A Model of a Cohesive Linear ∞-Topos via Orthogonal Site Structures

We construct an explicit example of a cohesive linear ∞-topos --- an ∞-topos equipped with both a cohesive adjoint triple (Π,♭,♯) and an idempotent product-preserving comonad Q^◇ with right adjoint Q_• forming a linear/non-linear adjunction, satisfying Beck--Chevalley compatibility. The construction uses a site whose objects are pairs (M,A) of a smooth manifold M and a finite-dimensional C*-algebra A. The cohesive structure arises from the projection to the smooth part, taking Mfd with the open cover coverage as the base site (so that finite limits are available). The quantum comonad Q^◇ acts by Gelfand duality on the algebra component. Because the cohesive modalities and Q^◇ operate on orthogonal coordinates of the site, the Beck--Chevalley commutations hold by direct verification on generating objects. The subcategory of Q^◇-connected objects is shown to be equivalent to the smooth cohesive ∞-topos. We discuss the scope and limitations of this model, including the finite-dimensional restriction and the triviality of the topology in the algebra direction. The orthogonal site technique introduced here may be applicable to other combinations of logical modalities.