Formal Derivation

Chapter 2: Formal Derivation

Mathematical Formalization of S_θ, σ_θ, and Time

2.1 The Recursive Entropy Functional

We define Recursive Entropy (S_θ) as the integral of the unbinding scalar over all space and recursion depth:

S_θ = ∫_Ω Σ_n=0^n_max σ_θ(x,n) d³x

Components:

Ω: Spatial domain (the region of interest)
n: Discrete recursion depth
n_max: Maximum recursion depth (Planck ceiling)
σ_θ(x,n): Local unbinding scalar at position x, depth n

Continuous form (using recursive time τ):

S_θ = ∫_Ω ∫₀^τ_max σ_θ(x,τ) dτ d³x

2.2 Derivation of σ_θ from Drift Divergence

The unbinding scalar emerges from the failure of recursive closure:

Step 1: Define the Ideal State

A perfect recursive step satisfies:

R̂(Ψ_n) = Ψ_n+1^ideal

Step 2: Define the Actual State

In reality, recursive operations produce:

Ψ_n+1^actual = Ψ_n+1^ideal + δΨ_n+1

Where δΨ represents the deviation from ideal recursion.

Step 3: Drift as Deviation Rate

𝒟(x,n) = α_M ‖∂_n ℳ_ij‖_F + α_Φ ‖∂_n ∇Φ‖₂

Step 4: Lock as Alignment Quality

ℒ(x,n) = ⟨Ψ_n+1^actual | Ψ_n+1^ideal⟩ / ‖Ψ_n+1^ideal‖²

Step 5: Entropy as Residue

σ_θ(x,n) = 𝒟(x,n) · (1 − ℒ(x,n))

2.3 Time-Linked Interpretation

Entropy Produces Time

The temporal functional from the Time Scroll:

T[Ψ] = ∫_Ω σ_θ d³x · t_P

Time Measures Entropy

Inverting the relationship:

T = ∫ dS_θ / σ_θ

Interpretation:

Time is the “bookkeeping” of entropy production
Where σ_θ → 0, time halts
Where σ_θ is large, time flows quickly

2.4 The Second Law (ITT Form)

From the definition of σ_θ and the properties of ℒ:

Claim: dℒ/dn ≤ 0 (lock degrades over recursion)

Consequence:

dℒ/dn ≤ 0 ⟹ d(1−ℒ)/dn ≥ 0 ⟹ dσ_θ/dn ≥ 0 ⟹ dS_θ/dn ≥ 0

This is the ITT Second Law: Entropy tends to increase through recursion, not as a statistical tendency, but as a structural necessity.

2.5 Summary of Key Equations

Equation	Name	Form
Unbinding scalar	σ_θ	𝒟(1 − ℒ)
Drift magnitude	𝒟	α_M ‖∂_n ℳ_ij‖_F + α_Φ ‖∂_n ∇Φ‖₂
Recursive entropy	S_θ	∫_Ω Σ σ_θ(x,n) d³x
Time functional	T	∫ dS_θ / σ_θ
Second Law	—	dℒ/dn ≤ 0 ⟹ dS_θ/dn ≥ 0
Information residual	I	I₀ − S_θ

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