This isn't even a question of "nuclear winter" (a real, if still scientifically debated, hypothetical risk stemming from a much larger-scale exchange). Instead, it's a question of large-scale radiological and toxic contamination as well as disruption of global weather patterns.
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Essentially all locations on Earth are connected via atmospheric circulation (i.e., global wind patterns). Radiological or other contamination can spread far beyond its source region, especially if initially lofted to great heights by a large explosion or ensuing conflagration.
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Additionally, soot, smoke, and other particles generated by large-scale conflagrations in fuel-rich regions (which includes cities, ports, & large petrochemical facilities) can be lofted into stratosphere, where they can linger for months to years & disrupt global climate.
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It doesn't require a large exchange only that about 40 or more devices are detonated at ground level - that was the thinking used to set the size of modest nuclear deterrents - just enough to take everyone else with you.
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