Segal Topoi and Natural Phenomena: Universality of Physical Laws

J. Lurie proved in Higher Topos Theory that for $K \in \text{Set}_{\Delta}$, $\mathcal{C} \in \text{Cat}_{\Delta}$, $f: \mathfrak{C}[K] \rightarrow \mathcal{C}^{\text{op}}$ an equivalence of simplicial categories, we have a Quillen equivalence $\text{St}^+_f: (\text{Set}_{\Delta}^+)_{/K} \rightleftarrows (\text{Set}_{\Delta}^+)^{\mathcal{C}}: \text{Un}^+_f$. We prove a partial converse to this theorem at the level of Segal categories, namely that if $L(\text{Set}_{\Delta}^+)_{/K}$ is isomorphic to $L (\text{Set}_{\Delta}^+)^{\mathcal{C}}$ in $\text{Ho}(\text{SePC})$, then $L \mathfrak{C}[K]^{\text{op}}$ and $L \mathcal{C}$ are equivalent as Segal pre-categories. We interpret this as indicating that the Segal category of pre-stacks $L (\text{Set}_{\Delta}^+)^{\mathcal{C}}$ on $\mathcal{C}$ is equivalently given by a choice of simplicial set $K$, relative to which phenomena in $\text{Top}^+ = L(\text{Set}_{\Delta}^+)$ are considered, a sort of relativity principle. If we further take the Bousfield localizations of $L(\text{Set}_{\Delta}^+)^{\mathfrak{C}[K]^{\text{op}}} \simeq L(\text{Set}_{\Delta}^+)_{/K}$ and $L (\text{Set}_{\Delta}^+)^{\mathcal{C}}$ with respect to $\tau$-hypercovers, $\tau$ a Segal topology on $L \mathcal{C}$, then regarding $L_{\text{Bous}}(L (\text{Set}_{\Delta}^+)^{\mathcal{C}})$ as the Segal topos of natural phenomena on $\mathcal{C}$, we also obtain an isomorphism $L_{\text{Bous}} (L(\text{Set}_{\Delta}^+)^{\mathfrak{C}[K]^{\text{op}}}) \cong L_{\text{Bous}}(L (\text{Set}_{\Delta}^+)^{\mathcal{C}})$ of Segal topoi of stacks giving the same representations of natural phenomena, concurrently with the equivalence $L\mathfrak{C}[K]^{\text{op}} \simeq L \mathcal{C}$, which we interpret as a weak universality of natural laws.