Information Physics: Notation & Symbols

A unified glossary of notation for Collision Theory, Electromagnetic Voxel Lattice (EVL), Information Physics (IP), and Entropic Mechanics (EM).

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Global Conventions

• Scalars: italic Roman (e.g., $t, x, y, z, E, S$).
• Vectors: bold lowercase/uppercase (e.g., $\mathbf{x}, \mathbf{V}$).
• Matrices / Linear maps: bold uppercase (e.g., $\mathbf{A}$).
• Operators (sets/classes): calligraphic $\mathcal{O}$.
• Specific operator instances: hatted lowercase (e.g., $\hat{o}$).
• Fields / continuous densities: Greek letters ($\phi, \psi, \rho$).
• Random variables / stochastic terms: sans serif or $\xi, \eta$ where standard.
• Units: SI unless stated; use $[\cdot]$ to denote dimensions.

Examples:

• Vector: $\mathbf{V}$
• Operator class: $\mathcal{O}$
• Specific operator: $\hat{o}$

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Core Symbols (Cross-Framework)

Symbol	Meaning	Type	Units
$c$	Speed of light	constant	$m\cdot s^{-1}$
$G$	Newtonian gravitational constant	constant	$m^3\cdot kg^{-1}\cdot s^{-2}$
$k_B$	Boltzmann constant	constant	$J\cdot K^{-1}$
$\hbar$	Reduced Planck constant	constant	$J\cdot s$
$H_0$	Hubble constant (today)	parameter	$s^{-1}$
$\Omega_m, \Omega_\Lambda$	Density parameters	parameters	—
$p_c$	Percolation threshold	parameter	—
$\ell_P$	Planck length	constant	m
$t_P$	Planck time	constant	s
$\ell_v$	Voxel spacing	model param.	m
$\tau_v$	Voxel hop time	model param.	s
$\eta$	Positional energy multiplier (EM)	scalar	—
$\mathbf{V}$	Intent vector (EM)	vector	—
$\mathcal{O}$	Operation class (COB)	set	—
$\hat{o}$	Realized operation	operator	—
$\phi$	Information density / potential	field	$bits\cdot m^{-3}$ or $J\cdot m^{-3}$
$\rho_{\mathrm{info}}$	Information-energy density	field	$J\cdot m^{-3}$
$\Gamma$	Bit processing rate density	field	$bits\cdot m^{-3}\cdot s^{-1}$
$\beta_0$	Information-reaction normalization	parameter	$s^{-1}$
$z_c$	Peak reaction redshift	parameter	—
$w$	Reaction epoch width	parameter	—
$q$	Power-law scaling exponent	parameter	—

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1. Collision Theory (CDE)

Primary PDE (Collision–Diffusion Equation):

$$\frac{\partial \phi}{\partial t} = D(z)\,\nabla^2 \phi - R_{\mathrm{info}}(z)$$

• $\phi$: information density (or potential)
• $D(z)$: effective diffusion coefficient vs. redshift
• $R_{\mathrm{info}}(z)$: information-reaction (Landauer-weighted) term

Information–reaction (Gaussian-in-redshift example):

$$R_{\mathrm{info}}(z) = \beta_0\,\Big(\tfrac{1+z}{1+z_c}\Big)^{q}\,\exp\!\left[-\tfrac{(z-z_c)^2}{2w^2}\right]$$

Percolation threshold: $p_c = 0.45$

From reaction to energy density:

$$\Gamma \;\sim\; \frac{R_{\mathrm{info}}}{\tau_v}\,g(\lVert\nabla\phi\rVert),\qquad \rho_{\mathrm{info}} \;=\; k_B T\ln 2\,\Gamma$$

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2. Electromagnetic Voxel Lattice (EVL)

Kinematics:

$$c \;=\; \frac{\ell_v}{\tau_v}$$

Pattern maintenance (Landauer identity view):

$$E \;=\; mc^2 \;=\; N_{\mathrm{bits}}\,k_B T\ln 2$$

Conservation of Boundaries (COB):

$$\mathcal{O} \in \{\mathcal{O}_M^{(1)},\;\mathcal{O}_J^{(2)},\;\mathcal{O}_S^{(3)}\}$$

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3. Information Physics (IP)

Memory compression efficiency:

$$\eta_{\mathrm{mem}} \;=\; \frac{I_{\mathrm{stored}}}{I_{\mathrm{total}}}$$

Bandwidth–capacity (informal scaling placeholder):

$$C_{\max} \;\propto\; B^{\alpha}\, T^{\beta}$$

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4. Entropic Mechanics (EM)

System Entropy Change (scalar form):

$$\mathrm{SEC} \;=\; \frac{\mathcal{O}\,\cdot\,\mathbf{V}}{1+\eta}$$

Entropic Gap (cosine-sim form):

$$EG \;=\; 1 - \cos\theta\big(\mathbf{a},\mathbf{c}\big) \;=\; 1 - \frac{\mathbf{a}\cdot\mathbf{c}}{\lVert\mathbf{a}\rVert\,\lVert\mathbf{c}\rVert}$$

Entropic Equilibrium (multi-agent):

$$\sum_i (\mathrm{SEC}_i\,W_i) \;\to\; \text{steady},\qquad \frac{d}{dt}\sum_i (\mathrm{SEC}_i\,W_i) \approx 0$$

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5. EVL-Specific Notation Extensions

Symbol	Meaning	Type	Units
$\eta_{\mathrm{lattice}}$	Positional energy multiplier due to lattice	scalar	—
$\theta_{\mathrm{lat}}$	Angle of operation relative to lattice field	scalar	rad
$B_{\mathrm{voxel}}$	Local magnetic field strength of a voxel	param.	Tesla (T)
$E_{\mathrm{trap}}$	Trap energy to dislodge particle from voxel	param.	J

6. CDE-EVL Model Parameters

Symbol	Meaning	Type	Units
$A(z)$	Activity kernel (unit peak)	function	—
$E(z)$	Early chemistry gate	function	—
$S(z)$	Percolation suppression factor	function	—
$D_0$	Diffusion scale (fitted parameter)	param.	$m^2\cdot s^{-1}$
$B$	Reaction normalization (fitted parameter)	param.	$s^{-1}$
$z_{\text{on}}, z_{\text{off}}$	Early chemistry transition redshifts	param.	—
$k_{\text{chem}}$	Chemistry transition steepness	param.	—
$z_t, w_t$	2D→3D crossover redshift and width	param.	—
$p_{2D}, p_{3D}$	Percolation thresholds (2D/3D)	param.	—
$\beta_{2D}, \beta_{3D}$	Critical exponents (2D/3D)	param.	—
$\alpha_{\text{hi}}, \alpha_{\text{lo}}$	Diffusion power law exponents	param.	—
$z_{\text{break}}$	Diffusion transition redshift	param.	—
$\lambda_{\text{obs}}$	Observed pattern scales	data	Mly

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Constants & Typical Numerical References

Symbol	Value
$c$	$2.99792458\times 10^8$ m·s⁻¹ (exact)
$k_B$	$1.380649\times 10^{-23}$ J·K⁻¹ (exact)
$\hbar$	$1.054571817\times 10^{-34}$ J·s
$G$	$6.67430\times 10^{-11}$ m³·kg⁻¹·s⁻²
$\ell_P$	$1.616255\times 10^{-35}$ m
$t_P$	$5.391247\times 10^{-44}$ s
$\Omega_{m0}$	$0.315$ (matter density parameter)
$\Omega_{\Lambda 0}$	$0.685$ (dark energy density)
$H_0$	$67.4$ km·s⁻¹·Mpc⁻¹ (Hubble constant)
$p_c$	$0.45$ (percolation threshold)
$p_{2D}$	$0.5000$ (2D percolation threshold)
$p_{3D}$	$0.2488$ (3D percolation threshold)
$\beta_{2D}$	$0.1600$ (2D critical exponent)
$\beta_{3D}$	$0.41$ (3D critical exponent)
$\beta_0$	$5.6234 \times 10^{-18}$ s⁻¹ (reaction norm.)
$z_c$	$5.3$ (peak reaction redshift)
$w$	$1.279$ (reaction epoch width)
$q$	$1.2$ (power-law scaling)

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Typographic Notes & LaTeX Snippets

• Vectors: $\mathbf{v}$, $\boldsymbol{\alpha}$
• Operators: $\mathcal{O}$, instances $\hat{o}$
• Fields: $\phi(\mathbf{x},t)$, gradient $\nabla\phi$, Laplacian $\nabla^2\phi$
• Percolation threshold: $p_c = 0.45$
• Planck-scale lattice: $c = \ell_v/\tau_v$

Example block (copy/paste ready):

$$\textbf{SEC:}\quad \mathrm{SEC} = \dfrac{\mathcal{O}\,\cdot\,\mathbf{V}}{1+\eta}, \qquad f(\eta)=\dfrac{1}{1+\eta}, \qquad EG = 1 - \dfrac{\mathbf{a}\cdot\mathbf{c}}{\lVert\mathbf{a}\rVert\,\lVert\mathbf{c}\rVert}.$$

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Versioning

• v1.0 — Initial consolidated notation for review.
• v1.1 — Added EVL-specific extensions.
• v1.2 — Updated CDE-EVL model parameters, added collision theory constants, corrected percolation formulas and parameter values to match implementation.