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We use the approximation method of Razborov to analyze the locality barrierwhich arose from the investigation of the hardness magnification approach tocomplexity lower bounds. Adapting a limitation of the approximation methodobtained by Razborov, we show that in many cases it is not possible to combinethe approximation method with typical (localizable) hardness magnificationtheorems to derive strong circuit lower bounds. In particular, one cannot usethe approximation method to derive an extremely strong constant-depth circuitlower bound and then magnify it to an $NC^1$ lower bound for an explicitfunction.  To prove this we show that lower bounds obtained by the approximation methodare in many cases localizable in the sense that they imply lower bounds forcircuits which are allowed to use arbitrarily powerful oracles with smallfan-in.

Localizability of the approximation method