Joy,
Interesting questions, only question 4 can I completely answer: short answer, it doesn't sqaure.
1) I don't know the details of the 6 differential structures on S10 either, or if they can be connected to the chiral-family physics distinction in the way I suggest. The topological transition would have to divide them in half, but the 3 remaining options would be required at the same time for the 3 families. I don't yet know if this is possible or even makes sense.
As S3/S7 are conceptually within the space of quaternions/octonions - giving a 12D space - I think I would first have to find the relation of my manifold - with its transition S10->S3*S7 - to this background - and find the time dimension in 12D - before I could consider the differential structure.
2) There is also a local-global confusion. Globally S3*S7 is a physical manifold where the spheres are curved - demanded by them being unified in S10 which isn't parallelisable. The KK identification for the construction of a local dimensionally reduced theory involves identifying the compactified space with a group space - this gives a flat S7 in the local theory, in which the rotation group space gives the other flat S3. Your analysis then requires the measurement spaces to be (flat) S3 and S7. This gives 3 types of spaces - physical, symmetry, measurement - to square with each other. If the physical curved space of my model is dropped, then the S10 "explanation" of the S7 to S3 map is lost, and so is the condition of 6 differential structures.
3) My model is simply constructed as an 11D extension of GR, with an initial radiation density of metric waves as extra-dimensional extension of gravitational waves. This requires the addition of an energy-density term with gravitational coupling, and using the same term necessarily involves the same assumptions. However, the definition of some physical quantities depends upon the number of dimensions, which changes with dimensional compactification (e.g. entropy). In 4D GR mass has a definition via the lst Casimir invariant of the Poincare group. In 11D the relation between the equivalent invariant mass and energy will be different from the 4D case, which raises questions about the relation between inertial mass and gravitational mass. The mass invariant in whatever number of dimensions will be the inertial mass, whereas the source of gravitational curvature in N-D GR is defined by the stress-energy tensor as being energy. The equivalence principle in 4D has a hidden assumption about the relation between mass and energy in 4D, but with dimensional changes the relation between mass and energy changes.
As the particle charges come from the topology of the space - including spin being a topological "charge" - they are independent of this mass issue. The incompleteness of the calculation of the reduced mass of the Planck scale particle black hole, makes mass the major unresolved issue of the model - the masses of the fermionic particles have to be added by hand from experiment, through a renormalisation process. In contrast, the W and Z boson masses are derived in terms of closed geometric formulae involving the electroweak scale - a Higgs boson mass of ½ electroweak scale is derived in the dimensionally reduced Lagrangian.
4) I hadn't followed Penrose's work of Conformal Cyclic Cosmology (CCC), but the first thing I note from is most recent arxiv paper is that the word "cyclic" is somewhat misleading - it is more iterative or recursive. My model won't square with CCC because mine is genuinely cyclic (it bounces).
Michael
