Hi Lawrence,
I agree with you in general, although I use different language and a different method of getting to the same conclusions. I always have to read your stuff over and over to make sure the connection is there, not because it's opaque but because you pack a lot of meaning into fairly spare symbols.
To this point of whether string theory solutions are real, or mathematical artifacts (quantum amplitudes), I think it just doesn't matter. That is, to make an analogy, in E = mc^2, c^2 is also a mathematical artifact; i.e., it doesn't affect the meaning of the statement E = m. Similarly, when we speak of the fundamental domain of string theory as a hyberbolic 2-space, the range of solutions can be infinite without violating the meaning of string theory as a unifying principle that retrodicts physical results. (By the same comparison, the theory of common ancestry known as evolution is a retrodictive theory, because we cannot observe random mutation and natural selection in real time, though we can model it and compare the model to past events.)
So a 10 dimension boundary condition should be sufficient to limit the domain solutions if the real solutions can be shown finite given the boundary -- which is precisely within the spirit of Einstein's "finite but unbounded" relativistic universe. A finite set of solutions obtained by construction rather than heuristics should give us the means to at least perform thought experiments and create accurate computer models.
So ... going back to my research, which I am in the process of expanding and trying to compact (wish I had your talent for that!) into more comprehensible and standard terms:
My energies have been directed toward a tight proof of the 10 dimension limit. A key idea is the construction of the modulus 12 subgroup of Sophie Germain primes (detailed in my time barrier paper) such that the 2 dimension submanifold self similar to the mod 12 subgroup possesses properties of linear independence over Q and infinite orientability on C*. This latter property allows n-dimension continuation over non-compact manifolds.
The role of the mod 12 structure is then clear, as an eternally recurring zero of an ordered set of kissing spheres admitting non-lattice numbers. That is, every sphere, S > 1, contains at least one 12-vertex lattice, S^2 --> S^oo.
To elaborate, supposes -- instead of analyzing by differentiation and integration -- we allow discrete elements of a contnuous function to self organize in the n dimension Euclidean space. The least 3 dimension sphere, S^2, with 2^n degrees of freedom -- 3(2^n) -- is the zeroth member of S^n, for being the least of the elliptic (zero) manifolds. Lesser Euclidean dimensions have hyperbolic (negative) characteristics and parabolic (positive) manifolds are hyperspatial.
Expressing as a table:
Geometry Order Dimension Topology
point - 3 0 none
line - 2 1 S^0
plane - 1 2 S^1
sphere 0 3 S^2
hypersphere 1 4 S^3
(I hope the table doesn't break up. I can't preview on this computer.)
Best,
Tom