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Agreed #0000 steel wool will not scratch chrome but it will further impregnate carbon into stainless steel causing it to rust even worse. Actually a really great way to clean chrome is a ball of aluminum foil, any light oil and a little elbow grease.
You cannot use carbon steel anything on stainless steel w/o out risking it rusting
The simplest condition under which rusting can occur on stainless steel is when a
piece of ordinary carbon or low-allysteel is rubbed against the surface of an otherwise
corrosion-resistant piece of stainless steel. See Figure 1. The iron from the ordinary
steel will rub off onto the stainless steel surface as a film of unalloyed steel, and, after
exposure to moisture in the atmosphere for a few days, that unalloyed steel film will
form ugly rust. This happens because the unalloyed steel film on the stainless steel
has little or no chromium, so the film transforms to ordinary “red rust."
I learned something. Thanks!
BTW, I remove all manner of surface mottling, haze, and other stuff from my stainless cookware using everyday non-abrasive copper cleaner, like Twinkle paste. I would try some of that. I discovered this use when cleaning the copper bottoms on my ancient Revereware saucepans and found that it works better than polishes and cleaners formulated specifically for SS.