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

On a closed almost-quaternionic Spinᶜ orbifold of real dimension 4n 12, with non-trivial determinant line bundle L and a compatible involution, two obstruction classes from independent traditions coexist: Swann's intrinsic torsion, which measures the failure of quaternion-Kahler holonomy, and the Bruinier--Funke shadow, which measures the failure of modularity of the associated partition function. We show that the U (1) -projection of the intrinsic torsion under the Sp (1) U (1) reduction induced by the Spinᶜ structure and the shadow of the equivariant elliptic genus are cohomologically equivalent classes in the -odd component of rational cohomology, normalised by c₁ (L) ; the proof uses Swann's absorption of almost-quaternionic torsion into hyperkahler holonomy, its orbifold extension, Leray injectivity, and the Bruinier--Funke characterisation of shadows. A functorial refinement --- a spectral push-forward into S₃/₂ (₀ (36), ), together with a compatible map of short exact sequences --- holds at the level of sheaves unconditionally, and pointwise under one explicitly stated structural hypothesis. On the twelve-dimensional orbifold K₈ = (CP² S²) ₖ (T²/Z₂) ^Spinᶜ the determinant line is O (3) by the Euler sequence, and the integer 3 coincides with the coefficient 3 in the completion E₂^* = E₂ - 3 ₂, the two anchorings independent. The oddness of c₁ (L) forces the involution to act freely on the shifted momentum lattice; constant configurations are consequently a symmetry-protected critical stratum of the spectral-rigidity functional on -even conformal deformations of the pillowcase factor, the kernel decouples, and the critical structure assembles multiplicatively to 6 2 = 12 = (K₈). The lowest eigenvalue of the Spinᶜ Dirac operator on the toroidal factor is exactly, independently of the modulus.