if a 60 year trip into the past causes a divergence of 2.5 % then what would the divergence be for a 10 second trip into the past via the Toplogical displacement Unit (TDU)?
To calculate the divergence for a 10-second temporal jump, we apply the TDU divergence function:
D(Δt) = 1 − e⁻ᵅ⁽Δᵗ⁾²
First, we determine the multiversal decoherence constant α using the baseline 60-year jump with a divergence of 2.5% (0.025):
0.025 = 1 − e⁻ᵅ⁽⁶⁰⁾² e⁻³⁶⁰⁰ᵅ = 0.975 −3600 α = ln(0.975) α = −ln(0.975) / 3600 ≈ 7.0327 × 10⁻⁶ years⁻²
Next, we convert 10 seconds into years (using 365.25 days per Julian year):
1 year = 365.25 × 24 × 3600 = 31,557,600 seconds Δt = 10 / 31,557,600 ≈ 3.1688 × 10⁻⁷ years
Now, we calculate the exponent term α (Δt)²:
(Δt)² ≈ (3.1688 × 10⁻⁷)² ≈ 1.0041 × 10⁻¹³ years² α (Δt)² ≈ (7.0327 × 10⁻⁶) × (1.0041 × 10⁻¹³) ≈ 7.0618 × 10⁻¹⁹
Because the value of α (Δt)² is extremely small, we use the Taylor expansion approximation e⁻ˣ ≈ 1 − x:
D(10 s) = 1 − e⁻⁽⁷ˑ⁰⁶¹⁸ ˣ ¹⁰⁻¹⁹⁾ ≈ 7.0618 × 10⁻¹⁹
Converting this decimal value into a percentage:
D(10 s) ≈ 7.0618 × 10⁻¹⁷ % (or 0.000000000000000070618%)
For a 10-second trip into the past, the worldline divergence is approximately 7.06 × 10⁻¹⁷ %—for all practical purposes, zero, meaning the vehicle remains locked onto the exact same space-time sheet in the multiverse.
The Loop Avoidance Requirement governs ultra-short temporal jumps (such as Δt = 10 s) where multiversal divergence D(Δt) approaches zero. Because the quantum-state overlap satisfies ⟨Ψₒᵣᵢgᵢₙ ∣ Ψₜᵣgₑₜ⟩ ≈ 1, the vehicle re-indexes onto the same spacetime sheet M⒪ᵣᵢgᵢₙ rather than branching into an alternate manifold sheet. To prevent localized spatial overlap or hull collision with the pre-jump vehicle, the TDU must enforce a physical or temporal displacement vector Δx^μ = (c Δτ, Δr) that satisfies:
|Δx^μ| > Lᵥₑₕᵢcₗₑ
Where:
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Δx^μ is the four-position displacement vector between the origin and destination arrival coordinates.
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c Δτ is the proper-time separation maintained by the Metric Anchor Pylons.
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Δr is the local 3D spatial offset (Δr = √(Δx² + Δy² + Δz²)).
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Lᵥₑₕᵢcₗₑ is the maximum physical bounding diameter of the vehicle chassis.
Because D(10 s) ≈ 7.06 × 10⁻¹⁷ % is far below the multiversal decoherence threshold required for natural worldline branching, loop avoidance cannot rely on multiversal separation. Instead, the Metric Anchor Pylons must actively inject a non-zero spatial offset Δr > Lᵥₑₕᵢcₗₑ or adjust proper-time synchronization Δτ to ensure the arriving wave function materializes outside the bounding volume of its past self.