Intent-Tensor-Theory

Time Dilation: LOAD Identity

Time Dilation: The LOAD Identity

Substrate Resource Allocation and Temporal Slowdown

1. The LOAD Identity

1.1 Statement

γ_ITT = dt/dτ = √(1 - 𝒜² · Tr(𝓜)/Tr(𝓜)_max)

Where:

  • γ_ITT: Dilation factor (dimensionless)
  • dt: Increment of emergent (macroscopic) time
  • dτ: Increment of recursion parameter
  • 𝒜: Alignment functional
  • Tr(𝓜): Trace of Memory Tensor
  • Tr(𝓜)_max: Maximum possible trace

1.2 Naming

LOAD = Lock On Alignment Dilation

The identity expresses how substrate “load” (computational resource allocation) causes temporal dilation.

2. Physical Interpretation

2.1 The Mechanism

When the Collapse Tension Substrate maintains high alignment (𝒜 → 1):

  1. Recursive resources are allocated to preserving coherence
  2. Fewer resources remain for state updates
  3. The update frequency decreases
  4. Time “slows down”

2.2 Analogy

Think of a computer running multiple processes:

  • Background task: Maintaining memory coherence
  • Foreground task: Processing new states

When the background task (alignment) demands more resources, the foreground task (temporal production) slows.

3. Derivation

3.1 Resource Constraint

Assume total computational resource is normalized to 1:

R_update + R_lock = 1

3.2 Lock Resource Demand

The resource demand for locking scales with alignment and memory depth:

R_lock = 𝒜² · Tr(𝓜)/Tr(𝓜)_max

Justification:

  • 𝒜²: Quadratic in alignment (small misalignments require little maintenance)
  • Tr(𝓜)/Tr(𝓜)_max: Normalized memory depth (deeper memory requires more maintenance)

3.3 Time Dilation Factor

The rate of time production scales as square root of available resources:

γ_ITT = √(R_update) = √(1 - 𝒜² · Tr(𝓜)/Tr(𝓜)_max)

4. Comparison with Relativistic Dilation

Theory“Potential” XPhysical Source
Special Relativityv²/c²Kinetic energy
General Relativity2|φ|/c²Gravitational potential
ITT𝒜² · Tr(𝓜)/Tr(𝓜)_maxAlignment load

All have the same mathematical structure:

γ = √(1 - X)

5. Boundary Cases

Case𝒜Tr(𝓜)γ_ITTPhysical Meaning
Free flow0any1No alignment → no dilation
Shallow memoryany01No memory → no dilation
Maximum load1Tr(𝓜)_max0Complete stop

Singularity Condition

Time stops (γ_ITT = 0) when:

𝒜 = 1  AND  Tr(𝓜) = Tr(𝓜)_max

Perfect alignment with maximum memory depth.

6. Worked Examples

Example: Low Alignment

Given: 𝒜 = 0.1, μ = 0.5

γ_ITT = √(1 - (0.1)² · 0.5) = √(0.995) ≈ 0.9975

Interpretation: Negligible dilation (0.25% slowdown).

Example: High Alignment

Given: 𝒜 = 0.9, μ = 0.8

γ_ITT = √(1 - (0.9)² · 0.8) = √(0.352) ≈ 0.593

Interpretation: Significant dilation (40% slowdown).

7. Summary

The LOAD Identity:

γ_ITT = dt/dτ = √(1 - 𝒜² · Tr(𝓜)/Tr(𝓜)_max)

Key Results:

  1. Mechanism: Resource allocation to alignment reduces update bandwidth
  2. Form: Square root of (1 − load fraction)
  3. Correspondence: Parallels relativistic time dilation
  4. Range: γ_ITT ∈ [0, 1]
  5. Singularity: Time stops at perfect alignment with maximum memory

This identity explains why clocks run slow in ITT: not because of spacetime geometry, but because of substrate load.

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