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It should not leave a scratch pattern, just a hazy, Kasumi finish. If it is scratching it may not be smooth enough.

Putting some flat carbon steel to it with pressure and some dish soap and water...

There are many long threads on Arks for razor use, basically lap flat with loose Silicone Carbide, beginning with 60 grit and finish on Wet & Dry up to 1 or 2k. Then, burnish with carbon steel, large...