What does this equation mean?

What a mask count actually costs is not one extra exposure. It is an extra circuit through the entire patterning loop: another resist coat, another exposure, another develop, in most multi-patterning schemes another etch and clean, and another round of the metrology that has to confirm the result before the next layer can be trusted to sit on top of it. If a critical layer’s target pitch is split across N separately printable exposures — as litho-etch-litho-etch and self-aligned multiple patterning both do, in different ways — and each exposure contributes an independent, zero-mean placement error of standard deviation $\sigma$ , the layer’s overlay-driven placement uncertainty combines in quadrature: $\sigma_N = \sigma\sqrt{N}$. Treat this as a simplified budgeting device rather than a physical law; real exposures on the same scanner and same reticle stage are not fully independent, and correlated error can be partly calibrated out. But the shape of the relation is the point. Going from one exposure to a four-way split does not merely quadruple the number of things that can go wrong in the abstract; under the idealised independence assumption it grows the placement-uncertainty budget by a factor of two, while genuinely multiplying the number of independent process passes — each with its own particle risk, its own drift, and its own measurement — through which the layer can fail. The International Roadmap for Devices and Systems’ 2023 lithography update draws the corresponding practical conclusion in its own language: “it has become clear that EUV will not replace multiple patterning. Both technologies will be used, and the choice of using multiple patterning or not will depend on level-specific details, yield, and cost” [ 3 ] . That is an institutional roadmap’s synthesis of the industry’s contributing members, not one vendor’s marketing position, and it is the closest thing to a consensus statement this comparison has.