  In the foundational logical framework of homotopy-type theory we discuss a
natural formalization of secondary integral transforms in stable geometric
homotopy theory. We observe that this yields a process of non-perturbative
cohomological quantization of local pre-quantum field theory; and show that
quantum anomaly cancellation amounts to realizing this as the boundary of a
field theory that is given by genuine (primary) integral transforms, hence by
linear polynomial functors. Recalling that traditional linear logic has
semantics in symmetric monoidal categories and serves to formalize quantum
mechanics, what we consider is its refinement to linear homotopy-type theory
with semantics in stable infinity-categories of bundles of stable homotopy
types (generalized cohomology theories) formalizing Lagrangian quantum field
theory, following Nuiten and closely related to recent work by Haugseng and
Hopkins-Lurie. For the reader interested in technical problems of quantization
we provide non-perturbative quantization of Poisson manifolds and of the
superstring; and find insight into quantum anomaly cancellation, the
holographic principle and motivic structures in quantization. For the reader
inclined to the interpretation of quantum mechanics we exhibit quantum
superposition and interference as existential quantification in linear
homotopy-type theory. For the reader inclined to foundations we provide a
refinement of the proposal by Lawvere for a formal foundation of physics,
lifted from classical continuum mechanics to local Lagrangian quantum gauge
field theory.
