Finite-Regulator Positive-Pushforward No-Go Theorem and Pass Programme
This paper is the sixth of the final eight papers in the VERSF Standard Model programme, and it tackles a different kind of problem from the first five. The earlier papers focused on the physical content of the theory: the electroweak vacuum and Higgs sector, neutrino structure, the Yukawa and mass matrices, the strengths of the gauge forces, and the emergence of the strong-interaction scale and confinement architecture. This paper asks the next unavoidable question: even if all of those ingredients are locally correct, do they fit together into one globally consistent quantum theory?
That matters because a gauge theory is not fully defined just by writing down equations that work near one ordinary field configuration. The theory must remain consistent across every allowed topological sector, every large gauge transformation and every globally distinct bundle of fields. It must also cope with the fact that non-Abelian gauge fields cannot generally be reduced to one unique global gauge choice. This paper therefore builds the global stage on which the previous results must live: a measure over physically distinct configurations, a treatment of the faithful Standard Model gauge group, a globally consistent fermion determinant, and a positive physical Hilbert space.
The paper also brings the strong-CP problem into the programme in its correct form. The physical quantity is not simply the QCD angle on its own, nor simply the phase of the quark mass matrices, but the combination
One of the important results is that the familiar matter–antimatter asymmetry in quark mixing is not the same thing as strong CP. VERSF can therefore allow a nonzero CKM phase while the combined quark determinant remains real. The present flavour stack already contains a conditional branch of exactly that kind, although the final physical quark determinant phase still has to be returned by the fully microscopic Yukawa calculation.
The paper’s sharpest new result is a structural fork. If the full continuum theory is genuinely the positive pushforward of VERSF’s operational probability measure, then a residual nonzero strong phase cannot survive. In that branch, VERSF must return
or else the positive global-measure construction fails and the theory encounters a named obstruction. The paper is careful not to overstate this. Positive probabilities for completed records do not automatically prove that the uncommitted quantum field history is itself described by an ordinary positive measure. That foundational step, together with the history-decoherence, transfer-operator, global-anomaly and topological-support calculations, remains the work needed for a strict physical pass.
Among the final eight papers, this one is therefore the global consistency gate. Papers 1–5 build the main physical sectors. Paper 6 asks whether those sectors can coexist inside one nonperturbative quantum theory without hidden topological, anomaly, positivity or strong-phase contradictions. If it succeeds fully, the programme will not merely have reproduced the local machinery of the Standard Model; it will have shown how that machinery can exist globally as one coherent quantum structure.