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):
- Recursive resources are allocated to preserving coherence
- Fewer resources remain for state updates
- The update frequency decreases
- 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 = 13.2 Lock Resource Demand
The resource demand for locking scales with alignment and memory depth:
R_lock = 𝒜² · Tr(𝓜)/Tr(𝓜)_maxJustification:
- 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” X | Physical Source |
|---|---|---|
| Special Relativity | v²/c² | Kinetic energy |
| General Relativity | 2|φ|/c² | Gravitational potential |
| ITT | 𝒜² · Tr(𝓜)/Tr(𝓜)_max | Alignment load |
All have the same mathematical structure:
γ = √(1 - X)5. Boundary Cases
| Case | 𝒜 | Tr(𝓜) | γ_ITT | Physical Meaning |
|---|---|---|---|---|
| Free flow | 0 | any | 1 | No alignment → no dilation |
| Shallow memory | any | 0 | 1 | No memory → no dilation |
| Maximum load | 1 | Tr(𝓜)_max | 0 | Complete stop |
Singularity Condition
Time stops (γ_ITT = 0) when:
𝒜 = 1 AND Tr(𝓜) = Tr(𝓜)_maxPerfect 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) ≈ 0.9975Interpretation: 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) ≈ 0.593Interpretation: Significant dilation (40% slowdown).
7. Summary
The LOAD Identity:
γ_ITT = dt/dτ = √(1 - 𝒜² · Tr(𝓜)/Tr(𝓜)_max)Key Results:
- Mechanism: Resource allocation to alignment reduces update bandwidth
- Form: Square root of (1 − load fraction)
- Correspondence: Parallels relativistic time dilation
- Range: γ_ITT ∈ [0, 1]
- 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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