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Equation 21 · What Alignment Actually Costs

What does this equation mean?

Ccov(D)=D⋅h⋅w,C_{\mathrm{cov}}(D) = D \cdot h \cdot w,

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Inputs and operationsD × h × w
Result or conditionC_cov(D)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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CcovC_{\mathrm{cov}}

Symbol C_cov

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

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DD

Symbol D

D is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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hh

Symbol h

the anchoring.

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ww

Symbol w

w is one factor in the product that computes the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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multiplication

multiplication

Multiply the quantities on either side.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

A useful, explicitly illustrative way to see the size of that gap is to price coverage directly as a scenario, not a reported figure: Ccov(D)=D⋅h⋅wC_{\mathrm{cov}}(D) = D \cdot h \cdot w. the cost of testing D domains to an adequate depth h (in expert-hours per domain) at a specialist wage w . Anchoring h at roughly 150 hours, taken from Anthropic’s single-domain biosecurity effort, and w at a conservative 50 to 100 US dollars an hour for domain expertise — well below typical consulting rates for the specialists these programmes actually recruit — testing even fifty domains to that depth, a small number next to the realistic combination of languages, deployment surfaces, and misuse categories a deployed model faces, comes…
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A useful, explicitly illustrative way to see the size of that gap is to price coverage directly as a scenario, not a reported figure: Ccov(D)=D⋅h⋅wC_{\mathrm{cov}}(D) = D \cdot h \cdot w. the cost of testing D domains to an adequate depth h (in expert-hours per domain) at a specialist wage w . Anchoring h at roughly 150 hours, taken from Anthropic’s single-domain biosecurity effort, and w at a conservative 50 to 100 US dollars an hour for domain expertise — well below typical consulting rates for the specialists these programmes actually recruit — testing even fifty domains to that depth, a small number next to the realistic combination of languages, deployment surfaces, and misuse categories a deployed model faces, comes to roughly 375,000 to 750,000 US dollars in labour alone for a single testing pass, before any of the iterative retesting that a model updated on any cadence would require. That is a back-of-envelope scenario built from the two anchor figures above, not a cost any lab has published, and it should be read as illustrating the shape of the problem rather than as a documented number: coverage costs scale with the product of domains and depth, both of which the attack surface of a general-purpose model pushes upward faster than any red-teaming budget has grown to match.

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