mathbf{I}} = \frac{\sum_{i,j} \theta\left(\kappa(\mathbf{I}*i, \mathbf{I}*j) - 1\right)}{\text{Volume} \times \Delta\tau} \quad \text{(Count of distinguishable states)} \] **Volume (Spatial Component):** \[ \text{Volume} = \int*{\text{Region}} d^3 I*{\text{position}} \quad \text{(Integral over positional axes of } \mathbf{I} \text{)} \] --- **7. Gravity (\( G \))** \[ G \propto \rho_{\mathbf{I}} \cdot \kappa_{\text{avg}} \cdot \frac{d\tau}{d\epsilon} \quad \text{(Density × contrast × progression)} \] **Average Contrast:** \[ \kappa_{\text{avg}} = \frac{1}{N^2} \sum_{i,j} \kappa(\mathbf{I}_i, \mathbf{I}_j) \] --- **8. Entropy (\( H \))** **Discrete Formulation:** \[ H(\tau) = -\sum_{i=1}^{N} P(\mathbf{I}*i) \log P(\mathbf{I}*i) \] **Continuous Formulation:** \[ H*{\text{continuous}} = -\int*{\tau} p(\mathbf{I}) \log p(\mathbf{I}) \, d\mathbf{I} \] --- **9. Causality (\( \lambda \))** \[ \lambda(\mathbf{I}_1 \rightarrow \mathbf{I}_2) = \frac{P(\mathbf{I}_2 | \mathbf{I}_1)}{P(\mathbf{I}_2)} \quad \text{(Conditional probability ratio)} \] --- **10. Consciousness (\( \phi \))** \[ \phi \propto M \cdot \lambda \cdot \rho \quad \text{(Mimicry × causality × repetition)} \] **Mimicry (\( M \)):** \[ M = \text{sim}(\tau_1, \tau_2) \propto \kappa \cdot \tau \quad \text{(Similarity between sequences)} \] **Repetition (\( \rho \)):** \[ \rho = \frac{\text{Number of repetitions in } \tau}{\text{Total states in } \tau} \] --- **11. Decoherence Rate (\( \Gamma \))** \[ \Gamma \propto \frac{\text{Entropy Exchange}}{\text{Edge Network Isolation}} \quad \text{(Quantum-to-classical transition)} \] --- **12. Energy (\( E \)) and Mass (\( m \))** **Energy as Information Transformation:** \[ E = \Delta H \cdot k_B \quad \text{(Entropy change × Boltzmann constant)} \] **Mass as Information Density:** \[ m = \rho_{\mathbf{I}} \cdot c^2 \quad \text{(Density × speed of light squared)} \] --- **13. Quantum Recurrence and Edge Networks** **Edge Network Topology:** \[ G = (V, E) \quad \text{(Nodes } V \text{ represent entities; edges } E \text{ encode informational links)} \] **State Change Dynamics:** \[ \Delta S = \frac{\partial H}{\partial t} \quad \text{(Entropy-driven state transitions)} \] --- **14. Non-Binary Probabilistic States (Patent)** **Analog Architecture:** \[ \mathbf{I}*{\text{analog}} = f*{\text{nonlinear}}(\mathbf{I}_{\text{input}}) \quad \text{(Thermodynamic encoding)} \] **Non-Destructive Readout:** \[ \text{Readout} \propto \nabla \cdot \mathbf{I} \quad \text{(Gradient-based measurement)} \] --- # **Key Dependencies** \[ \begin{aligned} \mathbf{I} &\rightarrow \kappa \quad \text{(Contrast)} \\ \mathbf{I} &\rightarrow \tau \quad \text{(Sequence)} \\ \mathbf{I} &\rightarrow \rho_{\mathbf{I}} \quad \text{(Density)} \\ \kappa \cdot \tau &\rightarrow M \quad \text{(Mimicry)} \\ \tau \cdot X &\rightarrow \rho \quad \text{(Repetition)} \\ M \cdot \lambda \cdot \rho &\rightarrow \phi \quad \text{(Consciousness)} \\ \end{aligned} \] --- # **Summary Of Core Equations** \[ \boxed{ \begin{aligned} &\mathbf{I} \in \mathbb{R}^n, \quad X \in \{0,1\}, \quad \epsilon > 0 \\ &\kappa(\mathbf{I}_1, \mathbf{I}*2) = \frac{\|\mathbf{I}*1 - \mathbf{I}*2\|}{\epsilon} \\ &\rho*{\mathbf{I}} = \frac{\sum*{i,j} \theta(\kappa(\mathbf{I}*i, \mathbf{I}*j) - 1)}{\left( \int d^3 I*{\text{position}} \right) \cdot \Delta\tau} \\ &G \propto \rho*{\mathbf{I}} \cdot \kappa*{\text{avg}} \cdot \frac{d\tau}{d\epsilon} \\ &H = -\sum P(\mathbf{I}_i) \log P(\mathbf{I}*i) \quad \text{or} \quad H*{\text{cont}} = -\int p(\mathbf{I}) \log p(\mathbf{I}) \, d\mathbf{I} \\ &\phi \propto M \cdot \lambda \cdot \rho \quad \text{where} \quad \lambda = \frac{P(\mathbf{I}_2|\mathbf{I}_1)}{P(\mathbf{I}_2)} \end{aligned} } \] --- # **Explanation Via Axes of \( \mathbf{I} \)** - **Spatial Axes**: \( I_{\text{position}} \in \mathbb{R}^3 \). - **Quantum Axes**: \( I_{\text{spin}} \in \mathbb{R} \), \( I_{\text{polarization}} \in \mathbb{R}^2 \). - **Temporal Axis**: \( \tau \) orders states \( \mathbf{I} \) without requiring \( t \) as fundamental. --- # **Edge Network Relationships** \[ \begin{aligned} \text{Edge Network} \quad G &: \quad V = \{\mathbf{I}_1, \mathbf{I}_2, \dots\}, \quad E = \{ \kappa(\mathbf{I}_i, \mathbf{I}_j) \geq 1 \} \\ \text{Entropy Gradient} &: \quad \frac{\partial H}{\partial \tau} > 0 \quad \text{(Time’s statistical directionality)} \\ \text{Quantum Coherence} &: \quad \Gamma \propto \frac{\Delta H}{\text{Isolation}(G)} \end{aligned} \] --- # **Quantum-Classical Transition** \[ \begin{aligned} \text{Quantum Regime} &: \quad \epsilon \sim \text{Planck} \quad \Rightarrow \quad \text{Non-local mimicry} \, (M) \, \text{dominates} \\ \text{Classical Regime} &: \quad \epsilon \gg \text{Planck} \quad \Rightarrow \quad \hat{\mathbf{I}} \, \text{collapses into discrete states} \end{aligned} \] --- # **Final Note** All constructs (gravity, consciousness, spacetime) are derived from \( \mathbf{I} \), \( \kappa \), \( \tau \), and \( \epsilon \), with no reliance on unobservable entities. The framework is **resolution-dependent**, unifying quantum and classical physics via: \[ \mathbf{I}*{\text{continuous}} \xrightarrow{\epsilon \text{ discretization}} \hat{\mathbf{I}}*{\text{observed}} \xrightarrow{\text{inference}} \overline{\mathbf{I}}_{\text{synth}} \quad \text{(Cycle of information dynamics)} \]