What is a CP number? In terms of CP-Symmetry and CP-Violation?

Question

When researching the origins of CP-symmetry and CP-violation, and the Fitch-Cronin experiment demonstrating 'indirect' CP-violation, I came across these statements:

https://www2.lbl.gov/abc/wallchart/chapters/05/2.html

Until 1964 it was thought that the combination CP was a valid symmetry of the Universe. That year, Christenson, Cronin, Fitch and Turlay observed the decay of the long-lived neutral K meson, , to p + + p -. If CP were a good symmetry, the would have CP = -1 and could only decay to three pions, not two. Since the experiment observed the two pion decay, they showed that the symmetry CP could be violated. If CPT symmetry is to be preserved, the CP violation must be compensated by a violation of time reversal invariance. Indeed later experiments with K 0 systems showed direct T violations, in the sense that certain reaction processes involving K mesons have a different probability in the forward time direction (A + B Æ C + D) from that in the reverse time direction (C + D Æ A + B). Nuclear physicists have conducted many investigations searching for similar T violations in nuclear decays and reactions, but at this time none have been found.

And:

http://large.stanford.edu/courses/2008/ph204/coleman1/

The first experimental test of CP violation came in 1964 with the Fitch-Cronin experiment. The experiment involved particles called neutral K-mesons, which fortuitously have the properties needed to test CP. First, as mesons, they're a combination of a quark and an anti-quark, in this case down and antistrange, or anti-down and strange. Second, the two different particles have different CP values and different decay modes: K1 has CP = 1 and decays into two pions; K2 has CP = 1 and decays into three. Because decays with larger changes in mass occur more readily, the K1 decay happens 100 times faster than the K2 decay. This means that a sufficiently long beam of neutral Kaons will become arbitrarily pure K2 after a sufficient amount of time.

What does CP = 1 or CP = -1 mean, exactly?

Answer 1

Background. P is an operation that mirror-reflects a state in space, and C an operation that converts particles into antiparticles, reversing their charge, flavor, baryon #, lepton #, fermion #, and so on.

Their eigenvalues are $\pm 1$, meaning some states/particles stay the same under these operations, except for an overall — sign in the second option. Before the mid 50s, it had been established that the strong and electromagnetic interactions preserved P and C, that is the left-hand side and the right-hand side of a reaction/decay, had the eigenvalues of C and P match across these sides.

But then it was discovered in weak interactions that the P and C eigenvalues of the left-hand side and the right-hand side did not match. P and C were thus said to be violated, or equivalently, the weak interactions are not P and C symmetric. Nevertheless, CP was still apparently preserved in these reactions, as formally proposed by Landau. This means that the product of the C and P eigenvalues, the eigenvalue of CP (which is therefore also $\pm 1$) stayed the same across a reaction, thereby restoring order.

So, typically, the CP eigenvalue of a neutral K-short is +1, just like the CP of two pions, so it weakly decays to such, if CP is conserved (not violated). A different combination of the very same quarks in a different arrangement, the K-long, flips its sign under the action of CP, i.e. it has an eigenvalue -1, just like the three pions it decays to. (Since both Ks are neutral, they can be CP eigenstates, of course.)

Now, in the mid-60s, in an epochal, imaginative, & delicate experiment, it was discovered, that, at a fantastically small rate, some long Kaons nevertheless could, rarely, also decay to two pions, the hallmark of CP=+1 states... So there was either a piece of CP=+1 in K-long, after all, or something "happened" in the decay to violate CP. In both cases CP was realized to be a good-but-not-perfect symmetry of the weak interactions.

This was subsequently explained successfully in terms of quark mixing of three generations, but this is a longer story. CP violation in the early universe is thought to be responsible for our universe being mostly made out of matter, and not antimatter, quarks and electrons, as opposed to antiquarks and positrons.