You are asking for a theory with three non-negative real parameters ($1/c$, $\hbar$, and the gravitational constant $G$) that is "continuous in the parameters" in some sense, but we only know good theories that live on the boundary. I don't see how categories would help here without some very strong uniqueness-under-deformation results.

Since the discussion is concerned with asymmetric encryption, shouldn't "polynomial time" have a dependence on key length?

@user253751 This sort of unbounded hierarchy of classes can be axiomatized using universes: en.wikipedia.org/wiki/Grothendieck_universe

You can find some overviews of the proof in Aschbacher's "Finite Group Theory" and Gorenstein's "Finite Simple Groups".

Thank you, my mistake.

I haven't thought about their paper in a while, but I think Oka's coherence theorem can be used on the dual module ... maybe?

Are you sure you want $\log d$ there? It's only less than 1 for $d=2$.

I cannot find a good reference with a proof, but some web searching suggests that if $\frac{\ell(X)}{\ell(C)} = d^{-1/2}$, then $X$ is contained in a body of constant width, and that volumes of such bodies are bounded above by that of the ball.

For $X$ a $d$-ball, $\frac{\ell(X)}{\ell(C)} = d^{-1/2}$ and $\frac{V(X)}{V(C)} = \frac{\pi^{d/2}}{2^d\Gamma(d/2 + 1)}$. For large $d$, this is about $\frac{1}{\sqrt{\pi d}} (\pi e/2d)^{d/2}$