The twistor lift of TGD led to the introduction of Kähler form also in M4 factor of imbedding space M4×CP2. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the self-duality of J(M4) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.
Can one say anything quantitative about these various breakings?
The determinant of CKM matrix is equal to phase factor by unitarity (UU†=1) and its imaginary part characterizes CP breaking. The imaginary part of the determinant should be proportional to the Jarlskog invariant J= +/- Im(VusVcbV*ub V*cs) characterizing CP breaking of CKM matrix (see this).
The recent experimental estimate is J≈ 3.0× 10-5. J/X≈ 0 .1 so that there is and order of magnitude deviation. Earlier experimental estimate used in p-adic mass calculations was by almost order of magnitude larger consistent with the value of X. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does nto distinguish between different meson so that it is better to talk about orders of magnitude only.
The parameter used to characterize matter antimatter asymmetry (see this) is the ratio R=[n(B-n(B*)]/n(γ)) ≈ 9× 10-11 of the difference of baryon and antibaryon densities to photon density in cosmological scales. One has X3 ≈ 1.4 × 10-11, which is order of magnitude smaller than R.
For a summary of earlier postings see Latest progress in TGD.