The Question

 Is any of what I'm writing useful?

I have no idea.

Mostly I think I have a useful framework to consider things from, and, thus far, an utter inability to actually successfully communicate that framework.  I can try to communicate the fundamental concept in math, but the math isn't, in fact, the fundamental concept; from the perspective of the concept, both of these equations are fine:

sin(ln(x))/x

(e^x*cos(sqrt(e^x)))/(sqrt(e^x))

Even this equation is kind of okay:

sin(1/x)/x^2

The point isn't the specific equation, it's the idea the equation represents.  But I've handed these equations to a number of people who don't quite get the idea.  So, well, I keep trying to figure out how to communicate the idea, which is not in fact about any specific equation, but a way of thinking.

The Newtonian equation for acceleration in a gravity well is a=M*G/r^2.  The units of G are m^3/(kg*s^2).  Einstein noticed you could pull c^2 out of it (also 8pi, but that's dimensional rather than unit analysis) to get m/kg; multiply this by mass, and you get meters.  Let's call this distance o, and we get (ignoring the constants) o/r^2.  We can form a triangle out of o, and if r is the hypotenuse, we get sin(theta)/r.

Okay, you may say, what was the point of that?

Well, there really isn't a point; it doesn't really serve any particular purpose; all I can really say is that it demonstrates that sin(f(x))/x can be equivalent to 1/x^2; that is, it can demonstrate that sin(ln(x))/x isn't immediately and obviously incapable of modeling Newtonian gravity over certain distances.  It definitely is not proof that sin(ln(x))/x can do so, it just rules out the immediate intuitive-no I've gotten.

It also happens to be the answer to another intuitive-no I've gotten, on whether or not acceleration can be modeled as rotation in time.

That's been a lot of the work I've put into this; answering the intuitive-nos with a "Maybe".

Thing is?  I'm already pretty certain I have a firm intuitive grasp on the shape of the universe.  It's kind of mundane and boring; sin(ln(x))/x, even if it isn't the correct equation, has, I believe, the correct shape.  So does sin(1/x)/x^2, for values less than 1.  So does (e^x*cos(sqrt(e^x)))/(sqrt(e^x)).

Physics is full of this shape.  Strong Gravity, as a contender for an explanation of color confinement.  The energy potential of the Higgs mechanism.  The nuclear forces.  Gravity.  The cosmological constant.  Hell, the hierarchy problem itself; Schwartzschild Coordinates, when you really start looking at them, dissolve the hierarchy problem entirely; if hydrogen atoms are a meter across, the hierarchy problem is really just a problem of perspective.

The idea isn't magical.  It isn't spectacular.  It is kind of anti-spectacular; you haven't truly grokked it until you realize how disappointing it really is.  General relativity has ripples, alright, great.

Still, maybe somebody else can find more use for it than I can.  If I can ever communicate it.