Notation and Units

Complete Symbol Reference for Technical Readers

Primary Temporal Quantities

Symbol	Name	Units	Dimensions	Description
T(Ω,t)	Temporal Functional	s	T	Accumulated emergent time over region Ω
σ_θ	Drift Scalar	s⁻¹	T⁻¹	Local temporal production rate
𝒟	Drift Magnitude	s⁻¹	T⁻¹	Rate of glyph field evolution
ℒ	Shell-Lock	—	1	Dimensionless stability coefficient ∈ [0,1]
τ	Recursion Parameter	s	T	Fundamental ordering parameter
t_P	Planck Time	s	T	≈ 5.391 × 10⁻⁴⁴ s
n	Recursion Index	—	1	Discrete update counter ∈ ℤ⁺

Symbol	Name	Type	Units	Description
Φ	Intent Potential	Scalar field	—	Latent permission field
F_i = ∂_i Φ	Intent Gradient	Vector field	L⁻¹	Directional intent
𝒞_i	Curvent	Vector field	L⁻¹	Recursive fold direction
𝓜_{ij}	Memory Tensor	Rank-2 tensor	L²	Recursive coherence structure
Ψ	Field Stack	Tuple	—	{Φ, F_i, 𝒞_i, 𝓜_{ij}}

Symbol	Name	Units	Range	Description
γ_ITT	Dilation Factor	—	(0, 1]	dt/dτ ratio
𝒜	Alignment Functional	—	[0, 1]	Recursive resource allocation
Tr(𝓜)	Memory Trace	L²	[0, ∞)	Total memory load
Tr(𝓜)_max	Maximum Trace	L²	(0, ∞)	Capacity limit
μ	Memory Saturation	—	[0, 1]	Tr(𝓜)/Tr(𝓜)_max

Symbol	Name	Units	Description
α_𝓜	Memory Coupling	—	Weight of ∂_t 𝓜 in drift magnitude
α_Φ	Intent Coupling	—	Weight of ∂_t ∇Φ in drift magnitude
κ_g	Temporal Coupling	J⁻¹	Appears in Delta threshold

Symbol	Name	Value	Units
ℏ	Reduced Planck constant	1.055 × 10⁻³⁴	J·s
G	Gravitational constant	6.674 × 10⁻¹¹	m³/(kg·s²)
c	Speed of light	2.998 × 10⁸	m/s
t_P	Planck time	5.391 × 10⁻⁴⁴	s
ℓ_P	Planck length	1.616 × 10⁻³⁵	m
E_P	Planck energy	1.956 × 10⁹	J

Symbol	Name	Definition	Description
R̂	Recursive Operator	Ψ_{n+1} = R̂(Ψ_n)	State transition map
∂_i	Spatial derivative	∂/∂x_i	i ∈ {1,2,3}
∂_t	Time derivative	∂/∂t	With respect to τ
∇	Gradient	(∂_1, ∂_2, ∂_3)	Spatial gradient
∇²	Laplacian	∂²/∂x² + ∂²/∂y² + ∂²/∂z²	Scalar Laplacian
Tr(·)	Trace	Σ_i A_{ii}	Sum of diagonal elements

Notation	Name	Definition	Application
‖v‖₂	Euclidean norm	√(Σ_i v_i²)	Vectors
‖A‖_F	Frobenius norm	√(Σ_{ij} A_{ij}²)	Matrices/Tensors
‖·‖	Context-dependent	—	Appropriate norm for type

Index Type	Symbols	Range	Usage
Spatial	i, j, k	{1, 2, 3}	Vector/tensor components
Recursion	n, m	ℤ⁺ ∪ {0}	Discrete state labels
Summation	—	—	Einstein convention (repeated indices summed)

Example: 𝓜_{ii} = Σ_{i=1}³ 𝓜_{ii} = Tr(𝓜)

Using dimensional symbols: L (length), T (time), M (mass)

Temporal Functional:

[T(Ω,t)] = [σ_θ] · [d³x] · [dτ]
         = T⁻¹ · L³ · T
         = L³  (or T when normalized)

Drift Production:

[σ_θ] = [𝒟] · [1 - ℒ]
      = T⁻¹ · 1
      = T⁻¹

LOAD Identity:

[γ_ITT] = [dt/dτ]
        = T/T
        = 1 (dimensionless)

All equations in this documentation use SI units unless otherwise specified.

In Planck units:

Simplifies equations but obscures physical magnitudes.

Common in general relativity. Length and time have same units.