Law extended problems (resonance framework)

Problem 4 – Multi-tier precedent resonance#

A legal system models three tiers of precedent:

$$ W_1 = D_3 ΛΘ, \quad W_2 = D_6 τ_r, \quad W_3 = X \sqrt{τ_r}. $$

The total precedent influence is

$$ W_{\text{tot}} = W_1 + W_2 + W_3. $$

If $$τ_r$$ increases by 20%, how does each term change qualitatively, and what is the net effect on $$W_{\text{tot}}$$?

Problem 5 – Regulatory compliance resonance#

Compliance cost is modeled as

$$ C = \frac{D_9 + ΛΘ}{τ_r}. $$

1. If $$τ_r$$ increases, how does $$C$$ change?
2. If $$ΛΘ$$ increases due to stricter oversight, what is the effect on $$C$$?

Problem 6 – Burden of proof resonance#

A burden-of-proof index is defined as

$$ B = X \ln(1 + D_3 τ_r). $$

If $$τ_r$$ increases by 30%, how does the burden change qualitatively?

Problem 7 – Procedural harmonics#

A procedural step has harmonic timing

$$ H = \frac{T_f^2}{D_6 + τ_r}. $$

1. If $$T_f$$ increases by 10%, how does the numerator change?
2. If $$τ_r$$ increases by 10%, how does the denominator change?
3. What is the qualitative net effect on $$H$$?

Problem 8 – Evidence decay under resonance#

Evidence reliability decays as

$$ R(t) = e^{-ΛΘ t / τ_r}. $$

1. If $$τ_r$$ increases, does evidence decay faster or slower?
2. If $$ΛΘ$$ increases due to environmental stress, what happens to the decay rate?