Another Update to Geometries

On further consideration, two of the three complex dimensions are basically identical, and correspond to the relationships between Xi and Yi with Zi.  That is, two correspond to curvature, and one corresponds to something else, potentially time.

Additionally, there are four potential positions of the origin.  Calling X and Y the orthogonal dimensions, and collectively referring to them as O, we have:
O, Z; O-,Z; O-,Z-, and O, Z-.  Each minus represents a symmetry flip; thus, O,Z and O-,Z- represent the same profile (let's call this matter), and O-,Z and O,Z- represent a flipped profile (let's call this antimatter).

I'm guessing the two identical profiles may represent spin; O,Z represents S+, and O-,Z- represents S-.

Note that this is a departure from earlier guesses that there may be movement in a closed dimension corresponding to Kaluza-Klein; this description implies instead that these are relatively fixed, mostly identical configurations.

Thus, antimatter is matter rotated by 90° in the complex dimensions, and equivalent spin is separated by 180°.

Additionally, I think I can rule out some lower guesses for b; in particular I don't think it can be lower than π^2/2.  π^3 or 3π^2 remain viable.  This is because there is no constant c such that sin(ln(c*r)/b) that is consistent with gravity at distance r for the solar system; in particular no constants are consistent with the orbital velocities of the planets while also being consistent with the existence of Earth.  (Specifically, gravity on Earth would be negative.)

ETA:

Pi^3 seems oddly feasible; I can get constants that are off, on average, by around 2.5% from observed velocities.  These deviations are small enough to potentially be explainable in relativistic terms (distance not being exactly as observed).

This corresponds to a far longer period than I find desirable - r*7*10^12 has some potentially great explanatory power for a variety of phenomena, ranging from the galaxy rotation curve to the Kuiper Cliff to a particular relationship between proton and electron mass-charge ratios.  However, explaining those things while failing to explain why Earth doesn't just fly apart isn't a good trade-off.

Pi^3 results in a period of around r*10^73.  Far, far higher than is desirable.

Note that by period I mean the distance between peaks, rather than the instantaneous period, which varies within a given period quite considerably.