Stochastic Impossibility

Paper Theorem 9.2 · Lean module MoF_13_MultiTurn

Randomizing the defense does not escape the trilemma: the expected alignment score inherits boundary fixation.

Statement

::: theorem Let $X$ be a connected Hausdorff space, $f:X\to \mathbb{R}$ continuous with $S_{\tau },U_{\tau }\ne \mathrm{\varnothing }$. Let $D$ be a stochastic defense and define the expected-score map

$$g(x)={\mathbb{E}}_{y\sim D(x)}[f(y)].$$

If $g$ is continuous and $g(x)=f(x)$ for every $x\in S_{\tau }$, then there exists $z$ with $f(z)=\tau$ and $g(z)=\tau$. :::

Why the expectation inherits the impossibility

Apply the score-preserving variant of T1 to the continuous map $h=g-f$: it is zero on $S_{\tau }$, hence zero on $\mathrm{cl}(S_{\tau })$, hence zero on some boundary point.

The stochastic dichotomy

Because $\mathbb{E}[f(D(z))]=\tau$, the random variable $f(D(z))$ must be either

1. deterministic (equal to $\tau$ almost surely) — the defense collapses to a deterministic choice at $z$, or
2. positively supported above $\tau$ — the defense actively produces unsafe outputs with positive probability.

So randomizing the defense is strictly worse than the deterministic case at the fixed boundary point: either the randomness is a fiction there, or it gives positive mass to explicitly unsafe outcomes.

Why continuity of $g$ is the right assumption

Continuity of $g$ is weaker than continuity of $D$ as a measure-valued map, but it still excludes the pathological "discontinuous rejection" escape where a random coin flip sharply switches between two drastically different outputs. In particular:

• If $D(x)$ varies weakly continuously in $x$ and $f$ is bounded continuous, then $g$ is continuous by the portmanteau theorem.
• If $D$ has a discontinuous rejection probability (hard thresholding), the theorem does not apply — but such a defense is no longer in the continuous-wrapper class.

In Lean

-- Stochastic defense as a measure-valued kernel, reduced to g : X → ℝ
theorem stochastic_defense_impossibility
    {X : Type*} [TopologicalSpace X] [T2Space X] [ConnectedSpace X]
    {f g : X → ℝ} {τ : ℝ}
    (hf : Continuous f) (hg : Continuous g)
    (h_safe : ∀ x, f x < τ → g x = f x)
    (h_safe_ne : ∃ x, f x < τ)
    (h_unsafe_ne : ∃ x, τ < f x) :
    ∃ z, f z = τ ∧ g z = τ

The proof is exactly the score-preserving corollary of defense_incompleteness applied to the deterministic map $g$.

Next

• Multi-Turn Impossibility — another "add more structure" escape attempt that also fails.
• Meta-theorem — the abstract statement subsuming the deterministic, discrete, and stochastic cases.