On the properties ${\mathrm{S}\mathrm{O}\mathrm{P}}_{2^{n+1}+1}$

We show that approximations of strict order can calibrate the fine structure of genericity. Particularly, we find exponential behavior within the ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_n$ hierarchy from model theory. Let $0$-$\mathrm{ð}$-independence denote forking-independence. Inductively, a formula $(n+1)$-$\mathrm{ð}$-divides over $M$ if it divides by every $n$-$\mathrm{ð}$-independent Morley sequence over $M$, and $(n+1)$-$\mathrm{ð}$-forks over $M$ if it implies a disjunction of formulas that $(n+1)$-$\mathrm{ð}$-divide over $M$; the associated independence relation over models is called $(n+1)$-$\mathrm{ð}$-independence. We show that a theory where $n$-$\mathrm{ð}$-independence is symmetric or transitive must be ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_{2^{n+1}+1}$. We then show that, in the classical examples of ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_{2^{n+1}+1}$ theories, $n$-$\mathrm{ð}$-independence is symmetric and transitive; in particular, there are strictly ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_{2^{n+1}+1}$ theories where $n$-$\mathrm{ð}$-independence is symmetric and transitive, leaving open the question of whether symmetry or transitivity of $n$-$\mathrm{ð}$-independence is equivalent to ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_{2^{n+1}+1}$.