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Equation 10 · Part 4 · Running an Interpretability Investigation That Holds Up

=

S(C,C′)=Δ(C)−Δ(C′).S(\mathcal{C}, \mathcal{C}') = \Delta(\mathcal{C}) - \Delta(\mathcal{C}').
=

What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

The defence against this is a control condition run alongside the target one, not after it: intervene identically on a component set C\mathcal{C}' matched to the target set C\mathcal{C} in size, depth and baseline activation statistics, but which the hypothesis explicitly predicts should have no effect, and compare S(C,C′)=Δ(C)−Δ(C′)S(\mathcal{C}, \mathcal{C}') = \Delta(\mathcal{C}) - \Delta(\mathcal{C}'). A specificity margin S that is small relative to the run-to-run variance means the intervention is detecting generic sensitivity to perturbation somewhere in the network, not the mechanism under test — which is exactly the failure mode the subspace illusion exhibits. Reporting Δ(C)\Delta(\mathcal{C}) alone, without the matched control alongside it, is reporting half…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.