Notation and Units

Notation and Units

Complete Symbol Reference for The Entropy Scroll

Core Entropy Symbols

Symbol	Name	Domain	Definition
σ θ	Unbinding scalar	[0, ∞)	σθ = 𝒟(1 − ℒ)
S θ	Recursive entropy	[0, Sθ,max]	Sθ = ∫Σ σθ d³x
S θ,max	Entropy ceiling	Fixed	nmax · ℓP² · Nfolds
𝒮	Saturation ratio	[0, 1]	𝒮 = Sθ / Sθ,max

Drift-Lock Components

Symbol	Name	Domain	Definition
𝒟	Drift magnitude	[0, ∞)	αM‖∂nℳij‖F + αΦ‖∂n∇Φ‖2
ℒ	Shell-lock	[0, 1]	Recursive coherence measure
α M	Memory coupling	ℝ⁺	Weight for memory drift
α Φ	Intent coupling	ℝ⁺	Weight for intent drift

Glyph Field Stack

Symbol	Name	Type	Role
Φ	Intent potential	Scalar field	Recursive direction encoding
C i	Curvent field	Vector field	Recursive flow vector
ℳ ij	Memory tensor	Symmetric tensor	State coherence storage
𝒜	Alignment scalar	Scalar [-1, 1]	𝒜 = ⟨Ci, ∇Φ⟩ / (‖C‖·‖∇Φ‖)

Recursion Parameters

Symbol	Name	Domain	Meaning
n	Recursion depth	[0, nmax]	Current fold number
n max	Maximum depth	ℕ	Planck ceiling on recursion
τ	Recursive time	[0, τmax]	Continuous recursion parameter
R̂	Recursive operator	Functional	R̂(Ψn) = Ψn+1

Symbol	Name	Definition
T	Time functional	T = ∫ dSθ/σθ
T[Ψ]	Temporal functional	T[Ψ] = ∫ σθ d³x · tP
γ ITT	Time dilation	γITT = √(1 − 𝒜²·Tr(ℳ)/Tr(ℳ)max)
t P	Planck time	5.39 × 10⁻⁴⁴ s

Temperature Symbols

Symbol	Name	Definition
T ITT	ITT temperature	TITT = T0 · σθ
T 0	Reference temperature	Planck temperature
T P	Planck temperature	1.42 × 10³² K

Thermodynamic Mapping

ITT Symbol	Classical Analog	Relation
Sθ	S (Boltzmann)	Sθ = kσ ln WITT
σθ	dQ/T	σθ dτ ↔ dQ/T
TITT	T	TITT = T0 · 𝒟(1−ℒ)
WITT	W (microstates)	WITT = exp(∫ 𝒟(1−ℒ) dV)

Fundamental Constants

Symbol	Name	Value	Role in ITT
ℓ P	Planck length	1.62 × 10⁻³⁵ m	Substrate resolution
t P	Planck time	5.39 × 10⁻⁴⁴ s	Minimum time step
ℓ P²	Planck area	2.61 × 10⁻⁷⁰ m²	Entropy quantum
ℏ	Reduced Planck	1.05 × 10⁻³⁴ J·s	Action quantum
G	Newton’s constant	6.67 × 10⁻¹¹ N·m²/kg²	Gravity strength
c	Speed of light	3 × 10⁸ m/s	Causality limit

Key Equations Summary

The Fundamental Four

Name	Equation
Unbinding identity	σ θ = 𝒟(1 − ℒ)
Entropy functional	S θ = ∫Ω Σ σθ(x,n) d³x
Time from entropy	T = ∫ dS θ/σθ
Entropy ceiling	S θ,max = nmax · ℓP² · Nfolds

The Second Law (ITT Form)

dℒ/dn ≤ 0 ⟹ dS θ/dn ≥ 0

The LOAD Identity

γ ITT = √(1 − 𝒜² · Tr(ℳ)/Tr(ℳ)max)

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