Equation 4 · Part 1 · The Safeguards Behind a Claude Refusal, and What They're Actually Trading Off

Symbol barc

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

\bar{c}

What this part means

the expected added cost per request.

Its job in the formula

barc is part of the quantity the equation computes from the expression on the right.

Full expressionSymbol barcArticle meaning

Where the article explains it

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 of traffic the probe escalates, the expected added cost per request is

The passage around this formula

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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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