ToV Proofs — Harm Attenuation Theorem (T10.2)
The Attenuation Theorem: under the Markov assumption, harm probability decreases exponentially with protective depth. Compute harm propagation, attenuation rates, protective depth requirements, and ve
5 toolstov-proofs
Harm Probability
Compute combined harm probability from a chain of propagation probabilities. Under Axiom 5 (Markov), ℙ(H|δs₁) = ∏ᵢP_{i→i
probabilities
Exponential Harm
Compute harm probability using exponential decay formula ℙ(H) = e^{-α(H-1)} where α is the attenuation rate and H is the
alphaharm_level
Protective Depth
Compute minimum protective depth H such that ℙ(H) < target_probability. Formula: H ≥ 1 + log(1/ε)/α. Answers: 'how many
target_probabilityattenuation_rate
Verify Attenuation
Verify the attenuation property holds: harm probability strictly decreases with each additional protective layer. Return
probabilities
Attenuation Rate
Compute the attenuation rate α = -log(P̄) where P̄ is the geometric mean of propagation probabilities. Higher α means st
probabilities