# **Appendix: Mathematical Symbols and Formalisms in Information Dynamics** --- ## **1. Fundamental Variables** **Existence ($X$)** - **Definition**: A binary predicate indicating whether an entity holds information states ($X = 1$) or not ($X = 0$). - **Formula**: $X \in \{0, 1\}$, inferred from $\mathbf{I} \neq \mathbf{0}$. - **Example**: - A particle’s wavefunction has $X = 1$. A vacuum with zero quantum fields has $X = 0$. - **Notation**: - Boldface denotes vectors (e.g., $\mathbf{I}$). - Scalar values are non-bold (e.g., $X$, $\epsilon$). --- ## **2. Universal Information ($\mathbf{I}$)** - **Definition**: A **multi-dimensional vector** encoding all possible states of an entity. - **Formula**: $ \mathbf{I} \in \mathbb{R}^D \quad \text{(Continuous, non-physical)} $ $ - $D$: Total number of informational dimensions (e.g., position, spin, energy). - **Component Example**: - Position: $I_{\text{position}} = I_1 \in \mathbb{R}^3$. - Spin: $I_{\text{spin}} = I_2 \in \mathbb{R}$. - **Observation Index**: - $\mathbf{I}_k$: The $k$-th observation (vector in $D$-dimensional space). - **Example**: $ \mathbf{I}_5 = \begin{pmatrix} I_{51}, I_{52}, \dots, I_{5D} \end{pmatrix}^T \quad \text{(5th observation with $D$ dimensions)} $ --- ## **3. Resolution Parameter ($\epsilon$)** - **Definition**: A scalar defining the precision of measurement, discretizing $\mathbf{I}$ into observable data. - **Formula**: $ \epsilon > 0 \quad \text{(Unit-dependent, e.g., meters, seconds)} $ $ - **Example**: - Quantum regime: $\epsilon \sim 10^{-35} \text{ m}$ (Planck scale). - Classical regime: $\epsilon \gg 10^{-35} \text{ m}$. - **Discretization**: $ \hat{\mathbf{I}} = \text{round}\left( \frac{\mathbf{I}}{\epsilon} \right) \cdot \epsilon \quad \text{(Observed data)} $ --- ## **4. Contrast ($\kappa$)** - **Definition**: Normalized difference between information states, quantifying distinguishability. - **Formula**: $ \kappa(\mathbf{I}_i, \mathbf{I}_j) = \frac{\|\mathbf{I}_i - \mathbf{I}_j\|}{\epsilon} \quad \text{(Euclidean norm)} $ - **Example**: - Two quantum states with $\kappa \geq 1$ are distinguishable at Planck-scale $\epsilon$. - **Single-axis contrast**: $ \kappa_d(\mathbf{I}*i, \mathbf{I}*j) = \frac{|I*{id} - I*{jd}|}{\epsilon} \quad \text{(Difference along dimension $d$)} $ --- ## **5. Sequence ($\tau$)** - **Definition**: An ordered list of information vectors, forming an emergent timeline. - **Formula**: $ \tau = \left( \mathbf{I}_1, \mathbf{I}_2, \dots, \mathbf{I}_N \right) \quad \text{(Cardinality: $|\tau| = N$)} $ - **Time Emergence**: $ t \propto \frac{|\tau|}{\epsilon} \quad \text{(Time scales with sequence length and resolution)} $ - **Example**: - A particle’s trajectory: $\tau = \left( \mathbf{I}_{\text{position}_1}, \mathbf{I}_{\text{position}_2}, \dots \right)$. --- ## **6. Information Density ($\rho_{\mathbf{I}}$)** - **Definition**: Concentration of distinguishable states over a region and sequence interval. - **Formula**: $ \rho_{\mathbf{I}} = \frac{\text{Count}(\kappa(\mathbf{I}_i, \mathbf{I}_j) \geq 1)}{\epsilon^D \cdot \Delta|\tau|} \quad \text{(Volume: $\epsilon^D$, sequence interval: $\Delta|\tau|$)} $ - **Example**: - High $\rho_{\mathbf{I}}$ in a galaxy’s stars explains gravity without dark matter. --- ## **7. Entropy ($H$)** - **Definition**: Disorder in information states over a sequence. - **Discrete Formulation**: $ H(\tau) = -\sum_{k=1}^{|\tau|} P(\mathbf{I}_k) \log P(\mathbf{I}_k) $ - **Continuous Formulation**: $ H_{\text{cont}} = -\int_{\tau} p(\mathbf{I}) \log p(\mathbf{I}) \, d\mathbf{I} $ - **Example**: - A broken egg has higher $H$ due to more positional and thermal state permutations. --- ## **8. Gravity ($G$)** - **Definition**: Emergent force from information density and sequence dynamics. - **Formula**: $ G \propto \rho_{\mathbf{I}} \cdot \kappa_{\text{avg}} \cdot \frac{d|\tau|}{d\epsilon} $ - $\kappa_{\text{avg}} = \frac{1}{|\tau|^2} \sum_{i,j} \kappa(\mathbf{I}_i, \mathbf{I}_j)$. - **Example**: - Galactic rotation curves arise from $\rho_{\mathbf{I}} \cdot \kappa_{\text{avg}}$, not dark matter. --- ## **9. Consciousness ($\phi$)** - **Definition**: Emergent complexity from mimicry, causality, and repetition. - **Formula**: $ \phi \propto M \cdot \lambda \cdot \rho $ - $M$: Similarity between sequences ($M = \frac{\sum \kappa(\mathbf{I}_i, \mathbf{I}_j)}{|\tau|}$). - $\lambda$: Causality ($\lambda = \frac{P(\mathbf{I}_b | \mathbf{I}_a)}{P(\mathbf{I}_b)}$). - $\rho$: Repetition fraction ($\rho = \frac{\text{Repeated states}}{|\tau|}$). - **Example**: - Human consciousness requires $\phi > \phi_{\text{threshold}}$ due to neural repetition and mimicry. --- ## **10. Change ($\Delta \mathbf{I}$)** - **Definition**: Transition between states in $\tau$. - **Formula**: $ \Delta \mathbf{I} = \mathbf{I}_{k+1} - \mathbf{I}_k \quad \text{(Difference between consecutive observations)} $ - **Example**: - A photon’s polarization change: $\Delta I_{\text{polarization}}$. --- ## **11. Repetition ($\rho$)** - **Definition**: Frequency of repeated states in $\tau$. - **Formula**: $ \rho = \frac{\sum_{k=1}^{|\tau|} \sum_{j=k+1}^{|\tau|} \delta(\mathbf{I}_k, \mathbf{I}_j)}{|\tau|} \quad \text{(Dirac delta for state equality)} $ - **Example**: - Earth’s daily rotation: $\rho_{\text{rotation}} \approx 1$ (repeats every $24$ hours). --- ## **12. Causality ($\lambda$)** - **Definition**: Directional dependency between states via conditional probabilities. - **Formula**: $ \lambda(\mathbf{I}_a \rightarrow \mathbf{I}_b) = \frac{P(\mathbf{I}_b | \mathbf{I}_a)}{P(\mathbf{I}_b)} $ - **Example**: - A billiard ball’s motion: $\lambda_{\text{collision}}$ predicts post-collision states. --- ## **13. Mimicry ($M$)** - **Definition**: Similarity between sequences or states. - **Formula**: $ M = \frac{\sum_{d=1}^D \sum_{i,j} \kappa_d(\mathbf{I}_i, \mathbf{I}_j)}{D \cdot |\tau|} \quad \text{(Normalized contrast across dimensions)} $ - **Example**: - Quantum entanglement: $M_{\text{entangled}} \geq 1$ at fine $\epsilon$. --- ## **14. Probability ($P$)** - **Definition**: Probability of an information state occurring in $\tau$. - **Formula**: $ P(\mathbf{I}_k) = \frac{\text{Frequency of } \mathbf{I}_k \text{ in } \tau}{|\tau|} $ - **Example**: - A fair coin: $P(I_{\text{heads}}) = P(I_{\text{tails}}) = 0.5$. --- ## **15. Informational Dimensions ($D$ and $d$)** - **$D$**: Total dimensions (uppercase). - **Example**: A particle’s $D = 3$ (position) + $1$ (spin) + $1$ (energy) → $D = 5$. - **$d$**: Specific dimension index (lowercase). - **Example**: $I_{\text{position}} = I_{d=1}$. --- ## **16. Edge Networks** - **Definition**: Graphs encoding informational correlations ($\kappa \geq 1$). - **Formula**: $ G = (V, E) \quad \text{where } V = \{\mathbf{I}_1, \dots, \mathbf{I}_N\}, \quad E = \{\kappa(\mathbf{I}_i, \mathbf{I}_j) \geq 1\} $ - **Example**: - Quantum entanglement forms edges $E$ with $\kappa \geq 1$. --- # **Notation Conventions** 1. **Vectors**: - Boldface ($\mathbf{I}$) denotes vectors. - Subscripts: $\mathbf{I}_k$ (observation $k$), $I_{kd}$ (dimension $d$ of observation $k$). 2. **Scalars**: - Non-bold symbols ($\epsilon$, $H$, $G$) are scalars. 3. **Indices**: - $k$: Observation index (rows in a dataset). - $d$: Dimension index (columns in a dataset). 4. **Functions**: - $\text{round}(\cdot)$: Discretizes $\mathbf{I}$ into $\hat{\mathbf{I}}$. - $\delta(\cdot)$: Dirac delta function for state equality. --- # **Expressions And Equations** ## **1. Contrast Between States** $ \kappa(\mathbf{I}*i, \mathbf{I}*j) = \frac{\sqrt{\sum*{d=1}^D (I*{id} - I_{jd})^2}}{\epsilon} $ - **Purpose**: Quantifies distinguishability across all dimensions. ## **2. Information Density** $ \rho_{\mathbf{I}} = \frac{\text{Count}(\kappa \geq 1)}{\epsilon^D \cdot \Delta|\tau|} $ - **Purpose**: Measures how tightly distinguishable states are packed in $D$-dimensional space. ## **3. Time Emergence** $ t \propto \frac{|\tau|}{\epsilon} \quad \text{(Discrete sequence length scaled by resolution)} $ ## **4. Gravity Formula** $ G \propto \rho_{\mathbf{I}} \cdot \kappa_{\text{avg}} \cdot \frac{d|\tau|}{d\epsilon} $ - **Purpose**: Unifies gravity as an informational effect, eliminating dark matter. ## **5. Entropy Gradient** $ \frac{\partial H}{\partial |\tau|} > 0 \quad \text{(Entropy increases with sequence length)} $ ## **6. Causality** $ \lambda(\mathbf{I}_a \rightarrow \mathbf{I}_b) = \frac{P(\mathbf{I}_b | \mathbf{I}_a)}{P(\mathbf{I}_b)} $ ## **7. Consciousness Threshold** $ \phi \propto \frac{\sum \kappa \cdot \lambda \cdot \rho}{|\tau|} \quad \text{(Requires $\phi > \phi_{\text{threshold}}$)} $ --- # **Key Cross-Concept Dependencies** 1. **Gravity Depends On**: - $\rho_{\mathbf{I}}$: Information density. - $\kappa_{\text{avg}}$: Average contrast. - $\epsilon$: Resolution defining sequence progression. 2. **Consciousness Depends On**: - $M$: Mimicry of neural patterns. - $\lambda$: Synaptic causality. - $\rho$: Repetition in brain sequences ($\tau$). 3. **Edge Networks Govern**: - $E_{ij}$: Links states with $\kappa \geq 1$. - **Example**: Quantum entanglement → $E_{ij} = 1$. --- # **Examples Of Symbol Usage** 1. **Planetary Motion**: - Observations: $\mathbf{I}_k$ (position, velocity, mass). - **Gravity**: $ G_{\text{planets}} \propto \rho_{\mathbf{I}} \cdot \kappa_{\text{position}} \cdot \frac{d|\tau|}{d\epsilon} $ 2. **Quantum Entanglement**: - **Edge Network**: $G_{\text{quantum}}$ links $\mathbf{I}_1$ and $\mathbf{I}_2$ with $\kappa \geq 1$. 3. **Consciousness in AI**: - **Threshold**: $ \phi_{\text{AI}} \propto \frac{\sum_{\tau} \kappa(\mathbf{I}_i, \mathbf{I}_j) \cdot \lambda \cdot \rho}{|\tau|} $ ---