You say "It appears to me that the people on this thread arguing that 1!=0.9999... have come up with a fully consistent, alternative way of defining equality...

You say "It appears to me that the people on this thread arguing that 1!=0.9999... have come up with a fully consistent, alternative way of defining equality..." but if you look, there are absolutely no mathematical proofs outside of the people showing that 1=0.9r. Instead, there are people blankly asserting that there's a 0.0...1 remainder, which as many have pointed out, is meaningless without a precise mathematical definition and perhaps a proof of existence within the set of reals. And the reason it's meaningless is because it's ill-defined and misunderstands infinite series, etc... I'm no math whiz, but the reasoning on display here isn't just sloppy, it's totally perpendicular to mathematical reasoning. Arguing against it is like trying to argue with someone who asserts that "the sunset tastes exuberant." The statement may look a lot like a meaningful phrase, but it's just nonsense.