The Zwegers shadow as $U(1)$-projected intrinsic torsion\\ on twelve-manifolds

On a closed almost-quaternionic Spinᶜ orbifold of real dimension 4n ≥ 12 whose singular stratum is the pillowcase, with determinant line bundle of odd first Chern class and a compatible involution σ, two obstruction classes coexist: the intrinsic torsion, measuring the failure of quaternion-Kahler holonomy, and the Bruinier--Funke shadow, measuring the failure of modularity. For a compact target the orbifold elliptic genus is a weak Jacobi form; the object studied here is the σ-odd twisted-sector projection of the equivariant Spinᶜ Dirac data, whose shadow we derive spectrally: a chiral telescoping at the four corners leaves one unpaired doublet, and the holonomy-weighted corner sum, passed through the Zwegers period integral, returns the cube of the Dedekind eta function at doubled argument --- a classical weight-$3/2$ unary theta series --- as a computed object. We prove that the $U(1)$-projected torsion class and the shadow class coincide on the σ-odd sector, identified by an equivariant corner-evaluation map between two-dimensional standard representations of the modular permutation of the twisted sectors, with the σ-even interior coefficient supplying the normalisation. A functorial refinement holds at sheaf level given one declared intertwining input, and pointwise under one further structural hypothesis, stated with its failure mode quantified. On the twelve-dimensional example the interior coefficient is the anticanonical three of CP², matching the constant of the E₂ completion at the stated normalisation, and the oddness of the determinant class pins the lowest Dirac eigenvalue on the toroidal factor to π, independently of the modulus.