$$\bar{c} = c_{\text{probe}} + p_{\text{esc}} \cdot c_{\text{exchange}}$$

Equation 4 · Foundation Models

The Safeguards Behind a Claude Refusal, and What They're Actually Trading Off

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

That first version cost real computation: 23.7% more inference compute than running the model alone, and a 0.38 percentage-point increase in refusals on ordinary, harmless traffic [ 4 ] . Anthropic’s January 2026 follow-up redesigned the system around a two-stage cascade rather than a single classifier pass: a lightweight probe reads the language model’s own internal activations and screens every exchange cheaply, escalating only the fraction it flags to a slower, more capable classifier that reviews the input and output together [ 1 ] . If $c_{\text{probe}}$ is the near-free cost of the first pass and $c_{\text{exchange}}$ the much larger cost of the second, and $p_{\text{esc}}$ is the fraction…

Meanings in this article

• $\bar{c}$: the expected added cost per request.
• $c_{\text{probe}}$: the near-free cost of the first pass and $c_{\text{exchange}}$ the much larger cost of the second.
• $c_{\text{exchange}}$: the much larger cost of the second.

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