process, machine, manufacture, or composition of matter, or any new and useful improvement thereof
A claim should fail 101 if it claims something that isn't one of the above. The were not claiming to invent a machine (the computer) They were not claiming to invent the process (hedging) They were not claiming to invent a specific improvement to a computer. (software added to the computer) Instead their claim was using a machine to solve a problem. That doesn't fall under 101 unless it falls under a patentable process. SCOTUS said that the 2 step process of (take exising process) and (apply it on a computer) is just a drafting trick to get a patent on old things. If you remove the drafting trick then you have a process with no steps. So they rejected it under 101.
Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof
So what do you call something that isn't a "process, machine, manufacture, or composition of matter"
SCOTUS gave it a name. They named it "abstract idea".
Using a computerto create and maintain “shadow” accounts amounts to electronic recordkeeping—one of the most basic functionsof a computer. See, e.g., Benson, 409 U. S., at 65 (noting that a computer “operates . . . upon both new and previously stored data”). The same is true with respect to the use of a computer to obtain data, adjust account balances,and issue automated instructions; all of these computer functions are “well-understood, routine, conventional activit[ies]” previously known to the industry. Mayo, 566 U. S., at ___ (slip op., at 4). In short, each step does nomore than require a generic computer to perform generic computer functions.
I think the above is one of the most important parts of the ruling. Basically stating that your patents stores data, or calculates something won't make it patent elligible if it wasn't patent elligible to start with. You could extend similer thinking to internet patents when talking about "well-understood, routine, convention activiteis"
Chemist who falls in acid is absorbed in work.