A Projected-Hessian Account of CKM Amplitudes, the C₃ Triangle Residue, Phase Provenance, and the Remaining Empirical Tensions
This paper is a careful stocktake of how far the VERSF quark-mixing programme has really got. Quarks come in three generations, and the weak force allows them to shift between those generations in very specific amounts. The Standard Model measures those amounts very accurately, but mostly treats them as numbers to be inserted. This paper asks a harder question: can VERSF explain the pattern, especially the rare first-to-third generation mixing and the small matter–antimatter imbalance that comes with it?
The previous paper, The Projected C₃ Hessian Return in VERSF, did the local calculation. It looked inside a minimal three-generation symmetry cell and showed that the relevant curvature is shared democratically across the three generation pairings. In simpler terms, it found the little “twist” in the quark family structure that could feed into the missing first-to-third mixing. It also showed that this twist, when combined with the already-known Cabibbo doorway between the first two generations, produces the right kind of correction rather than simply adding a number by hand.
This new paper takes the next step. It does not just say “we found the twist.” It asks what the whole quark-mixing sector now looks like if that prior result is accepted. The answer is a ledger: which numbers are inherited from earlier work, which pieces are newly returned by the C₃ Hessian calculation, which consequences follow exactly, which parts match experiment, and which parts still need proof. That is why the paper is valuable: it turns a complicated derivation into an auditable scorecard.
The encouraging result is that the amount of CP violation — the tiny asymmetry between matter and antimatter — comes out very close to the observed value, and the main triangle angles are mostly in the right region. But the paper is also honest about the remaining tension: one angle, called gamma, is still too high, and the strongest success depends heavily on a phase choice whose deeper origin must be independently proven. If that phase was chosen because it makes the answer work, the result would be much weaker. If VERSF can prove that phase from closure geometry alone, the result becomes much more powerful.
So the paper does not claim final victory. It claims something more disciplined: the quark-mixing problem has been reduced to a short list of named tests. The prior paper supplied the local C₃ curvature return; this paper assembles it into the full quark-sector ledger and shows exactly where the programme is strong, where it is exposed, and what must be proven next. That is real progress because the remaining work is no longer vague — it is local, named, and falsifiable.