The proposal has been that octonionic polynomials P(o) define the number theoretic holography. Their roots would define 3-D mass shells for which mass squared values are in general complex and the initial data for the holography would correspond to 3-surfaces at these mass shells. Also this assumption has problems. There is however no need for this assumption: the holography on the H side is induced by the M8-H duality!
The hierarchy of polynomials defines a hierarchy of algebraic extensions defining an evolutionary hierarchy central for all applications of TGD and one must have it. Luckily, the recent realization that a generalized holomorphy realizes the holography at the H side as roots for pairs of holomorphic functions of complex (in generalized sense) coordinates of H comes to rescue. It can be strengthened by assuming that the functions form a hierarchy of pairs of polynomials.
Twistor lift strongly suggests that M4 and space-time surfaces allow a Kähler structure and what I call Hamilton-Jacobi structure. These structures force a breaking of Poincare and even Lorentz invariance unless they are dynamically generated. It indeed turns out that M8-H duality generates them dynamically.
See the article A fresh look at M8-H duality and Poincare invariance or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.