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PATIENCE MEASUREMENT DOMAIN

**Patience ($F_{\text{Patience}}$):** Patience is measurable as the temporal horizon of coherent planning and the capacity to maintain coherence across delayed gratification intervals. It is the temporal extension of coherence.

Connections

Assumes

  • None

Enables

  • None
Physics Layer

The Patience Operator

\hat{F}_{\text{Patience}} = \int_0^{\infty} e^{-\delta t} \hat{U}^\dagger(t) \hat{C} \hat{U}(t) \, dt

Where \hat{U}(t) = e^{-i\hat{H}t} is the time evolution operator and \hat{C} is the coherence operator.

This measures the time-integrated coherence under evolution, discounted by \delta.

Mathematical Layer

Formal Definition

Definition (Patience Metric): Let \mathcal{A} be an agent with temporal modeling capacity. The Patience metric is:

F_{\text{Patience}}(\mathcal{A}) = \int_0^{\infty} e^{-\delta t} \cdot \mathbb{E}[C(\mathcal{A}, t) | \mathcal{A}(0)] \, dt

Where:

  • \delta > 0 is the discount rate (estimated from behavior)
  • \mathbb{E}[C(\mathcal{A}, t) | \mathcal{A}(0)] is expected future coherence given current state
Defeat Conditions

To Falsify This

  1. **Patience Without Long Horizons:** Demonstrate genuine patience in agents with provably zero future modeling capacity. This would show patience is not temporal.
  2. **Long Horizons Without Patience:** Show agents with extensive future modeling that consistently exhibit impatience. This would break the equivalence.
  3. **Impulsivity as Optimal:** Prove that maximal coherence is achieved through zero-delay gratification. This would invert the patience-coherence relationship.
  4. **Patience Independent of Coherence:** Demonstrate patience states that have no correlation with maintained coherence over time. This would decouple patience from coherence preservation.