Finally?

The development of the ideas related to the S-matrix has proceeded essentially by enthusiastic trial and disappointing error, the best scientific method that exists. It however seems that the big picture is now clear and stringy form for the S-matrix, or rather amplitudes for the generation of zero energy states from vacuum emerges although the interpretation is quite different as compared to that in string models.

Here I list just the main points.

1. The concrete application of Virasoro conditions on zero energy states combined with the requirement that the conformal parameters are rational numbers (required by p-adicization) implies that rational conformal field theories characterize the particle multiplets.

2. Massless sector cannot be understood in this manner since the conformal weights of all states are non-vanishing. Massless states would naturally correspond to c=0 representation without breaking of conformal symmetry and all states being null norm states with respect to the Virasoro norm. p-Adic thermodynamics determining mass squared values could be interpreted as thermodynamics describing massless zero energy states with massive ones.

3. Finite number for primary fields conforms with the vision that Jones inclusions N subset of M reduce the number of state space degrees of freedom to a finite number. An essential assumption is quantum criticality which allows to assign to physical states conformal fields expressible in terms of ordered exponentials of string fields.

4. Under very general conditions the amplitides for the generation of zero energy states from vacuum reduce to Lorentz invariant stringy amplitudes defined at partonic 2-surfaces taking the role of Euclidianized stringy world sheet. The reason is that the fields of minimal models are expressible in terms of ordered exponentials of stringy fields plus additional factors taking care of internal quantum numbers.

5. The analog of stringy perturbation theory could follow from the unitarity condition for the entanglement matrix between positive and negative energy states playing the role of S-matrix.

To sum up, just at this moment I cannot see any inconsistencies in the picture. But so I have believed also in previous summaries of the situation so that do not take me too seriously;-)!

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is just now.