The compilation-equivalence characterization conjecture for microcosms

Let $\microcosm{m}$ and $\microcosm{n}$ be microcosms. A measurable map $m$ is compilable in a set of measurable maps $N$ if there is a finite partition $X_1,\dots,X_k$ and maps $n_1,\dots,n_k\in N$ such that $m$ and $n_i$ agree almost everywhere on $X_i$. Write $M\prec_{\mathrm{c}}N$ when every element of $M$ is compilable in $N$, and define

$$\microcosm{m}\sim_{\mathrm{c}}\microcosm{n}\quad\Longleftrightarrow\quad \microcosm{m}\prec_{\mathrm{c}}\microcosm{n}\ \wedge\ \microcosm{n}\prec_{\mathrm{c}}\microcosm{m}.$$

Let $\mathtt{C}[\microcosm{m}]$ denote the complexity class associated with the microcosm $\microcosm{m}$.

Compilation-equivalence characterization conjecture. If

$$\microcosm{m}\not\sim_{\mathrm{c}}\microcosm{n},$$

then

$$\mathtt{C}[\microcosm{m}]\neq\mathtt{C}[\microcosm{n}].$$

The preceding theorem proves the inclusion $\mathtt{C}[\microcosm{m}]\subset\mathtt{C}[\microcosm{n}]$ when $\microcosm{m}\prec_{\mathrm{c}}\microcosm{n}$. The conjecture asserts that non-equivalent microcosms always induce distinct complexity classes, giving a converse characterization of the compilation preorder.