The Almost-Quaternionic Mock Modular Correspondence: Paper XXII in MEF Series

We consider Riemannian 12-manifolds carrying a Spinᶜ structure and an almost-quaternionic structure in the sense of Swann, motivated by Kaluza–Klein compactifications of the form M₁₂ = (S¹ × S³) × K₈ with K₈ a Spinᶜ orbifold of Euler characteristic χ(K₈) = 12. The intrinsic torsion of an almost-quaternionic structure, decomposed by Swann into four irreducible Sp(n)Sp(1)-modules, obstructs reduction to strict quaternion-Kähler holonomy; the Zwegers shadow of a mock Jacobi form obstructs classical modularity of the associated Spinᶜ-twisted partition function on K₈. We prove that these two obstructions are images of a single $U(1)$-equivariant torsion class: the shadow extracted by the Bruinier–Funke operator ξ1/2 coincides, up to normalisation, with the $U(1)$-projected component of Swann's torsion under the Spinᶜ-induced reduction Sp(1) → U(1). The proof uses classical Swann geometry, Bruinier–Funke theory of harmonic Maass forms, and Leray injectivity, and is unconditional. A numerical rigidity witness anchors the identification: the integer 3 in the non-holomorphic completion E₂^*(τ) = E₂(τ) - 3/(πτ₂) of the weight-2 quasi-modular Eisenstein series equals the first Chern number c₁(L) = 3h of the Spinᶜ determinant line bundle on K₈, with no adjustable parameter intervening. Dimension 12 is forced by Bott periodicity: Cl(0, 12) H(32) is the lowest dimension in which real spinors naturally carry H-module structure, localising the correspondence on 12-manifolds rather than generic $4n$-manifolds. A broader functorial framework lifting the correspondence to a commutative diagram of short exact sequences is developed; its section-level statements are explicitly conditional on a named auxiliary axiom, whereas the principal identification of the shadow with the projected torsion is independent of this axiom.