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Equation 7 · Frontier AI Model Comparisons: A First-Principles Introduction

What does this equation mean?

sobs=atrue+δcontam+εconfig,s_{\mathrm{obs}} = a_{\mathrm{true}} + \delta_{\mathrm{contam}} + \varepsilon_{\mathrm{config}},

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Inputs and operationsa_true + delta_contam + varepsilon_config
Result or conditions_obs
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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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sobss_{\mathrm{obs}}

Symbol s_obs

any observed score.

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atruea_{\mathrm{true}}

Symbol a_true

ata_true is one of the signed contributions combined to compute the quantity on the left.

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δcontam\delta_{\mathrm{contam}}

Symbol delta_contam

deltaca_contam is one of the signed contributions combined to compute the quantity on the left.

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εconfig\varepsilon_{\mathrm{config}}

Symbol varepsilon_config

noise contributed by harness, prompt, effort level, and grading choices, which can push the observed score in either direction.

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=

=

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

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addition

addition

Add the term after the plus sign to the term or group before it.

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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 slightly more general way to see the same point: any observed score can be written as sobs=atrue+δcontam+εconfigs_{\mathrm{obs}} = a_{\mathrm{true}} + \delta_{\mathrm{contam}} + \varepsilon_{\mathrm{config}}. where atruea_{\mathrm{true}} is the capability the benchmark intends to measure, δcontam\delta_{\mathrm{contam}} ≥\geq 0 is a one-directional inflation term from training-set overlap, and εconfig\varepsilon_{\mathrm{config}} is noise contributed by harness, prompt, effort level, and grading choices, which can push the observed score in either direction. Two systems’ sobss_{\mathrm{obs}} values can differ substantially even when their atruea_{\mathrm{true}} values are identical, and can appear equal even when their atruea_{\mathrm{true}} values are not, purely through the other two terms. Nothing about this requires…
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A slightly more general way to see the same point: any observed score can be written as sobs=atrue+δcontam+εconfigs_{\mathrm{obs}} = a_{\mathrm{true}} + \delta_{\mathrm{contam}} + \varepsilon_{\mathrm{config}}. where atruea_{\mathrm{true}} is the capability the benchmark intends to measure, δcontam\delta_{\mathrm{contam}} ≥\geq 0 is a one-directional inflation term from training-set overlap, and εconfig\varepsilon_{\mathrm{config}} is noise contributed by harness, prompt, effort level, and grading choices, which can push the observed score in either direction. Two systems’ sobss_{\mathrm{obs}} values can differ substantially even when their atruea_{\mathrm{true}} values are identical, and can appear equal even when their atruea_{\mathrm{true}} values are not, purely through the other two terms. Nothing about this requires bad faith on anyone’s part; it follows from the measurement being taken at all.

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