Independent finite simplicial arrangement theorem

Use Positive gallery category, cancellation and length for the finite central arrangement, open simplicial chambers, the positive gallery category $P$, its minimal-gallery congruence and the classes $u(A,B)$. Use Path presentations, congruences and formal inverses for the groupoid completion of this category. Products follow traversal order.

Use Antipodal transformation and finite-type Garside structure for the antipodal Garside structure and Categorical Garside structures and simple arrows for finite type, simple arrows, atomicity and the two divisibility-lattice conditions.

Use Complexified complements in an open convex cone with $T=V$ for the complexified complement $Y$, Fundamental groupoid of a space for endpoint-fixed path-homotopy classes and their composition, and Eilenberg--Mac Lane space for the $K(\pi ,1)$ convention. Choose a real basepoint $b_A$ in each chamber $A$. The groupoid on chamber basepoints means the full subgroupoid with objects $b_A$ and all endpoint-fixed path-homotopy classes in $Y$ between them, composed in traversal order.

Use Gallery fractions, fundamental groupoid and asphericity for the identification of the gallery completion with this full fundamental subgroupoid and for asphericity, under the same finite central simplicial hypotheses. The substantial arrangement-theoretic inputs are imported in the two supplier lemmas, whose source records give their locators and reading limits; this item composes their conclusions.