The Temporal Functional

Rigorous Derivation and Analysis

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1. Motivation

In classical physics, time t is a parameter—an independent variable against which we measure change. In ITT, we invert this: change is fundamental, and Time is derived from measuring change.

The question becomes: How do we construct a well-defined functional that captures “accumulated change”?

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2. Construction of the Temporal Functional

2.1 Setup

Let:

• Ω ⊆ ℝ³ be a bounded, measurable region (the “system”)
• τ ∈ [t₀, t] be the recursion parameter
• σ_θ: ℝ³ × ℝ → ℝ⁺ be the drift production scalar

2.2 Definition

Definition (Temporal Functional):

T(Ω, t) = ∫_{t₀}^{t} ∫_{Ω} σ_θ(x, τ) d³x dτ

This is a Lebesgue double integral over space and recursion parameter.

2.3 Mathematical Type

The temporal functional is a map:

T: 𝒫(ℝ³) × ℝ → ℝ⁺

Where 𝒫(ℝ³) is the power set of ℝ³ (restricted to measurable sets).

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3. Well-Definedness Conditions

For the integral to exist and be finite, we require:

Condition: σ_θ ∈ L¹(Ω × [t₀, t])

The integral exists if:

1. Ω is bounded with finite measure
2. σ_θ is bounded: sup_{Ω×[t₀,t]} σ_θ < ∞
3. t – t₀ < ∞

Under these conditions:

T(Ω, t) ≤ ‖σ_θ‖_∞ · |Ω| · (t - t₀)

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4. Properties of the Temporal Functional

4.1 Non-Negativity

Theorem: T(Ω, t) ≥ 0 for all Ω, t ≥ t₀.

Proof: Since σ_θ ≥ 0 by Axiom 4, the integral of a non-negative function is non-negative. ∎

4.2 Monotonicity in Time

Theorem: If t₁ < t₂, then T(Ω, t₁) ≤ T(Ω, t₂).

Proof:

T(Ω, t₂) - T(Ω, t₁) = ∫_{t₁}^{t₂} ∫_{Ω} σ_θ d³x dτ ≥ 0

Since the integrand is non-negative. ∎

Corollary: Time never decreases. This is the arrow of Time expressed mathematically.

4.3 Additivity in Space

Theorem: For disjoint regions Ω₁ ∩ Ω₂ = ∅:

T(Ω₁ ∪ Ω₂, t) = T(Ω₁, t) + T(Ω₂, t)

Proof: Direct from additivity of the Lebesgue integral over disjoint domains. ∎

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5. The Instantaneous Time Production Rate

Definition (Local Time Production):

Ṫ_Ω(τ) := d/dτ T(Ω, τ) = ∫_{Ω} σ_θ(x, τ) d³x

This is the spatial integral of drift at fixed recursion parameter τ.

By the Fundamental Theorem of Calculus:

T(Ω, t) = T(Ω, t₀) + ∫_{t₀}^{t} Ṫ_Ω(τ) dτ

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6. The Point-Density Formulation

Definition (Temporal Density):

ρ_T(x, t) := ∫_{t₀}^{t} σ_θ(x, τ) dτ

This measures “how much Time has accumulated” at point x by recursion step t.

Relation to Functional:

T(Ω, t) = ∫_{Ω} ρ_T(x, t) d³x

Evolution Equation:

∂ρ_T/∂t = σ_θ(x, t)

Key Result: The rate of change of temporal density equals the drift scalar.

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7. Comparison with Classical Time

7.1 Standard Physics

In Newtonian mechanics:

• Time t is a parameter
• We write x(t), v(t), etc.
• Time is “given”, not derived

7.2 ITT Approach

In Intent Tensor Theory:

• Recursion parameter τ is the fundamental ordering
• Time T is computed from the drift functional
• Time is “earned” by irreversible change

7.3 Correspondence Principle

In the limit of uniform, constant drift:

σ_θ(x, t) = σ₀  (constant)

We get:

T(Ω, t) = σ₀ · |Ω| · (t - t₀)

If σ₀ = 1/|Ω|, then T = t – t₀, recovering classical time.

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8. Summary

The Temporal Functional:

T(Ω, t) = ∫_{t₀}^{t} ∫_{Ω} σ_θ(x, τ) d³x dτ

Key Properties:

1. Non-negative: T ≥ 0
2. Monotonic: T never decreases (arrow of Time)
3. Additive: Over disjoint regions and intervals
4. Derived: From the drift scalar σ_θ
5. Local: Integral of a local field

This completes the rigorous foundation of emergent Time in ITT.

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