Causal diamond identified basically as intersection of future and past directed light-cones is basic notion in zero energy ontology. It has 4-D variant cd4, 8-D Minkowski variant cd8, and H=M4× CP2-variant CD= cd4× CP2.
Space-time surfaces in M picture are defined as roots for "real" or "imaginary" part of complexified octonion valued polynomial obtained by algebraically continuing first real valued polynomial with rational coefficients to complex valued polynomial by replacing real argument with complex argument $z$ commuting with octonions and then perfoming continuation to complexified octonion valued polynomial. One can continue at first step to a polynomial of z or of z** and complex conjugation has interpretation as particle- antiparticle conjugation.
To construct the solutions to the polynomial equations one must consider the equations near both boundaries of CD and glue them together smoothly. I have not consider this problem earlier. In principle, the polynomials associated with them could be different in the general formulation discussed in but they could be also same (see this and this). How are the solutions associated with opposite boundaries of CD glued together in a continuous manner?
- The polynomials assignable to the opposite boundaries of CD are allowed to be polynomials of o resp. (o-T): here T is the distance between the tips of CD.
- CD brings in mind the realization of conformal invariance at sphere: the two hemispheres correspond to powers of z and 1/z: the condition z*= 1/z at unit circle is essential and there is no real conjugation. How the sphere is replaced with 8-D CD which is also complexified. The absence of conjugation looks natural also now: could CD contain a 3-surface analogous to the unit circle of sphere at which the analog of z*= 1/z holds true? If so, one has P(o,z)=P(1/o,z) and the solutions representing roots fo P(o,z) and P(1/o,z) can be glued together.
Note that 1/o can be expressed as o*/oo* when the Minkowskian norm squared oo* is non-vanishing and one has polynomial equation also now. This condition is true outside the boundary of 8-D light-cone, in particular near the upper boundary of CD.
The counter part for the length squared of octonion in Minkowskian signature is light-one proper time coordinate a2=t2-r2 for M8+. Replacing o which scaled dimensionless variable o1= o/(T/2) the gluing take place along a=T/2 hyperboloid.
See the article About M8-H-duality, p-adic length scale hypothesis and dark matter hierarchy or the chapter Zero energy ontology and matrices.
For a summary of earlier postings see Latest progress in TGD.